Hey everyone, welcome to today's day trading recap for Wednesday, June 30th, a nice green day today, plus $1,933. I had a lot of fun today with the open house, letting non-members in to come in and check out how we day trade. So that was a lot of fun. Let's go to these trades. I'll show you what we did starting with our mighty 90s. Had two trades, one in Apple for plus $3.89 and then one in Beyond for minus $3.00.  No pairs trades today.  So we move that off the screen and let's go into the platform. I'll show you what we did. Let's start with the mighty 90s. And the first one was in Apple. So we booked $3.89 in Apple. Let me open up this chart here. So had this initial big push in Apple, nice spike in volume, got short right here and caught this down move there for a nice $3.89 in Beyond or one loser was in Beyond. And it could have easily been a winner, but had this initial strength and then it kind of pulled back gave us a nice little volume spike here. We got long right here and it continued to move lower added to it right here. It bounced. I got out of that ad and then it came back down and I added here and got out of that ad on this bounce. I was, it started to get weak here. I thought it was going to roll over. So I ended up just cutting out of it and taking a loss. As you can see five minutes later, this thing popped back up. So I would have booked a profit there, but ended up taking a loss on Beyond of minus 300 and then on the runners. All right. So let's start with Boeing. Boeing's still moving higher. So Boeing had a nice volume spike here. Just on this little tiny pullback got long, caught this move up there, got out of most of it. And then I think I got out of my, yeah, I got out of my last piece right up here at this pivot level. So booked 205 is small position in Boeing. I was going to add if it pulled back more, but never got, never got down there. So just small position, book 205, Disney. Disney was a nice one. Disney made us wait a little bit, but ended up being a beautiful one. So initial strength here pulled back. Again, I was going to get more size on here if it pulled back to the unchanged mark.  So plus 444 in Disney. Amazon, my biggest winner, plus 830, mostly just because Amazon's big boy contract, only did a couple contracts, but right here is it pulled back to yesterday's high, got long, popped up, got out of half and then it kept going and got out of, got out of the rest right at the highs of day booked 830 just on that little bounce in Amazon. And then Tesla plus 365. Tesla had this initial push higher pulled back. Let's see. Oh yeah. No, it's after this push pulled back to yesterday's high. I got in right here, I added one as it pulled back a little bit more and then finally it bounced up and closed at the last piece right up here. So plus 365. So plus 1,933 on the day. Good stuff. We'll be streaming live tomorrow, Thursday and Friday. Look forward to seeing you then. Talk to you then.


OK, so by now, you should understand what summary reports are, how Noisy is added to protect user privacy, and how Noisy can help you assess different noise strategies. So as you know, summary reports are noisy in order to protect individual user privacy. And how noisy a summary report is is going to affect how useful the data it contains is. So with Noisy Lab, you can simulate how noisy your summary reports are likely to be based on your own measurement data and your own parameters. But what does very noisy or not too noisy actually mean? How exactly do we measure noise in Noiselab? Well, this is what we're going to answer in this section. I'll be introducing you to two metrics we use in Noiselab to measure noise. Again, let me remind you, in a real system that's using the aggregation surveys and the real API, you don't know how noisy your data is. And this is by design. Noisy is added as a privacy protection mechanism. So in Noiselab, all we do is simulate. So we generate noise that will be similar to the noise you would get in a real summary report. And we also output these noise values for you. OK, so back to our question, how do we measure noise in Noiselab? First of all, what matters when you're using summary reports is not the absolute, the raw noise values. But what really matters instead is how much noise you get relative to the true real measurement data. And recall that ratio, noise to signal ratio, or noise ratio for a short. So signal is the true data, and noise is noise. So let me give you an example. Let's say you have a summary report that looks like this. This summary report here gives you the total preaches value across campaigns, geographies, and product categories. Now let's zoom in on one entry in this report, namely one entry is one bucket and its value. And by the way, maybe you've seen the term bucket and aggregation key, bucket and aggregation key mean the same thing. Now let's assume that this bucket represents the total preaches value for campaign one, geography, US, United States, and product category t-shirts. And let's say that for this campaign, the total preaches value for a t-shirt was about 50. Now this is the summary value. So some of that is actually noise. So the true preaches value is not actually $50. And because this data was simulated with Noiselab, here Noiselab can reveal how much of that is noise. Now let's assume that in that case, the random noise that was added had a value of 10, and the signal in that case would be 40. So for this bucket, the noise to signal ratio is going to be 10 over 40, which is 0, 25%. So really what this ratio is, is simply how big the orange bar is relative to the blue bar, which is the real measurement data. And this calculation here, this noise ratio, is for one entry in my summary report. But what we want is to assess the noise ratio for a whole summary report. So what Noiselab does to do that is simply calculate the noise ratio for each bucket here, and then make an average of these ratios over the whole report. What I've just described here is the simplest way to measure noise. And this noise metric is called average percentage error. That's APE for short. And it's noise to signal as a percentage. And APE is a nice way to measure noise because it's quite easy to reason about. But it also has a few limitations. First of all, if the true summary value for one of the buckets is 0, for example, if I have a bucket with 0 conversions, in that case, APE will be infinity because that's what happens if you divide by 0. And infinity is not a very informative piece of data. And the second issue with APE is that buckets with smaller sizes are going to have a disproportionate impact on the final value of APE. And that's because smaller buckets are more sensitive to noise. So as a result, comparing two APE values for two simulations can be a little bit misleading if you have a lot of small buckets. So APE is not an optimal noise metric. And for this reason, we've added a second metric that helps mitigate these problems. This metric is RMSPET, which stands for Root Means Square-Presentage Error with Threshold, which is a mouthful that I'm going to explain. What RMSPET does is it gives us smaller weights to smaller buckets. And because of this, this metric is going to be more stable and a little bit more reliable than APE when comparing two simulations. Another upside of this metric is that it's useful to data science teams because it's easier to manipulate if you want to do some more advanced calculations on noise. Its drawback, though, is that the formula is a bit more complicated, and it's a little bit less intuitive than the average percentage error. But overall, you can think of RMSPET as also a noise matrix that gives you an idea of your noise to signal ratios. OK, so let me summarize. There are many different ways to measure noise. And in NoseLab, we've picked two noise metrics. The first one is APE, which is simple, but can be unstable. And the second one is RMSPET, which is more complex, but also more stable. So both APE and RMSPET are going to be displayed for all simulations in the output bulk of NoseLab. And these two metrics have different properties. And really, there are just different ways to look at the exact same measurement data and noise data. The one thing we want to remember for now is we don't need to worry too much about the differences between them. But if we're going to be comparing two simulations, one with the other, we need to be looking at the same noise metric. In the next sections, we'll look at practical examples and which metric is more convenient, depending on the situation, is going to become clearer with these practical examples.


Everybody forced here up in 2022 Panini one football hobby 10 box number 13 Check the results that are on your screen anytime of late in sports cards comm results quickly It's at the bottom of the website page and the search bar is a great function to use to looking for results. Let's get started Not gonna recap this one guy is gonna be super quick Let's see One man show do you Andre Swift the Lions out of the 49 One man show Lions out of here. Yeah, Andre Swift Next you got browns Nick Chubb on the Lone Star patch auto eight of ten nice hit for the Browns here I think he's the only I think this is the only card on their checklist Congrats Cleveland Bradley are next up is a precision rookie patch auto jets Breeze Hall There might be the R from New York or it might be a nameplate. Honestly. There you go 14 of 25 jets Awesome Breeze man pull it jet. This guy goes crazy. Oh Here we go Niners formula one dual relic auto Nick Bosa Niners Nick Bosa one of 75 Player worn as well as from 2021 Bond gets it Jets again dual patch rookie auto took 49 Garrett Wilson now Grats again Edward Edward E to Edwards Edward E this one the Jets. There we go Saints crystal lava a 33 of 99 Ricky dual patch auto There you go Saints Sorry for the interruption Joe's been dropping the jets emojis in my chat Joe needs to be checked. Yeah, we'll have to talk to him tomorrow the parking lot before we go get to work I'm right rookie patch auto Trayvon Walker to 99 Jaguars Nice hit Jags a few more. Let's see a QB here. Hey, there's a QB Desmond Ritter day one rookie auto.  That's like adhesive. That's a scuff on the right though There you go. Congrats Falcons Another Trayvon Walker to 49 rookie patch auto for the Jaguars. This is the same one different parallel There you go Jags and we've got a 35 Seahawks Doug Baldwin quad patch auto The 2022 oh dang Doug Baldwin on card quad patch auto Seahawks That's pretty cool, man. He was he was underrated. All right, that'll do for the break Thanks again. Everybody the mixer is coming up next


What they gon' do with me now? I'm still a troke of the town I'm getting the sisters, I'm hoakin' them down Reachin' the smiles and the frowns What's going on? This Brian from Talk of the Town I'm here with my man Introduce yourself, brother What's up? This is White Rose Moxie From Detroit Detroit? Cool That's what I'm about to ask Where you from? I know you're not from New York So Detroit, how you grew up there All your life? Yeah It's funny I was born in Brooklyn Oh, so you were born in New York and then you moved over to Detroit when? Like super early Like I was a kid Do you even remember? No Your parents are together then?  Really my dad and mom listened to a lot of music like a lot Lauryn Heal, Jay-Z All of that Like a whole bunch of stuff though like uh What else? Outcast Like just I remember a whole bunch So not a lot of Detroit stuff? I mean that too But I mean it wasn't heavy into uh rap that I listened to if that makes sense like They a little older so they was on like All of the older stuff like big and all that Yeah that era like Almost like Big Daddy Kane and like Talib Kweli and like Missy Elliott almost like them type That's not even old bro But I see what you say Yeah Oh dude So then what got you started to music? Like how old was you? Like when you really started like saying like alright let me get into this music still Mm Like you mean just like taking it serious? Not really taking it seriously because you know I know like as an artist people continuously like grow and then there's like a pivot where like okay like now I really want to take this seriously you know I want to quit my job and do all things like that so like when you actually just started like saying like hey I like I like listen to music every day like maybe you know rapping at school or something you know freestyling at the table with your boys Oh yeah I started that like when I was like 8 I had this asthma attack put me in a coma for like a couple of days What word?  Um high school High school? How would you know? I'm 20 Oh so you just fresh out of high school? Yeah So cool And then when did you feel like you started getting some some support? Um Probably like 2019 I had dropped this song January 1st called White Goat Yeah Steal my biggest song on Spotify damn near That's the one with 8 million right? Mm-hmm I was tapped in I was heard I feel like I definitely feel like I've heard it before because um So then I guess growing that song do you feel like help you like the blogs promoting it and things like that?  Um I think it was just like we both knew about each other and I just hit them up like yo Oh so they was already tapped in? Yeah and they knew people I knew too like the Detroit like music scene like it's big and small I say like you know what I'm saying like if you know a couple people they probably know somebody else on some shit like that there's just no balls into everybody knowing each other mm-hmm I got you so then growing up then like what were some of your like influences inspirations as far as you know wanting to replicate them but you know uh it was a lot of people like a lot of people I say I just I was more like I was thinking about it I was more the type to just whatever song it was I wanted to know all the words like on some shit like that just repeat it over and over yeah like on some shit like that got you so then any artist like anything particular any artist that like you feel like is your absolute like um mm maybe like I don't know I was listening to a lot of Travis Scott in high school while I was taking it seriously Travis is my favorite artist all the time so I'm not even I feel you 100% he crazy how do you feel about everything that went down mm I don't know you think he's gonna have a big comeback you think he's done I hope he keep dropping music I hope he do he has that utopia coming out but so then you feel like you feel like that's one of your inspirations for real yeah and um I say future Wayne lucky uh yeah so then Jim collaborations any of them I would assume like Lucky Lucky would be crazy he's crazy right now he's going bananas facts and I've been listening to him like for a minute so that would be yeah he's been yeah hella respect to like Lucky I feel like mm I say he wanted the greatest if not the greatest right now rappers I would not I don't know some people say that's a hot take but you know I get what you say mm-hmm like he's tough tough uh my notes right here it says one of your big things is doing things intentionally like doing things with a purpose do you want to like break that down a little bit like how you like how do you see things um like I feel like it's a lot of shit doing just without intention and it don't necessarily be bad but I'm just trying to bring something different I guess like yeah I feel like mm a lot of success like even me like I've done like running gun videos before like for example but what I want to bring like and what I want to offer is more thought out like I want to bring everything like kind of thought out and not to the point where it's like oh my god like you know too crazy but just you know I really feel like we're damn near in a period where art coming back anyway so yeah I definitely see what you're saying and that's crazy that you said that because um the next thing I'm going to say was the waist side music video that you just put up that's the only thing on your Instagram right now so it's like so I tapped in and like I'm not even going to lie like it's one of the most creative like things I've seen just coming out the underground because like you have all the like what you're saying it's not really thought out so like I feel like people are just microwave music you know they're just put it in for like a second it's one minute songs you know what I'm saying yeah but with your music video you had all the references like the 50 reference Kendrick like all that you wanted to speak on the waist side video a little more like what made you think of that because I know you were working with that the director as well oh yeah I mean it was kind of like the same like a nigga walk in the studio and he'll be like make something like like make a little oozy song or like make a Playboy Cardi song like that should be so funny but people really be thinking like that so I just took that to like visuals like you know I feel like a lot of people mimicking sounds even down to me like I done seen bro I done literally seen moxie clones like that should get crazy like in my city in my city like which is crazier like if you really know but like I feel like yeah it's just kind of the way right now you know so I was just taking into the videos so like that it's funny you say that so like coming up then now that you say from your city like you don't feel like maybe you're also the blue prayer for a lot of people you don't want to take that responsibility of like you know like maybe you know obviously don't clone me don't copy what I'm doing but follow my footsteps cause I'm showing you how to do it you don't feel that way yeah no that's real no let me clarify like I mean like look wise like oh they just copy the whole style yes like crazy like you'll be surprised like that do look like you like on some shit like that but I mean as far as the music like I hope people definitely like see what I did like you know what I'm saying learn from it damn near like or get inspired or whatever cause I feel like it's been a minute it's been kind of rocky it's been a minute it's been a lot of shit going on and so you can take whatever from it for real I feel that so it also says right here that you're a you're a big book person who likes to read a lot yeah so talk to me about that like you feel like maybe reading books helps you with your creativity cause like I said like a lot of the songs that I heard like just completely different you know like we have a lot of joke people up here but like you're like you're like an artist artist I feel like you know so like you wanna like does books help you in that sense he said it twice shit I feel like I like reading I always did like I don't even know I just always was on that like even in like as a kid like I'd be reading chapter books like for real like at elementary school like just bringing it to class like I was interested in that shit like I don't know it was the story like telling a story I was always interested in good stories like I could listen to a real good story and you feel like that helps you at all in your music or yeah yeah I'd be absorbing that shit so it's like it'd be with me so then like okay so I didn't grow up reading I grew up just watching TV playing video games you know what I'm saying so it's like it's hard to like especially in this like social media area it's hard to maintain like information in your head do you feel like reading is easier to do then like maybe like watching TV or scrolling to TikTok because it helps your creative mind cause like I like that's different I feel like a TikTok could be like in my head like I could see it damn near again like I could hear that bitch like it's like I want to say it out loud but the reading should be more like when I'm talking to you like I just be talking like smoother I don't know like and if I don't read like I could hear it in the music like I'd be saying it'd be duller I don't know so I talked we talked about this earlier when you walked in but I was watching your real loud performance so that was your first time of rolling loud? no that was my second time but that was the first time I was on the YouTube with the sick got you so how'd that come so I guess how'd that come out the first time the first time shit no it was crazy Tariq just followed me on the gram and the next thing I know for real we was in LA and I had got the email it was like rolling loud in LA do that do the LA one and we was in LA like that shit was crazy and I was like bet and then we did it cool and then we came back for the Miami one and the Miami one was different it was a whole different level even though it was like four or five months it was just like I was ready that time like for real the Miami one was crazy you had all your people on stage with you I swear you feel me like I loved it swear LA one we had just hopped off tour so it was like that type of energy did some whole shit and then we hopped right back on the road and then Miami one was like okay okay everybody here like that was tough you feel like it's helped you a lot you feel like a lot more eyes are on you now um yeah but I don't know it just be it just be normal are you signed or yeah and who you signed to 300 300 okay yeah and how long did you sign I signed to 2020 2020 right pandemic time I want some crazy shit during the pandemic yeah but that's when you feel like maybe it's going crazy I mean like at the time probably you felt like it was going crazy but now 2022 after doing Rolling Loud and all this other stuff you probably you know it's always up yeah it's always up I mean like it wasn't even the same in 2020 get somewhere for real you know what I'm saying like everything all the success I had to my name was not music related like I had white gold but like that was it like you know what I'm saying and that's like SoundCloud era 2 so you know everybody just think everybody a SoundCloud rapper and I mean I love SoundCloud rappers I grew up listening to that shit but it's like damn that's not what I was trying to do with that like that wasn't what I was trying to be on so getting signed after that I had a lot of people that I told like watch this like you know what I'm saying and how's it feel now just to be able to say like look at me now mm I wouldn't say that shit I feel like but at the same time it feel like uh like that's part of the motivation so though like you can make something happen for real I was 100% like I got signed like in the pandemic like probably three weeks after I dropped my project and I feel like I don't know like I was just saying it you know what I mean like I was really saying like I was about to do that like it was crazy I was telling my my uh my homie like in here with me ever since like we met we was saying like alright but deal before I graduate high school and then that shit came just like that so you was always looking at a sign huh you was always looking at a sign mm yes because like at that time that was like how I was thinking and how do you think now like for the people you know for the young artists out there would you tell them the sign or would you tell them to stay independent I say just go with the like go with whatever feel right for you for real like don't be don't be like hard headed and pass up a bag but also know what you want know what you want and know what you need yeah so oh and get a get somebody who good at reading uh paper like the contracts yes yes even if you think you don't got to if it's a contract have somebody at least look over it because you will miss something even if you think you got it swear so I'm I'm gonna flip it right now it also says that I wrote this down O.D. long ago and it says your favorite shows Atlanta oh my god I've been watching that shit crazy recently Atlanta's like one of the I think Atlanta's one of the best shows that they like for anybody doing like a creative route at all bro it just shows the process completely he Donald Glover is crazy like I like a lot of his art and his music but once I seen Atlanta I was like okay and my friends was telling me to watch it they was like watch Atlanta like yes and I was like okay okay okay didn't watch it and then you just was like got to pick it up swear we was literally at the Clark Spillman Morehouse Homecoming and I was like chilling down there and I usually be moving but we was kinda chilling one day eating we was eating some lemon pepper wet wings that's what happened so then after that I was tired sitting back we watching Atlanta and I was like damn this shit crazy I finished it damn near before we left I got to like season three or something and I finished it when I got back still waiting for the next one man still waiting but I like it just because I just feel like it's just so spot on for young artists just trying to get in but it's from like it's also a manager's point of view you know cause he's like he plays as the manager so it's like it shows a little both sides like the business side but also like the rapping side and like it's crazy but how do you feel about the other one the little dicky show have you seen that Dave Dave I've seen a little bit of that I've seen a little bit of that too you don't mess with it no not even like that it's just I got a short attention span but yeah I've seen a little bit of that that's cool too got you definitely I see you I see you dripped out you big on fashion too yes I am heavily into fashion it's crazy you say that the hoodie though it's from my it's from my brother's clothing currency clothing 2K's currency clothing yeah there's some unreleased though coming soon that's crazy I was looking at it earlier what about the pants too not these I don't know what these is but this shirt is no rivals shout out no rival shop so you like some of your fashion influences though would you say tribe that's who you used to grow up in high school yeah I like he cool he cool for show I say I was looking at a lot of I was looking I'm not even gonna lie I was looking at a lot of like ASAP Rocky fits I remember that like specifically like 8th grade I look up like ASAP Rocky fits Tyler the creator fits in the end I don't know in 9th grade 10th grade 11th grade it was like I went to school with a uniform so it was like I couldn't even be on it like that so I feel like I didn't fall out of it but it was like I wasn't into it as much as I was and then but I was still like watching it like what was going on and then now it's like I mean I only like to wear like my favorite stuff now if that makes sense like I wear some pants like all week like I wear them on Monday, Friday, Saturday you on the move though swear so we're gonna play a quick game I like to call cookies you know so we're just gonna say some like your favorites and all that and you're gonna explain why okay but got you so favorite rapper call quickies bro you gotta okay okay okay favorite rapper uh Audemars lot why he's got gray lyrics gray beats got you favorite superhero hmm oh my god static shock static shock yes why he hard he hard best fashion show there was just a fashion show at Michigan State University not too long ago you said fashion show fashion trend oh best fashion trend by the way and it was crazy it's a article out if you check it out but um best fashion trend that's what you said right uh the Kanye fucking glasses oh the shades yeah yeah I messed with this okay worst fashion trend um frickin uh the uh the uh the uh the jersey wave jersey wave yeah like in rap music or yeah I feel like that called for a lot of bad outfits like some people turned it up but like other people I don't know bruh that was a dresser era best era for music 2016 2016 oh my god simply go to best year ever mm-hmm worst era for music I don't know you like it yeah favorite directors um Jordan pew nope get out love him uh I heard like bad things about note but uh favorite video game ooh Minecraft Minecraft yeah why uh soothing like you know relaxing exactly what you mean uh your favorite clothing brand um mm currency clothing two k's got you favorite childhood movie mm either chucky or the polar express probably your favorite food mm ribs barbecue ribs you heard the McRib is back right I don't know I don't know how I feel about the McRib I don't know how I feel about it but barbecue ribs with the bone in them there we go that's right your favorite drink my favorite drink yeah favorite drink Waikia water volcanic Hawaiian that's right I got two more your favorite quote mm I don't know my favorite quote mm it'll be you I got one I got one oh right right that's that's my favorite quote right there that's it that's you yeah alright and you lastly your favorite thing about yourself um show um my sister is this this yeah that's you how old is she she's not that's beautiful yeah let's see what I got last that's what I got so let's just end it off with your album talk to me when is it coming um human behavior comes out December early December um we gonna be dropping a couple more singles before then I got one dropping like a week from today but like probably like middle of November is that true any any features to expect stay tuned and a couple videos dropping soon so yeah I was about to ask are we gonna get a video for the single next week yeah the wayside video is crazy that's going to be hard to top but you know I think you got it coming soon we about to drop a trailer so for real type in so let them know where to find you um find me spotify apple music instagram whatever at white rose moxie


are new here, and I'm sorry about the early hours. I really should apologize for physics 209, which was the previous course. Of course, you might think you should blame that on Martin instead of me, but since I did the course assignments, I'm the one that put him in the eight o'clock slot, so it's really my fault. And I'm sorry about the eight o'clock hour, but it really works out best for the scheduling of GSI assignments and things of that sort to do at the day they have. Professors hate it as much as the students do, I think. All right, so this is the graduate course in quantum mechanics in the physics department, and my name is Robert Liljohn. My office is 449 Berge, which I'll put down here also. And the, I want to say, first of all, that if you received an email from me yesterday, it's because you were already enrolled in the course, and if you read that email, you'll know that it directs you to this website, which is the website for the course. If you did not receive an email from me yesterday, then please copy down that web address and go to that website, make sure that you can access it. Do it right away because it's important to get organized. And read what's on the website because it contains the organizational and logistics information about the course, things like grading policy and stuff like that, which I'm not going to go over in the course, in lecture. So that'll be the place where homework assignments will appear and all this stuff. You really need to know about that if you're going to take the course. Another thing is that I'm maintaining an email mailing list for the course, in which I hope to have all of your email addresses on a big long list, which I use to send you information if there's some emergency or if I'm sick and can't come to lecture or if there's an error in the homework and you didn't know about it, I'll send it to you on that email mailing list. So if you're taking the course, especially for credit, if you're auditing, you can sign up for this email mailing list too. I don't care. But if you're taking the course, you certainly want to be on the email mailing list. Now again, if you received the email from me yesterday, you don't need to do it because you're already on it. But if you did not receive an email, then please send an email to this address, physics221, invigorated.berkeley.edu, and just say, please put me on the email mailing list. You could also use that email address to ask me questions, but don't rely on it too much. I mean, don't think that an hour before the homework is due, you can send me a question and you'll get an answer because I don't necessarily answer that quick. It's better to make use of the office hours and the discussion sections for that purpose. Okay, so please make sure you're on the email mailing list. If you drop the course and don't care anymore about the course, send me an email to that address and say, we're moving and I'll take you off. All right, so that's all I want to say about organizational logistics matters. This course is a, well, it's a two semester course. The first semester of 221A requires as a prerequisite full year of undergraduate quantum mechanics. And if any of you have not had that, I ask you to basically don't have my permission to take the course unless you come see me to get that permission. So please do so if you haven't had a full year of undergraduate quantum mechanics.  I just need to talk to you about things because you make sure you know what you're getting yourself in for. All right, one of the purposes of 221A is to go back over this full year of quantum mechanics that you've had already and to organize it in a more logical and consistent way, paying attention to fundamental principles that maybe you didn't have time for before. And that's part of the purpose. Part of the purpose also is to review the material. We'll also introduce some new material in this semester too. So as part of this program, I want to begin my first lecture with first several lectures in fact on the mathematical methods or mathematical formalism and quantum mechanics. Most of this will be all the mathematics we'll cover in this semester. Later on we'll have to come back and do a little more. But this is most of it in this first week. And the mathematics mainly deals with vector spaces, in particular Hilbert spaces. Hilbert spaces is a particular kind of vector space and you can read the official definition of the Hilbert space in math books. I won't go into that. Let me just say that Hilbert spaces are the spaces of wave functions or let's say this, that the spaces of wave functions which occur in quantum science actually are Hilbert spaces and so we'll call them that. In any case, I want to begin with mathematics of Hilbert spaces. Now to get you oriented in this, I'm going to call on your experience with wave functions and things like that. I assume you know quite a bit already. So to begin with, and to get oriented, let's talk about wave functions and a one-dimensional problem. Let's see, I want to have this center board. This is the center board, yeah, okay. So let's begin by talking about wave functions psi of x and a one-dimensional problem. And you know pretty much what this is. You know in particular that swear of the wave function is the probability density for making new measurements is the position on the particle. Likewise, I hope you know that there's a momentum space wave function by a p and it can be given by what's essentially a Fourier transform of the position or configuration space wave function as it's called at the top, e to the minus i p x over h bar psi of x. And this also has a square of five p absolute value of square which represents physically the probability density in momentum space for making new measurements of momentum. Now, so there's only two wave functions psi of x and five p and they're related to one another by invertible transformations. That's really a Fourier transform. Now, let's suppose in addition that we've got some energy eigenfunctions, let's say we call them u in of x is equal to e in of u in of x for some problem. And let's suppose that they're discreeting on the generated eigenfunctions for simplicity. Then you know you can take an arbitrary wave function psi of x and you write it as a linear combination with expansion coefficients I'll call c in times the energy eigenfunctions. And you know that the expansion coefficients are given by an integral over x which is u in of x complex conjugated times psi of x. These are linear transformations that convert you back and forth between the psi and the coefficient c's. Well, these coefficient c's form a sequence that's called c1, c2, dot, dot, dot like this. And this sequence of coefficients can be thought of as quote unquote the wave function in energy space. Position space, momentum space, energy space. There are three different types of wave functions. Moreover, the square of these coefficients c in the square and I hope you know this is interpreted as the probability of making a measurement, when you make a measurement of energy it's the probability of getting the eigenvalue e n which is associated with that coefficient. So these are, what we have here actually are, these can be thought of as three different wave functions which describe the same physical system, the same state of the physical system. And from a certain point of view these are all equivalent descriptions. This is certainly true from the mathematical point of view because everything that one, all the questions that one wants to ask in the physical sense about a quantum system and ultimately boils down to probabilities of making measurements, all of these calculations can be done in any one of these three wave functions or as we say representations, position, momentum and energy representation of the wave function. And so from this strictly mathematical standpoint there's no reason to think that any one of these wave functions is privileged or special over any of the others. Now there's a psychological bias and think that the psi of x wave function is the privileged one and that it's better than the other somehow or that it's a starting point. Why is this so? It's partly because we're used to thinking about wave functions in physical space or functions scalar fields, field, scalar, vector fields in physical space, electric fields, pressures, pressure of fluids and things of that sort, things that we can measure in physical space. And so why isn't psi like that? Well, when Schrodinger was starting to write down the Schrodinger equation that's how we thought of psi as physical field in physical space. However, it's pretty clear that that's actually an interpretation that doesn't hold up because physical space is not the same as configuration space. Configuration space is basically x space. Why? Not in general, it's not. Why? Because you may have a wave function for two particles, in which case the wave function depends on the positions of both particles. So it's not a wave function in physical space, it's really a wave function on two copies of physical space. Configuration space you see is not the same as physical space. So that's one difference. Another difference is that if you get down to actually thinking about measuring a wave function psi of x, let's say for a single particle, let's go back to that, say that, just a single particle on physical space, think about measuring that. We'll talk about that later in the course. You'll find out the measurement process is really very different from the kind of measurements you would make of an electric field or a pressure and a fluid where you can just go to a point and measure it, measure the number. It's not so simple with psi. Certainly it's not hard to measure the square of psi in principle it's not hard because it's a probability density, make repeated measurements and just count how many particles lie in a small interval like you do for probability. The finding the phase of the wave function is a different matter, it's much more subtle. There's all kinds of subtleties that don't occur in the measurements of classical fields. So for many reasons, it's best not to think of psi of x as being the final physical space but rather on this x space or configuration space. All right. Now, so with that background in mind then, it's desirable in developing the formalism of quantum mechanics to use a notation which does not prejudice the choice amongst these observables. By the way, clearly these wave functions, clearly these observables are associated with clearly these wave functions and three wave functions are associated with three different observables, position, momentum, energy in this case and one can come up with other observables, angular momentum and anything else you wanna measure and each one of them has its own wave function. And these are all equivalent to those equivalent descriptions of the same reality, the same quantum reality. So in view of this democracy or covariance of quantum mechanics under changes of what we call representation, position, momentum, energy representation and others, this is a, these covariance principles were worked out in the very early days of quantum mechanics by notably von Neumann but others, Dirac and so on. In view of this covariance amongst these different descriptions, it's desirable to have a notation for describing the states of systems that doesn't prejudice the choice. And so that's what the Dirac notation does and that's what I want to talk about now. So in the Dirac notation, we speak of kets or also called ket vectors because they are vectors and a ket is denoted by this kind of notation like this and it stands, supposed to stand for any one of those wave functions up there. And it's customary to insert into a ket vector an identifying symbol such as psi, I'll put a psi in there. It doesn't mean that it stands for psi of x, it really stands for all of those wave functions. But it's an identifying symbol of the ket. And the ket vector belongs to a vector space which I'll denoted by a script E like this. It's a complex vector space which is called the Hilbert space. And as I say, I won't bother you with the official mathematical definition of the Hilbert space, although there is one, but rather just let's make a remark that Hilbert spaces are in quantum mechanics so the spaces of wave functions and that'll be enough for us. That's how we'll use the terminology. Anyway, it's a ket vector in this space. Now then, just to say a few words about the physical possible and to quantum mechanics, we'll get into this in more detail in about a week. Physical possible and to quantum mechanics assert in the formulation I'll present in this course they assert that by corresponding to every physical system there is a vector space, in fact a complex vector space and Hilbert space which is associated with the system and it gives ways of knowing what the Hilbert space is based on the kind of measurements you can make on the system. We'll go into this in more detail later on but in particular there's physical principles that tell you what the dimensionality of space it should be. You'll find for example for spin systems the space is finite dimensional and for systems that include position variable it's an infinite dimensional. In any case, Hilbert spaces may be infinite dimensional. To say that this is a complex vector space means that mathematically speaking you can form linear combinations of vectors with complex coefficients. That's what a complex vector space means. A real vector space means you're restricted to real coefficients when you make linear combinations. So in particular it means that it's meaningful to talk about linear combinations of ket vectors like this, psi with C1, psi with C2, psi 2. This is supposed to also belong to the Norbert space E where C1 and C2 are complex numbers all right this by saying that it's belong to the complexes using this mathematical symbol for the complex numbers. All right, now crudely speaking this is similar to the idea of linear superposition of electric fields in physical space. But remember the wave function really isn't in physical space so this is something else but it is certainly linear superposition of vectors. Moreover, there's complex coefficients and you might ask why do we need complex numbers? Why can't we do it with just real numbers? We'll see later on when we study spin system that it's impossible to do it. You can't satisfy the physical postulates of quantum mechanics with only real numbers. Okay, now people that study chemistry sometimes take courses in chemistry and they lose this because of most of the problems in chemistry involved in real wave functions so they have real coefficients but actually they have to be complex, all right. All right, so this is the beginnings of the ket notation and the ket vectors. By the way, the physical postulates that I'll say you'll have to speak about later on describe how the physical state of the system can be associated with a given ket vector. I'll just say one thing about that now however which is that the physical state is actually not associated with a ket vector but rather it's associated with what you call array. It's associated with array in the older space E. What an array is is it's not just the same as a vector if you make the origin in the vector space call it O and I draw a vector here like this is our vector ket vector psi. What the array is is it's a set of all vectors that can be contained by multiplying psi by any complex number. You can think of it as being all the vectors that lie in a one-dimensional subspace with it. The magnitude is scaled, also the phase factors get scaled. Array is a ket vector to it. It's a ket vector, let's put it this way. The state is a ket vector modulity overall normalization phase which doesn't have any physical significance. So the physical state is independent of normalization phase. So in any case the physical state corresponds to a whole array. One-dimensional subspace of the vector space. All right so that's a little bit about ket factors. Now you're no doubt roughly familiar with the idea that ket like psi or somehow supposed to correspond to wave functions psi of x that's the idea. Before I go into that I want to say that the program I'm trying to follow here is to start with the physical postulates and quantum mechanics which lead to ket vectors. We won't talk about wave functions. In effect what I'm gonna do is develop mathematics and ket spaces and then later derive them and derive wave functions from the physical postulates. So I will talk about wave functions at first except for purposes of illustration as I did above up here. As you know about wave functions already. Nevertheless behind the scenes we're thinking of ket spaces as wave function spaces. And I'm sure you know that ket is supposed to correspond somehow to wave functions. And I suppose you also know that bras that look like this are supposed to correspond to the complex conjugates of wave functions. Now so I want to introduce a subject of bras now. The question here is how do we, if I don't talk about wave functions how can I talk about a bra which is supposed to be complex conjugates? The answer is the following. As we say that a bra or a bra vector because a bra also is a member of a vector space. Just first of all is the notation. It's given in rack notation like this. You're the kind of a reverse kind of a reverse of a ket. And as in the case of a ket it's convenient to put in an identifying symbol for the bra just to identify it and label it, distinguish it from other bras. And what is a, what is a bra vector? It is a bra vector is a map described this way. This is it's a linear map as a matter of fact. It takes us from the ket space over to the complex numbers. And what that means is that if I have a bra which I call alpha like this it's something that can act on cats like this. And what does it produce? It produces a complex number. And you can write it like this way if you want. But we usually don't write it that way because it's too many parentheses. We usually just drop the parentheses and write it this way as alpha psi. So this is just the value of the linear map alpha acting on the ket vector psi. And then I guess I did do this wrong because I was supposed to do it another way. All right, let's do it this way. It's pretty easy to see that the bras is linear operators. And by the way, you're used to linear operators like Hamiltonian and the momentum and so on operators in quantum mechanics. Bras are not operators like that. Those usual operators like Hamiltonians in quantum mechanics are operators that act on the cats and what they produce is another cat. They map cats and cats into cats. A bras is a linear operator that's cats in the complex numbers. It's a complex value linear operator. It's kind of a simpler linear operator, actually. Now, bras vectors form their own vector space. You take the set of all bras like this. It's formed in its own vector space. It's easy to see how to multiply bras by scalars. You just multiply the answer by a scalar. And it's easy to see how to take linear combinations of bras. You just add up the answers for the different bras. I'll be explicit about this. C1 alpha 1 plus C2 alpha 2 is a linear combination of bras. And if we allow this to add up the cat, the answer is C1 alpha 1 plus C2 alpha 2 sign. And so this is, I'm just saying the parentheses is another bra. So, indeed, they form a vector space. And the set of bras like this is sometimes called the dual space to the original cat space. And it's done this way with an E star on a star indicating it's a space of bras instead of a space. If you're doing wave functions, you would say, oh, this is a complex conjugate wave function, just a wave function, so it belongs to the same space as the original space of wave functions. That's a simple point of view. And in fact, it turns out it moves deeper and more convenient, ultimately, to regard this as a separate space, which we call the dual space. This still doesn't tell us what's the analog of the complex conjugation operation. How do we complex conjugate a cat? Well, we don't complex conjugate a cat, but you do something similar in work like this. It relies on the fact that the cat space is, as to say, no work spaces, possesses a scalar in product, a way of taking scalar products as vectors. And this is related to the metric, which I'll tell you about now. The metric is a function, which I'll call g, and what it does is it takes two copies of the cat space and maps it into the complex numbers. This is map notation. If you're not used to it, what it means simply is that g acts on two cats, which maybe I'll call psi and phi, and what it produces is a complex number, like this, x of the pair of vectors. And this metric, g, is supposed to have certain properties, which I'll try to list here in case I'm going to go around a space. Properties of the first property is there's really two properties at first. One is that, call it 1a, is that g is linear, and the second opera image I'm calling phi there. So what that means is that if you let g act on psi, comma, phi, this is quite linear combination, c1, phi1 plus c2, phi2, if you let it act on a linear combination in the second slot, replace that by a linear combination, then you can spend it out just by linearity. So this becomes c1, g acting on psi, comma, phi1 plus c2, g acting on psi, comma, phi2, like this, that's the linearity. 1v is the second property, which is that it's anti-linear in the first operand. And I'll tell you now what anti-linear means is that if I replace the first operand by a linear combination less than only the second one alone, then we have g of, let's say, c1 psi1 plus c2 psi2, comma, phi. What is that equal to? Well, it's equal to c1 complex conjugate times g acting on psi 1, comma, phi plus c2 complex conjugate g acting on psi 2, comma, phi, like that. The term anti-linear, which we'll encounter once in a while in this course, means that when you apply something to a linear combination of what we have expansion with complex numbers as coefficients, then when you expand it out, you have to take the complex conjugates of the coefficients. Anyway, g is linear in the second operand of any linear in the first. So those are two properties of it. Let me try to see if I can put the other properties over here. There is a second property, which is called the symmetry property of g. And the symmetry property says that if I take the g of psi phi, a pair of cats, the answer is g of phi psi in the reverse order, which is a complex number, but I have to complex conjugate. This is the symmetry property. And the third property is called the positive definite property. And it says this, is that if I take g and let it act on two cats, which are identical, the same cat in both slots, well, first of all, before I write anything out, notice that by the property number 2, the answer has to be real, because it's equal to its own complex conjugate. So this is a real number. And in fact, it turns out it's always a positive number, so it's equal to 0. This is for all psi, for all states psi in the Hilbert space. And it equals 0 if and only if the same psi is equal to 0. It vanishes the g value of its metric, but the g with the same vector inserted in both slots is equal to 0 if and only if the vector is 0. All right, so those are the three properties of the metric. The metric in a real vector space is a way of doing dot products of vectors. And in particular, the metric to take the scalar product of the vector with itself, you get the square of the length. That's how it is in a real vector space with the Euclidean metric, which satisfied the real n log of these three properties. So roughly speaking, this g of psi psi here, which is a positive number, is interpreted also in quantum mechanics as the length of the cat's psi. Length, quote unquote, it's a geometrical analogy or metaphor, I think we've got the geometry of these complex spaces. But in fact, we're going to develop this cat formalism, and then later we'll derive wave functions from it, not treating wave functions as a starting point. Yes, all right. Now, next I need to tell you about something which in doctorized focus is called the dual correspondence. Actually, it's a fancy language for commission conjugation, which you know about already, but I'll tell you how it works. The idea is this, let's consider the metric acting on two cats. This is, like I say, in the interpretation of the scalar product of the two cats. Let's consider this metric acting on two cats. And let's regard the first one as being fixed, and the second one as being variable. So then this can be regarded as an effective function of just a single cat and a five cat. And moreover, it's linear in a five cat because that's the second operand, and it is a complex value. So this is a complex value linear operator acting on cat's psi. And so by our definition of a bra, it's actually a bra vector, or a bra acting on five. So this is some bra acting on five by the definition of a bra. And what is the bra? Well, the bra is it is a linear operator, but clearly it's linear operator. It's associated with the psi here that we fixed. So what we do is we write this this way. This is definition now. We write this as a definition of psi pi. But let's just say this is the definition of bra psi. We started given a cat's psi. And this is associated with a bra psi. So what this formula does is it defines a bra acting mapping that takes us from a cat's psi. This is called what soccer I call this is the dual correspondence. It takes us over to a bra psi. The definition of a dual correspondence is this formula. But in fact, after a while, we'll stop using a G notation when we end up just writing it this way. It means the same thing. All right. Instead of using DC, there's a more common notation, which is to use the dagger notation. That's to say the answer, which is the bra here. And we'll say that this is the cat's psi with a dagger on it. The dagger is submission conjugation, which is here being defined in its action on ket vectors. And what it does is it converts ket vectors into bra vectors, which are defined by the formula top. And this dual correspondence, if you like, you see is a mapping that takes you from the cat's space over to the dual space, the star. Did I mention that before that the dual space, each star is the space of bras. So naturally, the cat's the bras. If you're working on a space of finite dimensions and you have these postulates of the metric G, it's easy to show that the dual correspondence is invertible. The dimension of the bras space is the same as that of the ket space. And moreover, every bra, which is created in this bra, in fact, is the image of some ket under this dagger operation. So the cats and the bras are placed into one-to-one correspondence. And therefore, it has an inverse so that you can do a DC inverse notation from bras back to ket. And this inverse is traditionally denoted also by the same dagger notation. So what we get is this, as we can say that if I have a ket side and there's a bra side, and I get by applying the inverse of the dual correspondence, I just use the same dagger notation for it. This is really the same operator. It's really the inverse of it. But the notation doesn't cause me trouble because if you square it, you get the identity. You see it's a consequence of this that if I take side and dagger in twice, I'm just going to get the side dagger in. Well, in effect, we've defined the dagger notation now of both kets and bras, and it just amounts to switching one back and forth between the other. The dual correspondence is anti-linear.  This follows just from the fact that if the g operator is anti-linear in the first operand, that's what makes it a linear operation of the dual correspondence. The consequence of this structure that I've developed so far is the Schwarz inequality, which I'll tell you about now. The Schwarz inequality is important in quantum mechanics because it leads to the Heisenberg uncertainty relations, as we'll see a little later. If you apply the Schwarz inequality on a real vector space, you'll find that it's equivalent to the statement that the shortest distance between two points is a straight line. And in some sense, even in complex vector spaces, you can think of that as being the geometrical meaning of Schwarz inequality. But in the case, here's how it's stated. The Schwarz inequality says that if you take the scalar product of a phi with a psi that's a complex number, and take its absolute value squared, this is less than or equal to the scalar product of phi with itself times psi with itself. Both those numbers in the right-hand side are real and positive. So their product is real and positive. And the statement is that the scalar product is less than the product of these squares of the two vectors. We can think of psi psi as being the square of the vector psi real and non-negative number. So I won't use the Schwarz inequality yet, but at least let me prove it. It works like this. Oh, and by the way, there's an addition to this, too, which is that we get equals, instead of an inequality, it turns into an equal sign. We look at only if the kets psi and phi are linearly dependent. So that's the Schwarz inequality. The proof which I'll outline quickly on something like this is let's define a ket alpha, which is equal to psi, plus a complex number lambda times ket phi. We'll take a linear combination of the two kets, in other words. And we form the scalar product of alpha with itself, which, as you can see, is going to be psi scalar product of psi, plus lambda times psi scalar product of phi, plus lambda complex conjugated times phi scalar product of psi, plus the absolute value of lambda squared times phi scalar product with phi itself. Like this. And because this is the scalar product of a ket with itself, the result has to be greater than or equal to 0 by postulate number 2. In fact, I can't read it here. It's number 3, isn't it? Number 3, yes. Postulate number 3, this is greater than or equal to 0. Now, lambda is any complex number here. And it turns out, if you play with this for a while, that an interesting choice is to let lambda be equal to minus the scalar product of phi with psi divided by the scalar product of phi with itself. And I'll let you do the algebra. If you plug this in to these four terms here, what you find is that the last three terms are all equal except the middle two ones have got a minus sign on them. So it goes minus 1, minus 1, plus 1. And the result is this, is that alpha scalar product of alpha is equal to scalar product of psi with itself, minus the absolute value of the y-scalar product of psi squared divided by y-scalar product of phi. This is greater than or equal to 0. And just by multiplying through by this positive number here, you can see you get the short, second quality of the first form. Because a little extra work to do to show the second form that you get an equal sign if and only if you're linearly dependent on what you refer to the notes to see how that works out. So that's the result that we put here for this. It's something we can derive in this point. Now, next, let me move on and say something about linear operators. And I would like to call this, I would even access this back board. I would get both of the other. That's the reason I can't use that third board. I'm not supposed to be able to raise two of them at once, but apparently you have one of those in a very close up. OK, well, I'll just have to alternate back and forth. Next, I want to say something about linear operators in farm mechanics. Here, I'm referring to the kinds of things like energy momentum and Hamiltonian which we're familiar with. So the linear operator, as I'm mapping, it takes us from the cat space to itself. And it's supposed to be linear so that it has the obvious distributed law over linear combinations of cat vectors. It's also possible to talk about anti-linear operators where you need to take a complex conjugation of complex conjugative coefficients when you have linear combinations. I'm not going to say much about anti-linear operators now because most of the operators we deal with are linear. There's one important anti-linear operator in farm mechanics, which is time reversal. And we'll cover that later in the course. But don't worry about the anti-linear operators when we get to them. And so for now, let's just talk about linear operators which covers most of the cases, Hamiltonians, et cetera, or linear operators. Now, one of the things to say about linear operators, if you've got one, is that linear operator acts on cats. That's where they begin life. Can we also talk about linear operators acting on broths? Is this possible? I mean, if we do it, it isn't really the same operator. But I'll call it that anyway. Is it, can we define this? And the answer is yes, it works like this. So the idea is that if I have L acting on the side, and this is defined because L is a linear operator acting on the cats. And what I'd like to do now is to say, what is this? Am I let the linear operator act on a broth? What's the meaning of that? Well, if the linear operator acts on a broth, it's supposed to produce another broth. So this thing, psi L, is a broth. And therefore, it's a linear map that takes cats into complex numbers. And so therefore, I should be able to let this thing act on a cat by, and I should get a complex number. And in fact, I can define this broth by just saying what the complex number is and verify that it's in fact linear. That's the things I need to do. Well, here's a simple answer. The complex number is going to be this. It's going to be the broth psi acting on the cat L5. Because remember, we know what L does to cats. So this equation, in effect, becomes the definition of psi L. So I'll make this three little triple equal sign here because that means definition. Here it doesn't mean that here. So this is a definition of psi acting on psi L, or no acting on psi. As we say, it acts from the right onto a broth. And here it acts from the left onto a cat. Now, one of the consequences of this definition is that parentheses don't matter. So it's customary to drop them. And just to write this this way as psi L5. But lying behind me, this is the understanding that you can let L act either to the right or to the left. And it won't make any difference in the answer. But this is what you call a matrix element of the operator L, linear operator L. Tell you now about the outer product. Let's suppose we're given two cats which I'll call alpha and beta. You don't have to use Greek in these cats, but I have Greek labels that I'm doing that mostly here. Suppose I'm given two cats alpha and beta. Then I want to define a linear operator, which is called the outer product of alpha and beta, which is written this way, as alpha beta. It looks like a cat juxtaposed with a broth. So this is a linear operator. It's supposed to be a linear operator. And it's called the outer product. Well, I already said that, so let's say it again. It's called the outer product of cats alpha and beta. And how do we define it? Well, it's a linear operator. It means we can muddle an act on any cat. And we should get another cat. And what will the answer be? It will be this. It will be cat alpha times the same product of beta with psi. In other words, you can say the definition of the outer product amounts to just reading these three symbols, juxtaposed, with a different ordering of parentheses. You do it like this, or you do it like this. On the right-hand side, you see this now becomes a complex number of multiplying times a cat. On the left-hand side, it's an operator acting with psi. This defines an operator. It's a definition of it. So that's the outer product. The other product is essentially the same as what people call the tensor product or dyadic product if you ever study tensor analysis. That's what they call it. Now I want to say a little bit about basis vectors. You know from linear algebra that a basis is a set of vectors that expand the space. And that means that they're, well, they expand the space and they're also linearly independent. So they're linearly independent vectors that expand the space. So in a cat space, let's suppose it was implicit if we talk about a discrete basis. So we've got a set of 10 vectors I'll call here. And we're going to index by 1, 2, 3, something like this. And the number of such vectors is the dimensionality of the space, which could be infinite because cat spaces are oftentimes infinite dimensional. The basis may or may not be orthonormal. Basis don't have to be orthonormal. Usually they're not always. In quantum mechanics, we use orthonormal basis. So we said that basis is orthonormal, as I'm sure you know, that the scalar product of a pair of basis vectors is a product or delta like this. They're the most useful basis in quantum mechanics. So let's talk about those orthonormal basis. Then if I have an arbitrary cat, or as we say, state, psi, because this is a basis, that means I can expand this as a linear combination with some coefficients of the basis cats. And by orthonormality relation, the problems immediately, its expansion coefficients are just given by the scalar product of the basis cats with the original vector psi like this. I'm sure you're familiar with this sort of thing. Now in this expansion, I've got a complex number multiplied times a cat. When we write that, we can also put the complex number in the other side. It doesn't matter which side you put the complex number on when you multiply a complex number times a cat. But I'll put it over here on the other side because now what I want to do is take the expansion coefficient and plug it in here. And if you do that, then you get the sum of n of n cat n bra psi like this. So red one way, this is cat n times a complex number. But we can read it another way, which is outer product of n n acting on cat psi like this. In fact, let me move the parentheses like this because psi itself doesn't depend on n. And so what you see is that there's an operator which is in these parentheses here. And when it acts on an arbitrary cat, it reproduces the same cat all over again. Therefore, that operator must be the identity. So we say, we write it this way, is that the sum of the outer product of the basis cats with themselves is equal to 1, where 1 stands for the identity operator. It's common in quantum mechanics to write the identity operator as just 1. And this is called resolution of the identity, which is associated with the basis here, assumed to be a discrete basis. Later on, we'll generalize this in the case of continuous basis. But this is what it looks like. So this is an example of where the outer product appears in the resolution of the identity. Now, another topic. I've previously defined the dagger operation on cats, which maps them in the bras, and the dagger operation on bras, which maps them in the cats. Dagger operation is permission and conjugation. It's convenient to extend the definition of the dagger operation in complex numbers, in which case it's just interpreted as complex, same thing as complex conjugation. Now I want to extend this even further to define the complex, define the permission conjugate of an operator. Here's the idea. If I have a linear operator a, this is a linear mapping of the cat space into itself, I want to define the permission conjugate of a, which will be another linear operator on the same cat space. And the question is how to define this. And the answer is the following. It is that we say a dagger acting on psi equal to the first taken cat psi converted into a bra using the dual correspondence. You then let the operator a act on that. We know how linear operators act on bras. And then we take the whole thing and we finish with conjugated again. This is the definition of a dagger. Now, the question that arises is that if a is linear, is a dagger also linear? And the answer is yes. Because you see, when I first take psi and then I convert it into a bra by the dual correspondence, that's actually any linear operation. And then I'm going to let a act on the bra. But then our mission conjugated a second time, which is another any linear operation. And so the result turns into a linear operator. So a dagger is linear, linear operator. All right, so this is the definition of a dagger. There's a whole bunch of simple consequences of this definition, which I'll just do this down here. Let's see. So one of them is, is that if I take a and dagger it twice, I get a back all over again. That certainly implies in these other instances of the mission conjugate operation. Another property is that if you take a product of two operators and you dagger them, you get the dagger operators in reverse order. You have to reverse it. The third property is that if I take a matrix element of an operator a between two states, psi and phi, this is equal to, well, perhaps I'll write this one. Let's take the complex conjugate of that matrix element. This is equal to the original matrix element that's read in the reverse order where everything is daggered as you move through it. So we start with the right, I had phi, we dagger it, turn it into a bra. The operator a gets turned into a dagger. And that's psi gets turned into, excuse me, bra's like it's turned into that just reverse order. Here's another consequence of it. If I take the outer product of two cats, alpha and beta, and I dagger that, this is a linear operator, so I'll dagger it. What this turns into is the outer product of two cats in the reverse order. These are all easy consequences of the definition. Now, one result of these definitions is a general rule in quantum mechanics whenever you're dealing with cats, bras, linear operators, complex numbers all multiply together. The kind of multiplication can be ordinary multiplication. It can be operator's acting either to the right or the left or it can be the tensor product of two cats. All it has to be is just a meaningful expression. The rule is that if you want to take your mission conjugate of that, you read the expression backwards and you apply dagger to each item separately. Complex numbers go into their complex conjugate and so on. In fact, there's a couple of examples of that rule right here. You see this has been read backwards. This stars complex conjugations of the whole result, which is the complex number of the matrix element. Here, you're reading it backwards and daggering things. Reading it backwards and daggering. The rule applies quite generally. This is all part of the neatness or the advantage of the direct notation for calculations of quantum mechanics is that it makes all these operations quite natural in the notation. Now, as I'm sure you've done a way to make another definition, I'm sure you know that permission operators A is one which is equal to its own permission conjugate, definition of a permission operator. Permission operators are related to operators what are called self-adjoint operators.  But for practical applications and quantum mechanics, we can consider self-adjoint and permission as being essentially synonymous. If this worries you about it, you need to go study it. Of course, some functional analysis, I'll just put it that way. Actually, in finite dimensions, in a finite dimensional space, two-term submission and self-adjoint are equivalent. The only issue arises in infinite dimensional spaces. And I don't really want to get into the mathematics of technicalities of infinite dimensional spaces very much in this course. But in any case, for most practical purposes, we can regard these two terms as synonymous. Now, if you have permission operator, permission operators, of course, I'm sure you know this, are particularly important because they represent physical observables who gradually say more about why this is true as we go on. Right now, let me just say one more thing here. Yeah, let me just say one more thing here. Because if we say that a permission operator is positive definite, if we write this out, a permission operator is said to be positive definite if you have this following property, is that if you form the matrix element of your operator A to the same state on both sides, the answer is always non-negative. And furthermore, if you get equals to 0, if and only if the same side vanishes. This is a definition of positive definite. A positive definite operator, from another point of view, is an operator whose eigenvalues are all positive. One can also speak of a non-negative definite operator, and that's one whose eigenvalues may include 0s, but they don't include any negative numbers. And I haven't said anything about eigenvalues yet, but I will, I guess, in the next lecture, we'll talk about the eigenvalues and the spectrum of operators and things of that sort. OK, that's all for today.


Television is something we just take for granted, but what's behind it and how did we get to where we are today? In June 2006, an ITU conference of more than 1,000 delegates took place here in Geneva. That conference was to transform the world of broadcasting. It opened the doors to a new age of television. By the end of the conference, 104 nations of Europe, Africa and the Middle East had agreed a plan to introduce digital television. The target was to complete the task in just nine years. Deadline 1 a.m. 17th of June 2015. Although it would be a massive undertaking, it would mean more quality, more choice and more flexibility for viewers. It would liberate television. Before that, a plan was needed to agree what digital television transmitters could go where. That plan came out of the conference. Television transmitters illuminate a certain area of land, like a light beacon. In the area they light up, viewers can watch the TV services. Planning television services is like creating a giant jigsaw puzzle of transmitter coverage areas. They had to try to get for every nation the digital channels, the transmitters and the coverage areas they would like or come as close as possible. And at the same time, they needed to make sure that broadcasts from each transmitter didn't interfere with viewers in other countries. When the conference started, the participating nations made over 70,000 requests for transmitters, channels and coverage areas. Imagine being asked to create a jigsaw puzzle with over 70,000 pieces where each piece has to fit and respect a list of rules. It took enormous computer power to create the plan and as far as possible give the nations what they wanted. It was achieved by harnessing sophisticated computer software developed in the EBU together with the computer hardware power of the CERN. The success of the conference provided the impetus for the transition to digital television that was to happen all over the world. The advantage of having digital TV, it is your own TV. You can't control it by yourself, you can't put your content and nobody can control your transmission. Also transmit something much cheaper. For example, if you take a World Cup or something like that, it is a benefit for the UEE viewer. As we have digital broadcasting in Tanzania now, we have the free-to-have services that they used to get in the past including pay TV services. So we have more channels and more choice. In front of it, it was important both for broadcasting because it was enabling to introduce new services. Also because it was enabling to get the digital dividend and having the 800mg available for the mobile service. The digital dividend, in this case the freeing up of the spectrum will enable the national broadband strategy target of having 30% of the population broadband by 2017. It's important that both the broadcast services and the mobile service involve harmoniously. That's why we think that other nations can benefit from the observation of the problems we are facing throughout the process and the solutions we adopted. With the help of ITU BR, we're going to continue to assist those countries who are not going to achieve on June 15 by being sure that they will not be affected by the other countries who already met the deadline. I would like to congratulate the administrations which have successfully switched to digital television broadcasting. As for the others, they can be assured of the continued support of the ITU to carry out this process. I wish all countries a successful transition to digital television broadcasting. Because of the flexibility that digital offers, television has come a very long way from a small and simple TV set in the corner to multiple large screens with high definition images. And digital technology also allows TVs to connect up to the internet to create a world where television networks and internet networks understand and cooperate with each other. And tomorrow may bring even further steps to reality the new worlds of ultra-high definition television and immersive audio. What will all of these achievements mean for the way we live our lives? Well, whatever they are, they're all the legacy of their ITU conference in 2006.


How about the reaction that you guys played in the majors? You hear the cheers? You hear the booze fans have a right to what they want to do with? We're a little shocked that it was that extreme that the national fans, now Philly fans, they get a bad rap, but the national fans came on strong. Well, I was definitely took back by that. You didn't even hear a golf clap out there for him. It was no mixture of any kind of sincerity for him. I was shocked. He's a rookie of the year, six-time all-star MVP. He made that team nationally relevant. They made multiple playoff appearances and he played hard for that organization. So I was a little took him back with the energy level and shocked that they just collectively all booed at him for that long, that many reps. And it's going to be like that all season long. And he's a calmer trader. Listen, 300 million is a tough, tough to turn down when it's on the table like that.  Come on, Dunstrell. The guy's in his prime. He's a franchise player. He takes a contract, basically an hour up the road. You know, they get a lot from Bryce Harbour. Bryce Harbour did a lot for them. But I expected the booze at night because he's in his prime. He left with us on the court.  Everybody in that stadium will go an hour up the road if they gave 300 million money in that kind of money. Exactly. It's called chips with dip. So that's a tough situation. And he also saw the way that Phillies were going in the direction that. So maybe he just wanted to change. So it's nothing personal, but I really thought they would honor him a little bit better than that. But hey, it was very well- They gave a video, Montage. I need a little more in the video. Well, I mean, I've got it. They gave me a video. I ain't won no money, Pete. It's nothing personal, just business and $330 million worth of business. Is he a guy who will put the Phillies over the top? They contended last year. And we know they have other issues. But can he be that big of a difference-maker because the Phillies just don't want to hang around? With that kind of money, they want better results. He's never won a playoff series, pressure. I think he could be the guy because he takes pressure off the supporting cast. And you're already seeing the dividends of that. Everyone's taking their shots. You're seeing Franco, Real Muto, Hoskins, very comfortable in the lineup and it forces teams to pitch to the collective mass. And they don't understand that situation. You saw in screen training, they swung the bat well. They hit a lot of home runs. So I think with the ad of him taking the pressure off the rest of the team, he's going to be the focal point of booze everywhere because of the contract. So it's not just going to be Washington yet. People are going to give their energy to him. I think he's going to be just fine. I think he's going to definitely get them to the postseason and go deep in the run. Well, I just think he made the right decision because Philly's building around him. He took a, I would say, superstar-friendly contract for Philly. They got McCutcheon. They got Segura. They got the guys that you mentioned. They got a lineup that's very long length. They got Franco hit on the eight hole. This team's going to be there all year long. It all comes out of pitching for the Philadelphia Phillies. Tied for the most ever by a starting staff in a season's first five games, but now you can tack on another one. That's a dozen with the sales struggling here. Yeah, I'm a little concerned. I agree a thousand percent they heard. It's one thing for him to be getting beat, throwing 95, 97 miles an hour. He's throwing 88 to 89. That's very concerning of his velocity. It was showing fatigue. He didn't really have a deep spring training to get feel and really get stretched out. So I'm definitely individually concerned about Chris Hale and just how he feels moving forward in the season. I've watched him pitch for years. 89 is not going to get it done for him because that 94, 95 mile per hour fastball with that electric breaking ball makes him almost unhittable. So something's going on with him right now. He could be a little burned out in an extra month and a half pitching in the playoffs and the World Series, but he'll be back. He's got his pros. Last year he was up around 95. So you think, what we call a World Series hangover, but you pitch those extra weeks, those extra innings that does have an effect when you start the next season. Absolutely, yeah. When we won the World Series, I only had a month off until I had to get back to working out back on the mound after December, January goes like that. And then you're back in the spring training and he didn't get a full spring training. So I'm a little concerned, but the velocity needs to come back. You can get beat all over the field, but they're going to look at this tape and be worried about his velocity being down so far. Yeah, so not just for sale, but you got Rodriguez, Porcello, we saw Price the previous night. Avaldi is that wearing on all of them, but you're not concerned long term. That's a good staff. That's a good staff and the velocity has been fine except sale. He's the one, he's the callus. They all have to play off sale because he's the one guy in that rotation that can go up against anybody's ace and win that ballgame for you, which takes pressure off the rest of the rotation. Everybody else's velocity seems to be fine. They're just missing in spots and not being able to follow the glove, but sale needs to get his strength back quickly. And it's the first time Chris Stiller's going this deep in the playoffs too. So it's a new field for him. They're going to have to use Johnny's staff. They're going to have to bring some guys up, maybe give a couple of Phantom DLs and get these guys some rest, but they'll be back in due time.