Alright, it's about that time to get the show on the road, so everyone please give a very warm welcome to Chiyago who will be talking about hacking PLCs. Sorry, I have a little experiment here, and it took a little longer to set up than I was expecting. But I am very, very happy to be here at DEF CON today, and I was honestly expecting like 50, maybe 100 people. So thank you guys for taking the time to be here on Saturday morning and listen to me. I am Chiyago Alves and a little bit about me. I am a PhD student at UAH, and I also work at the Center for Cybersecurity Research and Education. And yes, this is what I do. I love to hack stuff and I try to break as many things as I can. And I'm also the creator of the Open PLC project. If you guys haven't heard of it, you will hear about it a little bit today. Heads up if somebody have heard about Open PLC before. Wow, thank you. That's great. That's great. Thank you. Also today, it's a very important day for me, because this trip to Vegas marks our ninth year anniversary. Me and my wife were together for 90 years. And yeah, I used to have more hair back then, but she's still pretty, the same way. So I'm a lucky guy. And she always complained about being like slowing me down because now we have kids and I have other things to focus on. But I keep telling her that if you want to go fast, you go alone. Life is not about speed.  That's why I'm here. That's because of you. And I'm very happy I invited her to be here at my talk. Thank you. And she has a very important job, actually, because I have really bad jokes and she laughs of all of them. So if you hear her laughing, just please laugh as well. So we have a nice presentation. So a little outline about today. I'm going to start with a background info, a little bit about PLCs. If you guys know about PLCs, I'll try to go fast. If you don't know about PLCs, please pay attention because I will go fast. Then we'll go and talk about a little attack vectors, some stuff we can do to break those things.  So hopefully things will go out fine. So what is a PLC after all? It is, that's what it looks like, right? It's a brick. And it stands for Programmable Logic Controller. And basically it's a digital computer, right? It's used on automation since 1970. So it's a long time ago. And basically it is an embedded system with a CPU, some RAM, some ROM, right? Some Flash nowadays. And you put your program in there, you run your program. And your program decides what to do with the input data it senses from the input modules that PLC has.  I have three PLCs set up here trying to heat water. And they're doing a poor job doing that. But then you program those devices to any logic you want. And it has output modules to control real life stuff, right? So for example in this setup it's controlling the heater based on the temperature sensor that is attached to the input module. And you will see PLCs in many different places. So here I have like a dam, a substation, a gas distribution system in a petrochemical facility. PLCs are there. And you imagine what happens if you hack into those PLCs and make a crash, right? Or change a setting. So for all those situations a disaster will follow if you break into a PLC. Because they are controlling large, huge stuff that can cause even physical damage and life loss, right? So what is the problem with the current PLCs we have?  You know, if you're building an embedded system it shouldn't cost that much. Nowadays there are some companies selling what they call cheapo PLCs. You can buy one for like $100 or $200 but then they get you on software. There are companies selling software, programming software for PLCs for up to $5,000 per station. And that is ridiculous, right?  And also they rely on very legacy technology. PLCs were built like 40 years ago. And that's also a major problem because this legacy technology was created at a time that internet didn't exist. So they use unsecured protocols to communicate. Most companies rather patch their devices instead of redesign. We are in a situation where we need redesign and their closed source. So this is a problem if you are a researcher and trying to figure out how they work. So I came up with a solution. If you guys know about it, open PLC, that's my thing, right? Open PLC is the world's first and only one PLC available with the full source code. There are other approaches that people try to develop their own PLCs. But it is not close to what a PLC should look like. PLC, there are many standards that define what a PLC should do and how it should behave. And I try to follow those standards close so that it will look like a real PLC and behave as much as possible to a real PLC. And I make the source code available, anyone can go take a look at the code and use it. It is free, right? And basically open PLC has three main components. The runtime, that is the big thing that I'm creating, is the thing that goes into the embedded system, embedded device. And initially I started creating my own hardware, but then I gave up and started using Raspberry Pi's, Arduino's and stuff that everybody have. So you can load the runtime on your embedded system and have it read your PLC programs. You create your PLC programs on the editor, that's supposed to run on your computer, right? On your Windows, Linux machine, some folks have ported it to Mac OS as well. And the third portion of it is KWR, which is a GUI builder that allows you to create nice animated graphical screens for your PLC programs, right? So for the open PLC runtime, it runs on many different platforms. These are the platforms that are currently supported, officially supported. Like I said, Raspberry Pi, Arduino's, but also there are some industrial devices that people created that also runs open PLC like UniPi and Pi Extend. This is really cool, you can go and buy those devices and install open PLC today if you want. There are other devices that are not here, they're not officially supported. They were created by the community. So I have a forum with lots of people in there and they make all sorts of questions. And they also contribute to the project and they created unofficial patches that make open PLC run on like Orange Pi and other platforms that basically run Linux. This is the editor, what it looks like. It supports five programming languages. These are all the five languages defined on the standard. And this is KWR, right? This is a real nice graphical screen created on KWR, so it can create many things. This is the water temperature control you're going to see today. So let's going to talk about PLC protocols for a bit. I don't know if you touched on SCADA protocols before, but what it looks like, there are problems with those protocols. Basically, most protocols today are derived from legacy serial networks. So if you're old enough, you might remember those RS485 networks where you hook up a bunch of devices on a serial cable and all the devices receive the same message and they decide if they want to accept it or not. This is great for hacking, right? You just listen to the bus and you get everything. There is no authentication, of course. There is no integrity. And when I mean integrity, I'm not talking about error correction. I'm talking about if somebody tempers the message, is that going to be figured out at the other end or not? So they don't have any integrity mechanisms. And there's no confidentiality. So the message goes on the field with no confidentiality, which means it goes in plain text, right? So the most popular SCADA protocol is Modbus. It's about 90% of the PLCs support that. Actually, I haven't found any PLC that does not support Modbus. They usually support their own protocol plus something else plus Modbus. Everybody talks Modbus. And in essence, Modbus is very simple. There's nothing very complicated with that. It is, of course, based on serial networks. And the commands, basically how it works is that the commands for the protocol are encoded into function codes. So you have different function codes for different things you want to do. And it's open. That's probably why everybody supports Modbus because it doesn't require any licensing. You can go and pay Modbus Foundation to have your device certified so that they make sure that your device is actually following the full standard, but you don't have to do that to support the protocol. There's no licensing, right? Licensing fee. And this is the Modbus frame. That's what it looks like. Basically, the first byte is a slave ID. And the slave ID is a unique address because it was based on serial networks. So you have to have a way to address the device. The function code is what I was talking about. It tells the slave device what to do. It's like the command it should obey, right? And the function code varies from reading or writing to memory areas. Data. Well, data is data, right? It depends on the function code you're using. And finally, CRC. It's just error correction verification. Again, this is not checking integrity of the message itself. It's just checking if there are any physical errors that happened on the transmission, right? If somebody can tamper at this frame, change the data, recalculate CRC, and send it through, it will go, right? It will be accepted by the PLC. So those are the most used function codes. There is function codes to read digital outputs, to read digital inputs, and also to read and write registers. Basically, all it is, is working with memory. So Modbus gives you free access to the PLC memory. You can read whatever you want. You can write whatever you want with no authentication. Who loves that, right? They updated Modbus to make it compatible with TCP, IP networks. What it means is that they added a frame on top of the Modbus frame. So basically, it has a transaction ID to keep track of all the transactions you're making. So if the host makes a transaction, it makes a random number on that transaction ID. So when it receives a response, it might receive a bunch of response from the same slave device, and it will match the transaction ID to make sure that that response is related to that query it was making. Protocol ID. It seems that originally Modbus on TCP was meant to work with other protocols as well. But they created this field so that different values on that field will mean different protocols. I've never seen anybody using anything different than 00, and 00 stands for Modbus. So I think they field on this. Length is length, right? How many bytes there are after it. And after that, it's just plain serial Modbus frame as it was before. They kept the slave ID byte on that frame because some people would just convert between TCP IP networks and serial networks. So you have all devices talking serial and new devices talking TCP. So keeping that byte over there means that serial devices will also be able to receive that message, right? Even though if they are addressed on TCP. For TCP only PLCs, usually, that byte means nothing. They will accept anything. Tax scenarios. What can we do with this? You can create a few different attacks on that. Because the message is so simple, you can just interrupt the communication and cause a denial of service. There are many ways to do that. You can intercept the message and read the contents because there is no encryption, no confidentiality at all. You can also modify the message. You get the message, you get a bump in the wire or some sort of eater cap or poisoning hack. And you get the message since it's plain text. You modify the contents, send it back. Yeah, that is pretty disastrous. And my favorite, injection. You can only just send a freaking frame. The PLC will accept you, right? It accepts everybody. So let's demo it. Hopefully it will work. So I have here skatobr and let me start the last PLC. I didn't have the time to do it. Hopefully I will now. Let me get back to skatobr. So this is skatobr for you to get familiar with it. It's an old tool but it works very well and I incorporated it into the open PLC project so that we can update it and renovate it over time. So here I have all my data sources. These are all my PLCs. So here on the table I have three different PLCs. The first one here is open PLC, running on a UniPy platform. This is an industrial platform. The second one is an Allen Bradley MicroLogix 1400 series. And the third one is a Schneider modicon M221. So let me go ahead and try to turn them all on. Hopefully they are good. So let me start this PLC as well. In the meantime while this thing is starting let me show you the graphical screen and see how it looks like. So this screen shows my system, my setup here. So what I have as I said before, I have a cup filled with water and a water heater element inside it and a temperature sensor. The temperature sensor is connected to the PLC so I have three identical setups and since PLC talks ladder logic I just created the same ladder logic for all the three PLCs. So they are running identical logic programs. Of course each architecture will just interpret that logic differently internally. Some will compile that to binary code that's in case for open PLC some will just convert that to virtual machine that will just run interpreted instructions. So it varies by vendor but they all in essence doing the same logic. I just copy paste the same ladder diagram to make sure they are all running the same thing. And on that configuration I have a flexibility here that I can set the PLC for manual mode turn on the heater if I want manually so you can see the heater going up here just a little break to turn on the PLC, sorry. And I can turn the heater off I can put in auto mode so the auto mode will try to keep the temperature around 40 degrees Celsius and you know temperature is a slow thing so you might go up slowly and go down slowly but in essence all the PLCs are trying to keep that around 40 except this third little one because it's not on yet. Sorry about that. Let me just log in here and in one second I will turn it on. So my job here is try to attack these PLCs once I am in the network I will show you how easy it is to create a Modbus frame and attack it, right? Last click. Not yet. Okay should be good to go. Let me start it. Yes I want. Okay this is so machine basic this is the software used to program the Schneider Modicon M221 we will have a lot of fun with that software today. Okay so let me start my terminal here. So I have an injection attack ready what does injection attack does? It tries to send Modbus packets that will switch the PLCs in manual mode and keep the heater on, right? And this message will be sent over and over and over again so that the operator will lose control of the PLC and will not be able to turn it off, right?  This little attack has two parameters I created this software by the way everything is available on my GitHub so if you want to get those tools go ahead you have it. So it has two parameters the first one is the host 100, 100, 100 because I'm really bad with numbers I had to choose a good one. And the other parameter is frequency I won't use it the default frequency is a thousand messages per second so let's try that on open PLC go all right cool heater is on let's turn it off well it doesn't work so let's put it out of mode it doesn't work either I'm screwed so that's probably what the operator will be thinking you see the temperature going up and you can't do anything about it and all it takes is sending freaking modbus frames that's all it takes what I'm doing here is riding to a portion of the memory of the PLC that I know it's storing the setting for manual or auto mode and the other area that is storing the setting for the heater if it's on or off that's all it takes and I can guarantee you that this kind of attack works with 100% of the PLC that I know of so okay let's stop this because I have boiling water already let me put back on auto mode we'll shut off the heater now let's try the same thing with our friend Alan Bradley it is at the same IP except 101 I know I'm bad with numbers so let's try that again boom oh gosh no it's on doesn't work auto mode also does not work temperature will go up and yeah it works the same way you see different brand of PLC freaking same modbus protocol that's the culprit let me stop this auto mode again last attack let's try that on the Schneider modicon please misbehave okay it's on even though the temperature is also over the set point it does not work either so that's how simple it is you can today start creating your own PLC attacks okay let me stop this now this was this was pretty fun and I had a lot of fun creating this but I have one more thing for those of you who are fan of Steve Jobs this is where the best part happens right so let's talk about micrologics micrologics is a PLC from Alan Bradley and rock ultimation I don't know the history but they keep both names in so I just say both names but what are the fun facts about micrologics 1400 series you should be buying one today to hack it's a pretty fun device right so first of all it runs vxworks vxworks is one of the most popular real-time operating systems on the industry it's on everything including PLCs and there are a bunch of vulnerabilities reported for that device guess what I bet most of those works with Alan Bradley as well right it is one of the most popular Alan Bradley PLC you see that everywhere and it supports modbusdcp that's why the injection attack worked this is my favorite feature it can be killed remotely by sending a bad modbus packet this is great so let me talk about this what I call micrologics deadly packets basically all it takes it's to trick the memory memory allocation algorithm of the micrologics PLC so what I do is I start a new transaction ID protocol of course is modbus 00 and on the length field I say I will need 20 256 bytes but I only have 6 alright so it might try to allocate that much memory but it's only receiving less than that and right immediately after I start a new transaction ID where the length is the length of the previous packet and it's incomplete it stops at the unit ID so it makes the PLC confused with this memory management system and I cause a buffer overflow writing that message on a different area of the device crashing the device and making it unrecoverable so let's try to do that alright so just to check here I should have everybody running wow I left that PLC on for too long alright so we are targeting ellen bradley now it should be working fine so I'll just put in manual mode to prove like the heater is on heater is off it should work beautifully auto mode should keep the heater off let me just keep that on the heater on so that will be more fun let me go it's hard to type when your hands are shaking it should be ellen bradley exploit this is also available on my github so all the only argument it takes is the IP address of the PLC so all it does is send those two deadly packets and let's see what happens when I do it dead so you'll see right now that the PLC is not communicating anymore you see the exclamation point all around dead how cool is that right you can even try to press the buttons over there it's completely dead and you know the fun fact of it is if you power cycle it's still dead I'll prove it I just power cycled it and it will boot again it managed to boot again but default LED keeps on for some reason so yeah it is booting up and you still see the exclamation points here you will see it till the end of the presentation so the problem is that application got corrupted so the PLC cannot launch it again the default mode is always on the only way to make it work again is to reprogram it with the ladder logic this is creepy right so if you lost the ladder logic you screwed please don't send it on the wild there are too many micrologics PLCs out there alright let's behave okay let me see if I have something else this is a problem that affects all micrologics 1400 series I don't know about others because I'm not rich to buy all the PLCs but I bet it also affects some other related series like 1100 and so I talked to Alan Bradley through ICS cert we published this vulnerability and they are really nice they responded back and tried to work with me to create a patch so they released recently about a few months ago a firmware update this firmware update fixes this vulnerability although it's a pain to update it I spent like half an hour trying to do that I'm supposed to be a security expert imagine people all in the field what they will do to try to update this thing they won't they won't stop their factory to update their PLC firmers hopefully the new PLCs that will be sold now will have this new update already built in there are other mitigations where please disable Modbus DCP this is not a fix dude if you're disabled in the protocol you're not fixing it you're just disabling it so this mentality is common for all the PLC vendors that I usually talk to minimize network exposure but PLC is behind a firewall so basically what they're telling you is the security is on you we don't provide security you're freaking alone so try to do your best these are good practices but I honestly believe that a PLC the device itself should provide any sort of security and one last thing alright Schneider no reason why I brought the Schneider PLC with me the Schneider has a protocol that is common to all the Modicon family I think other families also use this it's called Unity protocol at least me and some other researchers are working with this call it Unity you might hear both names and this protocol is obscure it is undocumented it runs on top of Modbus DCP so you have a normal Modbus transaction but one you have a function code 0x5a this is x a decimal for 5a this triggers the sub protocol that runs on the data field of the Modbus frame for the Modbus documentation 5a is a reserved function code for vendor specific functions this is a nice part it is used to configure and monitor the PLC down to the operating system level you can do a total memory dump using this protocol you can get the operating system image using that this is crazy it's proprietary and undocumented so they try to make secure by obscurity but that won't prevent us from figuring that out how it works so basically Unity have a bunch of function codes so it's a function code inside a function code you have the Modbus frame and then you have the Unity frame and these are the function codes you can play with and I have to thank Luis Litas he is a researcher on the same field we started talking and communicating like one year ago because I figured out he was working on the Unity devices as well and he is contributing to each other and he has a very nice blog LitasEnlated.xyz you go, if you want you can go and check it out he has a step-by-step how he managed to get those function codes and reverse engineer the DLLs for the so machine basic and get that information so this is pretty cool and basically you can take a look at some messages here they are pretty cool they are a block this is my favorite but today given the time constrained we will only play with a few of those so basically we will start a communication we will read a project info to read some data about the project this is also cool take PLC reservation it means that the PLC needs to be connected to someone someone needs to reserve the PLC if you take the PLC reservation you are taking the other party down this is DOS built in the protocol dudes this is what so you just keep sending this take PLC reservation and I think nobody else can connect so release PLC reservation and you are releasing it because you just don't want to communicate anymore and start PLC and stop PLC will basic main functions to start and stop the PLC alright so let's play with it I will just keep this slide open so we can play a little bit with how it works is that up? ok so I created also a little software modicon tester I don't remember the name of the executable ok so this software was created in C sharp in a hurry please don't look at the code it's messy but the good thing is it works and all I did was to create this thing to help me reverse engineer the protocol so it basically encapsulates the messages on the modbus frame with the function code 5.8 so I can send anything and it will be accepted as unity messages so all you do you connect to the PLC it's already set up with the M221 IP address I connect and then it gives me the hexadecimal output and the ASCII related to that so let's start by starting our communication so every function code has 001 and I'll end it with 00 so it just gave me back a hi I'm here that's fine hello so let's go ahead and release PLC reservation if somebody was talking to it sorry and then I'll take the reservation 10 00 and now the take reservation is an interesting thing sorry I should have make it bigger but I'll promise I'll read it right the message is sending me back with the take PLC reservation it ends with a magic byte in this case it's 8e even smaller for me so 8e I mean it's this magic byte that I need to send before any message that I want to send now on now that I have a session reservation I have reserved the PLC for me I'll have to use this magic byte that means that before people were trying to do replay attacks so the replay attacks would be successful just because they didn't have any session management so you could just get messages that the main station was sending and just replay that and it would work now you need the session number but that won't prevent us from sending stuff right so now let's play with this I have 8e as my session number you can see here urgent Alan Bradley is not talking still sorry sorry dude you can see that the modicon is still working fine temperature is 40 degrees celsius I can turn the heater on manually and off right it works so what I'll do here is stop PLC command so I need to start with my 8e and then the stop message is 41 it ends with ff00 and let's see stopped now the PLC can still communicate but the heater is off I cannot turn it on I cannot turn it off I cannot put it in auto or manual mode it's off how cool is that so just by sending a few messages I can have full control of the PLC but here comes my favorite feature of this protocol this is really interesting so I'm going to use so machine basic here for a second let me just create something new so I'm just using their application to talk to this PLC over here right and when I click here it already identified PLC I try to log in and it's trying to log in but this application is password protected how come so I cannot access my PLC without the password if I try to type anything it won't give me access alright what can we do about it okay let me get back to my unity tester here so remember that message read project info for some reason that thing doesn't even require session so I'll go ahead and send this zero zero no session zero three is the function code for that and zero zero zero zero will just get the header so let's see what comes out of it my header for the project that is stored on the PLC what is this yes the freaking password right so if I were Steve Jobs and I was working at Schneider I would say this is not a buck this is a feature what if you forgot the password the device needs to tell you that right so let's try it defcon is my password so when I type boom I mean super hard now I have full control of the PLC again using their own software just because I stole the password and it works I've tried with different slightly different types of modicon M something serious and the same thing works so yeah you can play with that I can start the controller because I stopped it before and I can see the programming on it yeah I have full access so this is pretty hilarious now that's to show you how secure your PLC is and imagine that those things are controlling your nuclear power plants and your water filtration systems I'll just go live in a jungle so Schneider also they were really nice I submitted this through ICS cert and they talk with me and we try to come up with some mitigations and they recommended the following things disable unity protocol this is the best mitigation ever so again you see a different vendor with the same mentality just turn that freaking thing off it doesn't work right they also recommended because of that password vulnerability and all of that they recommended to store your project files in secure access restricted locations and encrypt your project files with reputable third party file encryption tools again they're giving you the responsibility to fix stuff to secure your settings and this is pretty it gets me worried right and a little bit about what I plan to do in the future by the way I haven't showed you this this is what Open PLC looks like this is a web interface that is running on the device so once you install the runtime you have that interface up and you can see your programs you can see your hardware you can even mess up with the code live right so it's pretty cool and this thing is also vulnerable and insecure I created it to be like that because I'm trying to mimic what other vendors are doing so we can kind of research about it but my PHD thesis is about creating a secure version of Open PLC that will try to prevent those types of attacks from happening so it was really fun to be here at DEF CON today it was really an honor to me to be here and thank you very much for coming and staying with me this Saturday morning if anybody have any questions


Greetings, we began discussion on Fano resonances and we have already come a significant way in this discussion. We have discussed the Fano parameters related them to the general formula coming from collision physics. And we began to discuss the Helium atom spectrum as a prototype of a many electron system. So, this is a two electron system, two electron is the smallest many electron system. So, it is best to develop the formalism in terms of a two electron system. And I will spend some time at the beginning of today's class to discuss the Helium atom spectrum. So, here you have two electrons and these are the singly excited states of the Helium atom. So, you have got one electron in the 1s and then the second can go into the 2s or the 3s or the 4s or it can also go into the 2p, 3p, 4p or 3d, 4d, 4f and so on. So, as you go from left to right in this diagram you are going to higher orbital angle of momentum states. So, l equal to 0, l equal to 1, 2, 3 and so on. Now, this is part of the story because here you have in the two electrons one electron in the 1s and the second in the 2s. But then there is there are two possibilities because the two electrons can pair into a singlet or a triplet. So, this is the singlet panel and there is a likewise a triplet panel. So, you can have singlet as well as triplets. So, you can have these two combinations of any two electron system. So, I am going to look at some of these diagrams a little closely magnify some parts so that we can focus attention on details. So, here are the possible configurations of the two electron system. You can have the 1s 2 two electrons in the 1s then you can have one in the 1s the other in the 2s, one in the 1s the second in the 3s and then in the 4s and so on and then finally, into the continuum which is indicated by EC. But then it can go to l equal to 1 state. So, you have the 1s 2p, 1s 3p and so on these are the things that I showed in the previous diagram. And I will strongly recommend this book by Macy and Burhup volume 1 which is a very good reference for these details. So, these are the different configurations based on the 1s state, but then you can have you know these are the singly excited states. So, there is a bound to bound transition and over here there is a bound to continuum transition. So, these are the singly excited states one of the two electrons is excited to the other electrons, but then you can have excitations of both the electrons and then you have doubly excited states. So, these doubly excited states you can have both of them into 2s 2 then one in the 2s the other in the 3s, one in the 2s the other in the 4s and then one in the 2s the other in the continuum s, but then you can also have one in the 2s and the other in the 2 in the 3p this should be 3p and not 2p no it can be 2p and then you can have the 2s 3p that and then finally continuum p as well right and then you can have 3s 2 and so on. So, all of those I have not shown. So, there are these several singly excited state configurations and several doubly excited state configurations and here two quantum numbers are involved two principal quantum numbers are involved both are bound states. So, they have the principal quantum number and one is indicated by an uppercase n and the other by a lowercase n. So, here is another look at the same diagram and here you have got the 1s 2 and these are the singly excited states of the helium. So, this has got an ionization threshold of 24.6 electron volts then you have the doubly excited states and these are based on the 2s 2 2p 2 and so on possibilities right and then you have some more of these like the doubly excited states based on the 3s 2 and so on. So, they have different ionization threshold. So, this is 24.6 this one is 64.6 65.4 this is 79. So, these are the different thresholds for these limits then again as I showed in the previous diagram you can have the orbital angular quantum number moving from left to right. So, you have the s states over here the p states over here and then d over here and so on. So, as you move to the right you can go to higher angular momentum quantum states. The labeling as you notice is done in terms of independent electron picture. So, the quantum numbers are the hydrogenic quantum numbers although this is the many electron system and the hydrogenic quantum numbers are not really good quantum numbers. So, what gives you a good quantum number is a configuration interaction between these states. So, the real wave function will be a superposition of these different states and some of these could also include the continuum states and it is this configuration interaction which is of great interest to our discussion today. So, now in these diagrams which I showed until now today I have not shown the continuum but I will show it in the next slide which we had a glance at toward the end of the previous class and this is from Fano and Cooper's review of modern physics which is very awfully cited literature in this and here this is the same spectrum except that energy is moving from energy increases from left to right in the previous diagrams it was from bottom to the top. So, I have rotated this diagram through 90 degrees and here energy is going from left to right. So, this is the first ionization threshold which I showed you in the previous figure which is 24.6 electron volts and then you have the second one at 65.4 and so on right. So, these are the ionization thresholds which move from left to right as you go to higher ionization thresholds. Furthermore you have we have shown also in this these discrete bound to bound states and you will notice that they have an overlap with the ionization continuum of the singly excited states. So, there will be configuration interaction between the doubly excited states and the singly excited continuum. So, let us have some further look at this now there are some more details coming up in this diagram and that is the reason I am showing this diagram in pieces you know part by part because if you put everything in the same figure all at once there is so much information over there that we tend to miss out on some of the details. So, here we have in this diagram we have shown the 1s2 ground state of the helium atom. So, these are the singly excited states as we have seen. So, this is the singly excited state continuum this is the doubly excited n equal to 2 continuum right. So, the 2s2p states and so on are over here and then you have got the continuum for this and then likewise for the n equal to 3 states. So, these are the ionization thresholds and you can have the ionization threshold is of course given by the usual Rydberg-Bammer formula. So, you can just put it over here and you will get the corresponding ionization thresholds. Now, for helium of course z is equal to 2. So, you get 4 over here z square and that is that is what it is. So, here is the energy scale and this is not to so these scales are different. So, you have got 24.56 over here which comes here then you have the n equal to 2 limit which is here shown on the scale over here and then you have got the n equal to 3 and you have the doubly excited discrete bound states all of these are embedded in the continuum on the left side. So, all the doubly excited states on the right are embedded in the continuum of what is on the left and that is the configuration interaction in which you will have a bound to bound configuration having a configuration mixing with a bound to continuum. So, here is the magnification of this part alone. So, this is the singly excited state configuration right converging to the series limit at 24.58 and it is this part which is magnified over here, but now I show not only the 1 S 2 S and 1 S 2 P separately, but I also show the singlet and the triplet. So, there are actually if you notice over here there are 4 rid bulk series, but they are so tiny that one could miss out on that and that is the reason I have magnified this diagram.  So, you have a number of possibilities. So, all of these are built on the 1 S 2 bound state. So, this is the magnification of this part which is the singly excited configuration. Now, if you look at the doubly excited configuration. So, here is a magnification of this part of the doubly excited configuration based on 2 S 2 S 2 S 2 P etcetera, but then again the 2 S 2 S you have a singlet and a triplet and likewise for the 2 S 2 P also you will have a singlet and a triplet P. So, you will have a number of rid bulk series coming out of this. So, here is a so this is again a magnification. So, there are actually 4 rid bulk series in the double excitations. So, this is what leads to the configuration interaction between the bound to bound and bound to continuum because all of these discrete states doubly excited discrete states they are embedded in the continuum. So, here you will have a similar spectrum also for the negative hydrogen atom which is again a 2 electron system. So, I am not going to discuss this in great length, but just to point out that the hydrogen minus will also have similar very similar rid bulk series. So, when you work with photo detachment and so on you will be using a spectrum of this kind. So, now this is the main question of interest that you have these 2 electron configurations and a many electron system you can you can discuss important correlations in terms of these 2 electron configurations. So, when you have the energy of a doubly excited state in 2 possible configurations 1 of which is a bound bound and the other is bound continuum then you will have resonances and what is going to happen is that the electron may spend some time in the excited state configuration before escaping into the continuum. So, the electron can of course escape into the continuum because energetically that is possible. So, it would happen, but before it does so there would be a little bit of time that it spends in the doubly excited state and what this time delay will do is to cause an additional phase shift which will be a resonant phase shift because the double excitations belong to the discrete bound states and they are therefore, at very sharp not not really sharp they have their lifetimes, but they are at specific energies at the resonance energies. So, there will be a resonant phase shift coming from this and it is this that we are going to learn from Fano how to analyze this. So, we mentioned in some of the earlier classes that there are these 2 kinds of resonances the Feshbach resonance or the Fano Feshbach resonance as I normally refer to it and the other is the shape resonance. So, the Feshbach resonances are because of configuration interaction between the bound to bound and bound to continuum configurations. So, the shape resonances are due to the form of the potential. So, this is when the electron can tunnel through a potential barrier and then it finds itself to be exposed to 2 different alternative states one which is trapped in the inner well and the other after tunneling through the barrier. So, these are the 2 possibilities and Feshbach resonances will therefore occur below the ionization thresholds where a shape resonances will be above the thresholds. So, these in both cases you do undergo a phase shift through pi as you go through the resonance. So, the Fano Feshbach resonances are relatively narrow and this is just a gross feature one has to get into the details because within the family of Fano Feshbach resonances there are very many different kinds of shapes and widths that we have already talked about in our previous class. So, what happens is that our interest now is in studying the electron correlations the configuration interactions which lead to these Fano Feshbach resonances and they lead to auto ionization these resonances that we are talking about are more specifically called as auto ionization resonances. And what happens is because the discrete bound state is embedded in the continuum at some energy the energy difference between the 2 states becomes exact and then it becomes impossible to use ordinary perturbation theory because in perturbation theory you often have the energy difference in the denominator and then things blow up. So, you need special techniques and that is what Fano introduced in this marvelous paper. So, let us refer to this configuration interaction between discrete and discrete and continuum. So, the discrete state I will indicate by phi d. So, get familiar with the notation I am using the continuum state by psi e the discrete states are a square integrable. So, I normalize them using the usual integral normalization integral and the dimension of the discrete states are of course, l to the minus 3 by 2 the continuum states however are not square integrable and these are then normalized on the delta function like a delta function scale. You can normalize them on a k over 2 pi scale in which this is how the normalization would go or we have discussed this normalization earlier as well in scattering theory and you can also normalize them on the energy scale and if you normalize them on the energy scale then the dimension of the ray of the functions would be e to the minus half l to the minus 3 by 2. So, mind you the dimensions of the discrete states and the dimension of the continuum states are different because of how we have chosen the normalization. So, these dimensions are different and that is something that you must keep in mind. So, this will also be similar case for the negative hydrogen ion and you can think of this 2 electron system whether it is a negative hydrogen ion or a helium atom as a 2 electron system and if you ignore the interaction between these 2 electrons then it is as if you are having a Hamiltonian for one hydrogen atom and another hydrogen atom. It is almost like that. So, the 2 energies would add up if you ignore the interaction between the 2 electrons. So, you have got the kinetic energy operator for one electron and the kinetic energy operator for the second electron and the potential energy of each of the electron in the field of the nucleus, but you have ignored in H 0 this is the unperturbed Hamiltonian the interaction between the 2 electrons. So, this is your unperturbed Schrodinger equation and it gives you an energy which is just the sum of the 2 energies. Now, you can write the 2 electron wave function as a product of these hydrogenic states and your product then becomes your 2 electron wave function becomes a product of these 2 and these are obtained as solutions of these 2 independent Schrodinger equations. Now, this is the case when both the electrons are in the discrete, but we considered various possibilities you can have 1 s, 2 s, 1 s, 3 s and so on, but both are in the discrete, but what if 1 is in the continuum and this is the 0 order energy for the 2 electron system when both are in the discrete, but then you can also have 1 in the continuum and the first order correction when you do include the interaction if both the electrons were in the discrete would then be given by the matrix element or the perturbation Hamiltonian in the unperturbed states which is phi d which is just the product of those 2 1 electron hydrogenic states. So, this would give you the first order correction. So, this goes into the energy matrix E phi. So, we are going to construct the energy matrix for various possibilities. So, you have now the perturbation H 1 and E phi will give you the energy correction due to the electron-electron interaction if both the electrons are in the discrete. So, this is the possibility we have considered over here. So, E phi is the correction and of this H 0 is nothing but the sum of the 2 energies that we got earlier and H 1 comes from this correction over here. So, this is what gives you the E phi which goes into the correction. So, this is when both the electrons are in the discrete. Now, what if one of the electron is in the continuum? Now, if one of the electron is in the continuum then one electron state is represented by the bound state which is chi and the other will have an energy which is E minus epsilon 0 with a position coordinate R 1. So, the, so you still expect the 2 electron wave function as a product of the 2 wave functions, one for the discrete state and the other for the continuum. So, now you have this continuum and the discrete state is embedded in this continuum. So, this is now your energy matrix. So, you have got the energy contribution when both the electrons are in the discrete that is phi D which is what we indicated earlier. When you have one in the discrete and the other in the continuum right then you have V E and then here this is just the matrix element of the Hamiltonian in the continuum states. So, this will have a delta function popping up right because both of these energies are in the continuum. So, this is now the framework of the energy matrix. Now, the thing to remember as I mentioned earlier is that this is got the dimensions of L to the minus 3 by 2 and this will have a dimension of E to the minus half L to the minus 3 by 2. So, what is the dimension of V E? The dimension of V will be E to the half. So, this V has got the dimension of root energy and this is to be remembered because when we take its square you get the energy width which has got the dimensions of energy. So, that comes out as V square in our analysis. So, just remember these dimensions. Now, the configuration interaction wave function will be a superposition of the discrete state and the continuum state and the continuum you will have an integration over a range of energies right DE prime right. So, this is your configuration interaction wave function and now our task is to determine these coefficients A and B and go from there ok. So, here we are. So, this is our Schrodinger equation. Now, let us project this on a discrete state because this is the Schrodinger equation for the two electron system and it will have components on the two electron bound states or one in the bound state and the other in the continuum. So, in this case we project it on the discrete state possibility. So, let us take the projection of this Schrodinger equation on the state phi d. So, we have got two terms one coming from the discrete and the other coming from the continuum right. So, these are the two states. Now, we already know this what is it? This we have already determined because this is E phi right. This is our E phi which would go into the energy matrix. So, this is E phi. So, we put this over here and now you have got the matrix element of H in this continuum and phi d. So, this H over here is nothing but E psi right. So, I pull E common and now you have got a projection of psi on phi d which you get from here which will give you A E. So, now just I have written this Schrodinger equation. So, that on the left you have got E psi on the right you have got H operating on the superposition of the discrete and the bound states. So, this is the projection that we are considering right on phi d. So, here you get the E phi which is here. So, E phi comes here this coefficient is A E coming from here and then this factor is nothing but V which we have written earlier right. So, you get V over here and this is the projection of psi on phi d. So, the projection of psi on phi d is this coefficient A E which comes here right. So, you have got E A E equal to A E E phi plus this integral. Now, I will refer to this equation with this green star because I am going to need it in subsequent analysis. So, whenever we need to refer to this or I will pull it up with a reference to this green star it is like giving it an equation number. So, here you have got this configuration interaction. Now, we will this time project it on the continuum. Now, this continuum state is a mix of this is the configuration interaction state it is a mix of a discrete and a continuum right. So, it has got both the components and we now project this Schrodinger equation on such a state which is a mix of discrete and continuum. So, this is our projection now look at it term by term. So, this is the projection on the discrete part and here is a projection on the continuum part. So, now what do we get? We look at each term separately A E we can pull out as a we can as a factor we can factor it out and then you have got the matrix element of H n psi n phi d which is nothing but V. So, from this term we get A E V E and likewise from this term we will get a Dirac delta because these two states are in the continuum. So, you get a Dirac delta there. So, now you have this relation and now we have the left hand side which is E and these two will of course give you a Dirac delta you can see it coming and then on the right hand side you have this delta function integration over E prime. So, only the term in E prime equal to E double prime will survive. So, you get a coefficient B with E E double prime and this E becomes double prime that is because of the Dirac delta integration. So, here are the relations we have. Now, this is the result that we got on the previous slide and in this I now put this superposition on the right side. So, now I get two terms and from the first term E and A E are scalars they will come out and you will get a projection of the continuum state on the discrete, but they are orthogonal. So, that will vanish and then from here you will get the scalar product the scalar product of these two which will give you a Dirac delta. So, let us put both of these over here and now you get from the first term you get 0 from the second you get a delta function you carry out the integration you get only the term in E prime equal to E double prime. So, you get this equation here and this is the one which I will refer to as a red star. So, now we have got two equations that I am going to be referring to one with the green star and the other with the red star. So, these are the two relations one with the green one with the red. So, these are the two relations that we are now going to analyze. Now, V is real and here we have used two primes. So, there is no need to use two primes we will use only one prime. So, I will rewrite them without the double prime and writing them as real numbers. So, what do we get from this relation red star? From here we get V equal to A E V E divided by E minus E prime which is fine as long as E is not equal to E prime. So, all this will work as long as E is not exactly equal to E prime, but that is just the case we really want to handle that is precisely what we want to handle. So, to be able to include the E equal to E prime Fano used a very beautiful technique which was devised originally by Dirac and this is the Fano Dirac technique. So, they make use of principal value integration to address this situation. So, I will show you how it is done. So, the purpose is to address this particular case of E equal to E prime and this is the Dirac Fano technique and what this technique does is following Dirac and this is really beautiful device to handle such situations that instead of writing B like this you write it as 1 over E minus E prime. So, the A E V E prime is here which is here. So, this is the A E V E prime that factor is here and then over here you add a function z multiplied by the Dirac delta because you are going to use it in some integrant. So, that will leave a value only when E prime is equal to E which is just what you want. So, that is the trick. So, this is to be used in principal value integration. So, this is the Dirac Fano trick and you will see that it really does wonders for us and the function z is so chosen that it meets the appropriate boundary conditions because the boundary conditions are known to us. So, by putting appropriate boundary conditions you can determine the function z and now whenever you have BE in the integrant then you can use this Dirac Fano substitute. So, this is the brilliant mathematical device which Fano used in this context. So, this is to be done using the principal value integration whenever BE appears in the integrant and the principal value integral which I am sure all of you would have worked with in your mathematical physics courses or some other courses and collision theory and so on. So, whenever you evaluate an integral from E 1 to E 2 and then you have a problem over here when E prime goes to E then you go close enough and then hop over that point and then do the rest of the integration. So, that is the principal value integral that is how it is defined. So, this is our Dirac Fano proposal and what is understood is that the principal that whenever you have this type of an integral to be determined then you should carry out a principal value of this integration and use z appropriate to boundary condition. So, these are the two things which go into this technique. So, it is the idea is very simple but extremely beautiful and very powerful. So, let us see what it does for us in this particular case so here we have the principal value integral defined and our interest is in this relation here which we have identified as a green star and over here this integral will be replaced by the principal value integral. So, this integral this BE is to be replaced by this factor here coming from the principal value integration trick of Dirac Fano and we will work with this relation instead of this. So, here we are so now this is the relation that we have agreed to work with this in this we have already employed the Dirac Fano trick now first we cancel out AE which is common in all the terms. So, AE is cancelled out we need not carry it any further the rest of the equation is here after cancelling AE now in this we have got three terms one is E phi which is here then you have this term which is here and then you have got the third term which is integration over E prime Z V E square and the Dirac delta. So, there are these three terms in this expression are we all together. So, now here what do we get we make use of the principal value integration from this term you have got a Dirac delta integral. So, this is the Dirac delta integration and this will give you a value which is appropriate only for E prime equal to E because the integration is over E prime. So, you get a value which is appropriate only for E prime equal to E. So, you get Z V E square from this term and here you will now carry out a principal value integration because the value for E equal to E prime is already taken care of over here that is already taken care of. So, your problem is done actually it is really brilliant idea. So, now it also tells us what Z should be because Z is now a solution of this. So, this principal value integral you represent for compactness by a function E this is a function some function of E there is no difficulty in evaluating it there is no singularity in that right. So, you evaluate this integral and you represent it by some function of E and that tells us what Z should be it is given by this. So, now we have almost all the pieces that we need and now it is just a matter of rearrangement of these terms to extract our final state configuration interaction wave function. So, here the Z is given by the Dirac Fano trick the configuration interaction is this as we know. So, what we are going to see is that this particular state which is in the continuum. This will look very similar to a continuum state solution that we have worked with in scattering theory all along you will remember that this goes a sin k r plus delta right. So, you have got the sin k r k depends on energy. So, it is sin k r plus delta which is a background phase shift, but then there is an additional phase shift which is coming from the configuration interaction this is the resonant part. So, there is this additional phase shift which is coming because of the time delay that the electron may get bound into the doubly excited bound state for some time before it escapes into the continuum. So, you have got a background phase shift and then a resonant phase shift due to the configuration interaction. The resonant phase shift it will turn out and you will see this coming in the next you know as we go further in this class that this resonant part of the phase shift will turn out to be given by negative trend inverse pi over z that is what it will turn out to be, but that is a result which you will see coming. So, now we have to determine this coefficient a according to normalization and this is our normalization here. Your configuration interaction state is you have a combination of the discrete state and the continuum state. The discrete state is weighted by this coefficient a. This is what we want to determine. These states are of course Dirac delta normalized. So, if you take this Dirac delta on the left, then on the right side you have got a projection of this on itself. So, you will get A e star A e from the first term phi and psi are orthogonal and then you will get the B e star B e times this. So, we have used these orthogonalities earlier as well. So, this is the left hand side is nothing but the Dirac delta and then the right hand side you have got the A e bar A e. There are two different energies I am considering mind you both are in the continuum e and e bar. So, you have got two coefficients A e bar and A e and over here because you carry out this delta function integration over e double prime. So, that the only term that survives is the one for which e double prime is equal to e single prime. So, you have got the e bar e single prime and then the B e and the subscript here is again e single prime. That is the Dirac delta integration. Now, here is the relation which we have obtained earlier which is from the previous slide which I indicate by this diamond here. The only thing I have done in moving this to the next slide is I take this A e bar on one side. So, I have delta e bar minus e over here. So, this is the relation I get from the previous slide. This is the Dirac-Fano trick. So, we have putting all our pieces together. Now, in this relation which is indicated by this diamond relation here, this over here is I use the Dirac-Fano inclusion of e equal to for equal to e prime because both are here you have got e prime over here. So, you have one for e bar and the other one for e. So, you have two substitutions over here one coming from this relation and the other coming from the corresponding relation for e bar. So, this is for e and this is for e bar. So, these two go into these two pieces. So, I substitute this and now I have got an equation which looks slightly large, but we can handle it on the screen. So, remember that you will get quadratic terms when you multiply you will get a z over here and another z over here you have a v over here and another v over here. So, you will get quadratic terms and z and v. So, let us write all of these terms. So, you have the a e v e and a e v e prime over here. So, there is an a e bar and a e. So, a e bar comes here the a e comes here the v e bar the v e prime is inside the v e prime is inside because it is under integration. So, it does not come out and then you have two terms over here. So, the v e prime remains under the integrand. So, you will get v e prime over e bar minus e prime here. Then you have got the second function there is a Dirac delta here and then you have got a v e prime sitting on top of this e minus e prime and then this function z and then this Dirac delta which is here. So, now you have got 1 2 3 4 4 terms. So, 2 terms multiplying 2 terms over here. So, you will get 4 terms. So, they are here 1 2 3 and 4 you have just multiplied them out and I think this is when I was preparing the slides I realized that it is really nice to be able to use in this technology enhanced learning program that we have the provision to flash these slides at the click of a mouse. So, that we can concentrate focus of a discussion on the physics and not get lost in carrying out mere substitutions that itself takes such a long time on a blackboard if you were to write it using a chalk. So, this comes at the click of a mouse and we can actually concentrate on the physics and on how the analysis is done. So, here we have these 4 terms right and you can of course, look at the pdf at leisure and make sure that everything is done right. So, here you have 4 terms 1 2 3 and 4 out of these 4 terms these 3 have got Dirac delta and you are carrying out integration over E prime. So, they will give you terms in which only E prime equal to E will survive right and then there will be a 4th integral which will be left coming from this quadratic term in V right. So, those are the 4 terms. So, here is the result of those 4 terms. So, here you have got the first one you have got this integral over E prime this is quadratic in V. So, here it is and from the remaining 3 terms you carry out the Dirac delta integration and you are left with only those terms corresponding to which E prime is equal to E bar in this time over here E prime must be equal to E and over here E prime must be equal to E bar. So, those are the 3 terms which you are left with, but remember that in the last integral you have carried out integration over E prime. So, E prime must be equal to E bar. So, when you write this Dirac delta E prime has been written as E bar. So, that is something that you have to keep track of. So, now we have got all the terms and I have now I have got these 4 terms, but I rewrite them. So, this first term which is indicated by this 4 point green star which comes here. So, it is just a rearrangement of terms, but the color code will show you which term is where then I have this term coming over here with the difference that here I had E minus E bar, but now I have E bar minus E. So, I have plugged in a minus sign to compensate for that and then I have this term which comes here as it is and then I have this term with the Dirac delta E bar minus E coming here, which is a quadratic term in Z and V, good. So, now we have a double singularity here. So, what do we do? We use identity which is well known in mathematics. So, when you have a double singularity of this kind you use this identity that which lets you take advantage of the Dirac delta integrations. So, I will not prove this identity I will use it. So, this is the identity that we will use and this was used by Fano in this work. So, instead of this 1 over E bar minus E prime multiplied by E minus E prime you use the right hand side of this and this will give you the Dirac delta or product of Dirac delta in the numerator and then the rest of the terms. So, we will use this identity over here and using this identity this integral now takes this form. So, instead of this integral you have the D E prime V E prime square coming here the D E prime V E prime square and here this 1 over double singularity is now written in this beautiful bracket which is this entire right hand side of this identity. So, this is the relation that we now have to work with. So, I brought it to the top of this slide now. Now, here again if you play with the Dirac deltas you can show that the product of these two Dirac deltas is completely equivalent to this. So, we will use that and using this combination instead of this we simplify this further and let us now analyze this relation. So, this is the relation that we are going to analyze and I will I have used most of the discussion is from Fano's paper there is some from Kavans paper and some from some other sources. So, I will cite all the references and it is good to use them in combination. So, in addition to Fano's paper I recommend Kavans book theory of atomic structure and spectra and the notation is slightly different which is why I am alerting you to that that the principle value function this integral has the same notation in both Fano and Kavans, but the difference is here that what Fano calls as z instead of that Kavans uses eta, but his function differs from Fano's by a factor of pi. So, that is the only difference and if you are using a combination of these two sources then you have to be careful with the literature. So, here what we will do is we have this relation here in which you have an integral of 0. Integral over E prime of a quadratic term in V over E bar minus E and this is where you can use the principle value integral which is represented by the function F. So, we will use F of E instead of this term over here because the E equal to E prime or E equal to E bar we have already taken care of using the Fano Dirac trick. So, now by using this F instead of this we now get all of these terms are now over here. So, first I write these two terms which come here. So, these two terms come here then you have this term these terms in which you have got V square over E bar minus E. So, these are replaced by the principle value integration F E minus F E bar. These are the two which are coming here and then you have this term which comes here and then what are you left with? You left with one term over here which has got the pi square V square. So, the V square is here A E bar A E is here and the pi square is here and then you have got the E prime equal to E. So, instead of E bar minus E prime you have got E bar minus E because E prime has to be equal to E. So, that is what you get here. So, now I think we are getting it in a mathematical form which is becoming compact and we can address it in a tractable form. So, notice that these terms both have got this factor common this is A E bar A E over E bar minus E this is A E bar A E over E bar minus E. So, you can combine these two terms you can factor this out and write it write the remaining terms separately. So, here the principle value how are we including E equal to E prime because the delta function is equal to E prime is here it goes into the Z that is taken care of already. So, when you take the principle value integration you are hopping over the singularity you go as far as the singularity and then pick up the integration from the point next to it and the point of singularity is taken care of by the function Z. So, here you are and now you have got this term as common in this factor. So, we can factor it out and I have factor it out and written the remaining four terms. So, they are now in terms of f and f E bar. So, here I have here I have f and f E bar and here I have Z V square and here I have Z E bar V E bar square. So, these are the four terms which will be in the remaining factor. So, those are the four terms in the remaining four in the second factor this has been factored out and this is the residual part which we had written here which had the Z square plus pi square and the Dirac delta. So, here it is. So, you have the A E bar A E V E bar V E Z square plus pi square the Dirac delta. So, we have got all the terms Z we know we already have seen that Z is given by this relation and we are now able to anticipate a very important very nice result. So, this is the value of the function Z at E, but we have two energy indices one is without the bar and the other is with the bar. So, this is Z E bar and you put both of them. So, you have got two expressions because you have got Z V E square and then a Z E bar V E bar square. So, you have to use the one with appropriate notation and the difference between these four is then nothing but E minus E bar because of this cancellation because when you cancel this the E phi drops out and then you have only the E minus E bar. So, now we have got a simpler relation. So, this is the left hand side which comes here and the right hand side you have got A E bar A E or E bar minus E which is here what is under this red under lining is now just the difference between E and E bar which is here and then you have got these two terms over here which is Z square plus pi square direct delta multiplied by A E bar A E and V E bar V E. Now, I rewrite this by moving this term to the left. So, I have got on the left hand side this term minus this, but I have it with the plus sign because I have E minus E bar in the numerator and it is negative in the denominator. So, I have got A E bar A E with the plus sign on the left. So, this is the relation that I will bring to the top of the next slide. So, here we are. So, we have got these two terms of the left. Now, this is the relation that we have referred to earlier from slide 68 the which I had indexed by a diamond and now if you look at these two relations it is obvious that this factor over here must be equal to unity the left hand sides of both the equations are the same the right hand sides are very nearly the same except for this factor which must be equal to 1 and that tells you what the coefficient A E must be this is what we were looking for. We wanted to know what is the weight factor in the configuration interaction mixing. So, this is the weight factor and this now comes out as a square root of 1 over V square Z square plus pi square. So, here you can write Z because we know what Z is Z is E minus E phi minus F over V E square. So, you can rewrite in terms of this function of E instead of Z. So, this is your weight factor A. So, we now have this and now we have taken care of the E equal to E prime and now we get the configuration interaction we have all the terms we have the B this mixing we have got this coefficient. So, we now get the complete wave function and in this we make use of the principal value integration as we have done. So, you have the complete configuration interaction wave function in which you have the A E phi D which is here and then you have this piece over here. But because of this Dirac delta integration you have got only the term corresponding to E prime equal to E which is why you get this Z E V E psi. So, this is the configuration interaction wave function that we have now reduced and we will use this in our next class when I will discuss the resonance phase shift. Because I mentioned that what happens as a result of the configuration interaction that the electron which has the possibility of escaping into the continuum because one of the two configurations belongs to the continuum. It has the possibility of spending some time in the doubly excited bound state. So, this is the bound to bound and bound to continuum configuration interaction and because it spends some time over here that time delay manifests as an additional phase shift. That is the resonant phase shift which I had anticipated little bit toward the beginning of this class. But we will see it more explicitly in the next class. Any question?


Thanks to great, so climate and climate change and how to change that has come, has arrived in the middle of our society and Tom is going to talk about the role of digitalization and in the climate protection. Okay, now it's working. Yeah, all fine. And just a brief thing beforehand. We have a brief technical break. We're very sorry for that. We're going to have some intermission activities here. And let me ask you questions about the sleeping habits of the audience. Show of hands, please. Who has slept more than the required six hours? Show of hands, please. Oh, that's great. So who has slept between four and six hours? Yeah, less people. Two hours or less? All right, that's okay. Now the talk is going to start. Okay, very awake audience. It's nice. Technical mistakes or problems are always an ever occurring problem for us at Fridays for the future. And I'm a participant for Fridays of the Futures for about a year. And I came today. Please delete the internet.  Of course, we're not against digitalization. No, no, no, we really like to use digitalization ourselves. Here's a demo from Berlin with Greta Thunberg, who had, who held a very powerful speech one year ago. And that was very inspiring for me. So that I wanted to become part of this movement and not just part of the general climate movement. And what do we want? We talk about climate justice. So it's a typical chanting on the demonstrations. Climate justice now. And of course the 1.5 degrees objective is something that a lot of people talk about. But it's still very abstract. And what you can do of course is to look actually into the literature of climate justice. And we still try to figure out what climate justice specifically means. So there's still some room for interpretation. But there are also other topics that we care about. And the climate protection that we do is anti-Nazis and feminist. And we also have some additional demands for Germany, which have received some response. So we demand a zero, so netto-zero, two emissions. And so if you if you remove trees, you should also replant trees so that you always balance things out. And that's our demand for 2035. And because of the current trends, we think it should actually be earlier now. We also demand getting out of coal power by 2030. And we demand 100% renewable energy sources until 2035. And we consider digitalization to be essential and necessary for this. And what we demand until the end of this year is the end of subventions for fossil fuels. And that one quarter of all coal power plants shall be shut down. And all of this should happen as quickly as possible. And the CO2 tax should be increased to 180 euros per ton as quickly as possible. So we did a lot of agenda setting. So these type of pictures are something that a lot of people have seen for the first time this year. And I actually saw this myself three times this year live. And it's it induces so much rage and fury to see these things and to talk to people who actually live near these excavation sites. So this image, this is from the Hambach excavation site next to the Hambach forest just to really illustrate the local destruction of nature due to coal excavations. But of course here we want to talk about digitalization. We like to call on, we like to refer to the science. We, so there's the scientists for futures as the science, the group scientists for future. And they said that digitalization is, they say that digitalization is not the problem, but that it is a tool that we must use properly. If it only serves the economy and the increase of economic activity, then of course we will have a rise in emissions. But this is not what we should aim for. And it's important to really use digitalization properly in the interest of nature. And this is not only and we're using digitalization for our own purposes. And but in my opinion, the economic factor is one of the greatest victories for the Fridays for the futures movement. And we are a lot of people so and what is really amazing is that these protests happened in more than 600 locations. So we had a lot of technical troubles to have the audio at all of these demonstrations and to make sure that everything is coordinated. So so what I consider the positive political message is that that there was this survey. We see that a lot of the population in Germany is very much for climate protection. And I think this is really impressive. And I don't know how the current numbers look like. And tears like a little equation that I thought about it yesterday. It should not only be about economic economy, economic activity. It should not only be about surveillance. So if we have if you have less surveillance, then of course we have less emissions. And this is something we can all in this room together agree on and fight for. So in total, we had tried we tried to not follow this these steps, but instead we we tried to stay under the 1.5 degrees limit.  And I have to admit that this talk well it that this talk has been following the tendency of first procrastinate and then improvise. Because after powering through all these weeks for the strikes, I'm like exhausted. And this is not only about the strikes themselves, but also about the communication. And if you look at these people who are on this picture, all of them or pretty much all of them have been there every week. And it's not like just one big demonstration, but just but recurring demonstrations every week. And we see that there's a lot of messages. And it's very difficult to filter what is good information, what is bad information. And in some cities we have decided not to go on the streets every week. And we have to improve this. And I want to include this here, because maybe here are people in this room who have a solution for better communications in the movement. But of course, so easy it will probably not be, probably it will not be so simple because this is how Fridays for the future looks in Germany. And it's very hard to see the forest for the trees. And the communication with such a massive number of people is really difficult. And there are a lot of critical points. And one example is like the coordination of this talk, so that I am giving this talk is actually a privilege for myself. And that I am that I have grown up in Germany and that I have a bit more experience than other people. But but it's also because I know specific people and of course we can't coordinate with everyone in the movement who actually stands here and is giving this talk. And of course there are disputes internally. But through these disputes we actually grow. But people who actually go onto the streets, those are the people who just said like one year ago, yeah, something has to be done about the climate. But now, but now people actually start questioning economic systems. They really grew and we reach out to international organizations and and now it's young people have learned to give themselves a voice that hasn't been heard before. We wouldn't be us if we weren't just continuing and I'm going to reiterate that tomorrow there's going to be another demonstration and I'm going to participate and I will be very very happy about every single person who joins us. I just really really rushed through my talk right now. But maybe that's not so bad after all. We're going to be here. We're going to be very happy to be here at events like this. What other things are we planning for next year? As I already told you, we are concerned about coal a lot and next year there's that in four. That's a coal plant in NRV who burns a stone coal and this coal isn't being mined in Germany. Maybe some of you remember that Steinmeier gave a very moving ceremony for the last bit of coal mine in Germany and now we have a power plant that is going to come online for the first time in Germany and just consider this and you can be sure that Fridays for Future is going to put a lot of pressure on this event. We're going to be at this event and we're going to be joining other organizations with the similar goals and this year we have learned how it works to network with other actors and organizations and have mutually beneficial arrangements. We've been in the Rhineland where there's civil disobedience incidents and on the other side there's young people who live near these mines who are taking to the streets and then there's demonstrations such as all the villages stay who were also there demonstrating against the coal plants and so we've met people young and old local from abroad and we've joined together and we've protested and these images of joint protests were carrying out into the world to show people and I've showed you this map in Germany with the pins and on a European level we aren't as connected to each other. There's going to be a convention in Turin next year called Smile where we are going to look at the European level and building up these European networks. The number of participants in the demonstrations may have risen but we have sorry they may have gone down but we have built something unique in my opinion. We are not just people going out to demonstrate or me giving talks and you're listening but we are coming together. People are coming together under this umbrella so I'm very much looking forward to the Q&A day later and to show you a picture again. It's just not 200,000 people in Berlin but every week at least 100 youths meet to plan these demonstrations and a lot of this happens digitally via WhatsApp and this digital mobilization is what we need in these climate protests. We have built a website where you can see and look for cheap flights to book but they weren't real of course that was to point out that flying is bad for the climate and this is a way of taking protests from the streets to the internet and what makes me really very happy is when I talk to people about Fridays for Future is when they tell me what they think about our organization. So we tend towards having our own opinions even if it's just an audience Q&A so can we start that Q&A now? Thank you for our talk. So there's a lot of time now for questions. There will be the first question from the internet will come right up who if anyone else has questions please do approach the microphones so this is a question from the internet who clearly was at these demonstrations he's sick of being protested against by people who are on the other side of the political field so that's actually a difficult topic we always have scenarios where people are hateful towards us I've seen approach I've seen passerbys spit at people we had a case in Berlin where right extremists proclaimed to proclaim they would destroy this demonstration there are these structures you can reach to which protect you against these right extremist people they are always open to help you they are always friendly towards you we are currently trying to build up a Germany-wide help group to help you against these right-wing approaches maybe that isn't quite built up yet so maybe you should approach local sources this will be treated with respect wherever you go just make sure wherever you go is a place you can trust so microphone number one please my question is and that is what I call you is that really a good idea to make people pay money for CO2 because that would also that would also encourage economic opportunities in that I did vote against the green point which encourages further economic development in the trash-handling business maybe you have another idea of how to solve this so I believe in the trash that we burn and we blast through the atmosphere we don't see that maybe I think about I often think about whether the this CO2 tax is a sensible solution but I think it may be a sufficient tool to make companies not single persons but established companies be worried about be worried about doing more emissions so we we haven't come as far yet as to the point where coal energy doesn't doesn't add up anymore so we would have to improve these prices we found out that there is actually there is actually a tax thing a tax return where money is being paid back to the people so that there isn't any bad notices around these CO2 tax there's there's this giant carpet of solutions to this this is one of the things we do view as sensible microphone number two please do you also discuss in some form of base or in some form of voting circles whether you do want to protest in schools as well so maybe you would even strike inside the schools themselves maybe you would build up a media for that well I'm no longer a student I do not know that there are always suggestions and feedback there's always suggestions and there's always feedback in that topic you may you may intend on asking for about something else where there would be gigantic meetings and gigantic opportunities for people to strike in schools my idea would be to spend your time that you usually would spend on the streets to invite teachers invite older people to discuss with them on a less on a base that is less violent than we are trying to move away from that Fridays in school thing so in Berlin right after that disobedience we talked about about topics and all other all sorts of other things to to make to educate us and to make us more knowledgeable so we do want to do that but it's not been done until now hello this is the circle I do want to ask you whether you want to try and bring this into parliament or fund a party for this as right now it's only on the streets there are the greens there is the green party but in the german parliament and in the european parliament there isn't really that much climate climate oriented parties we are not we do not support any party as an organization there is a discussion internally and how to approach these topics that is handled on an individual scale we don't we do not want to make a party we are organized very differently from parties and we do have some base basic democracy democratic base work but we are different from parties in many ways we don't really fit into that role quite well we're not really institutionalized we're really official we're not a party many of us are political many of us do engage in these political topics but and we also do speak to MEPs or MDPs in that in those fields but overall there is not really this organization we do see that in small bits though another question from the internet so you spoke about IT infrastructure there was a talk yesterday about the IT infrastructure of extension rebellion and do you exchange in that topic and do you use your other systems right up until now we have our separate infrastructure I do have to admit though that I'm not in that field of knowledge I don't deal with these website related things in this detailed communication there are people who do that here you can visit them of course we try and of course we try and cooperate as right now we're not in that much stress as we're not protesting every Friday most things are signal groups which are joined between extension rebellion and Friday's future we as a we as a organization use email if you whenever you read an article or a thread on Facebook Twitter whatever about you there's always there's always people who laugh about your organization who always complain about you being not that climately reasonable as well there's people who get really offend by that do you have a strategy on how to deal with this do you think this is okay or do you think that people don't really know they can play a non-roll well we do want to we do try to get people to moderate these comment sections we haven't we haven't found a good solution yet we have to view this organization in Germany as only being one year old so in some parts we're very much overwhelmed by the situation in some places we should treat these matters more accurately though mainly on Twitter and Instagram we try and communicate with people who have who have experience in these themes so that we can successfully do this social media thing we should of course work on that for longer another question from the internet well the internet is emotionally quite distressed how does this look on the personal level to to go with this organization and to be faced with this ignorance on climate change there's of course there's people who think people just want other people just want to see children suffer how do you view the situation I can only answer this personally for myself for myself I whenever I whenever I see this acutely whenever I see this as a whole whether that's racism or sexism I do notice that very much in the beginning I do see this pretty much immediately I may also be able to speak for this entire haul that this is the same with climate topics it's even more abstract there's not that many people who actually do hit you that's more easy doing this is more easy when you only treat numbers but for me personally it's just retreat retreat to the front lines that's why I'm active that's why I do this if I just sat on my ass then I wouldn't really I would explode inside there are many institutions who do who deal with this there for example there is psychology for future which are psychologists who who deal with topics such as the aspect of hate the aspect of activism in that field this is very this topic is very frustrating I do not really want to do this another question you said you try you were trying to be above party level there are there are requests to diverge to change article 20a to make it German zero to make it to German zero law I have to say I do not really understand this this German zero thing we do not have an opinion on that as a movement but the people who did this who made this possible already had successful campaigns there was a people's vote in Berlin which was successful so this is not going backwards there is development in that direction our next if that's going to be our next coup I don't know do you do you also have people who care about the technology behind that of course you can say we should all exit coal but that would also mean we would have to build that many other power plants or x amount of wind power plants so these propositions look very small and abstract but we but we've been doing this for months and we talked to so many different researchers about this there are many many there are many informations on this on our website if you go to the FAQ and visit the forum then you will see that everything we request from governments and everything we want to pull through with is actually possible there are many people invested in actually thinking about what we would need for this change you mentioned multiple times you had extremely difficult and spaced out decision processes can you explain how this works and can you explain how you do this first of all we're decentralized in these local groups the best thing about that is that they are autonomous on themselves they can do almost anything they want they can do what they want basically these groups all have all have representatives who do a weekly phone conference within different regions where you can manage these certain these certain events and what you want to do this will be carried over by the representatives into the local groups and this will be decided about there this isn't the optimal process because this takes some time for the different groups there are also self-explanatory things that are predistinguished by their work fields this isn't perfect I believe this is really difficult to get perfect we've been discussing about this manner for a very long time we have a structurization spreadsheet which is kind of which I as myself regard as a very unnecessary measure yeah we just have to think about this microphone number one next question again thank you for starting all this movement I just wanted to tell you and this whole hall there is actually developers for future there are so many different groups who are invested in this climate thing I wanted to ask how known this is there could you maybe do an announcement for that there are parents for future artists for future all these different groups I would be interested what has to happen so that everyone wakes up everyone gets involved in this when I listen to politics and science who always talk about these how much CO2 do we have left how much space of movement do we have it's always been talked about seven years we have seven years we have seven more years this all doesn't sound that dramatic I just want to be active want to know how to spread this information further on what does what has to happen what do we have to do this is a question to everyone because this responsibility isn't only with Fridays for Future this is for everyone applause and I support that what is important for me is that this has to carry on what can we do is also a question we have to ask ourselves I'm very happy about these initiatives but we didn't really start with this climate justice movements have been going on for such a long time and we don't know them we don't we don't name these names we don't name these activists who actively fight people and other people in the rainforest who actively get murdered by corrupt governments for fighting for their own for their own survival so all these people who are locked and who are under suppressed and forcefully denied are our founding fathers so to say I don't want to say that we as Fridays for Future started this as it is just not justified next question is from the internet again on the different levels of Fridays for Future how are demonstrations and protests managed are there any problems regarding that that is very depending on place in some parts these weekly strikes have been have developed into critically close groups it's just it's just basically it depends on the location there are many places where there is breaks in the way of this path you always get these questions and bad things to hear from even policemen and police women we can't really say that we are all pretty much chill with that we have some sort of natural distance from that and we just have to see how this changes further on one more question microphone one thank you very much for your engagement thank you very much for pushing this topic into everyone's mind to he can hear it my practical question when you look into energy more you will see that regenerative technologies and regenerative electricity doesn't run continuously and that this is a very difficult solution well I don't want you to answer this I would want politicians the ones who we have to put this responsibility upon to answer this do you have any knowledge that you could put on politicians desks of course we didn't write this out in letters there are magnificent and wonderful people who help us with this knowledge who help us communicate this knowledge I myself study at the TU Berlin and there's a group which is called colexed they deal with how the exit of coal energy go in detail we aren't on that topic we aren't researching this but we do have some ideas we're not on the stand where we can say we could do that tomorrow but this development we want to watch more where there's where we see these regulations where Peter Admeier wants to have a thousand meter distance between solar no wind power plants and villages we just have to continue on there are multiple different ideas for this and you can speak to that later now we are running out of time you can ask speaker later after this talk and Koli invited to keep this discussion up thank you for listening to the English