Thanks for checking out this convention review video. This is for the Monster Mania number 44 Which was in Hunt Valley, Maryland I've been doing the reviews for all of these that I go to and Sadly, I could not make it to hunt Valley last year because I really wanted to because they had some people from Twin Peaks And I love Twin Peaks. So that was a sad time. I just happened to be out of town at that time But this time I was able to make it and I was glad that I was I always like going to Monster Mania Um There were a few issues there, which I will talk about at some point during this But for the most part really good really well run and really enjoyed it As usual and the celebrities are always the big draw for me personally because I like to get things personally signed and be able To display those in my house. It's just really cool. So I Was excited to go the big headliner for this one was Bruce Campbell After that there were a lot of good celebrities But Bruce Campbell was clearly the biggest focus for everyone So I'll just kind of talk a little bit more when we go into it I showed up super early for this one Because I wanted to see if I could get a ticket for Bruce Campbell now the way they did the Bruce Campbell thing was it was like this There was bad communication first of all I bought my ticket for this show a year in advance because they announced Bruce Campbell about a year ago Which is great like announce them as soon as you can that's awesome And so I bought it about a year ago And then they didn't communicate specifics of what people really needed to know about the Bruce Campbell autographs until the week of The week of the convention they then say oh by the way He will only sign a certain number of autographs per day It's not a situation where come and get in line and they'll get through as many as possible within that You know time period within the day. It was he was he has a certain number that he does per day So they said they were gonna be doing a ticketing system So you have to show up in the morning to get in the line to buy a ticket for the autograph now the smart thing about that is That they were having you pay up front and they would write your name on the ticket And so that you would show up at a later time because Bruce didn't start signing actually until 12 That day when most celebrities start at 10 So you could just show back up at 12 and then they would you as you got through the line You would just hand your ticket to Bruce Campbell with the item you wanted to get signed and he would you know take your name off there to personalize it to you and everything and That made it more smooth like it went more smoothly after that one of the problems though Is that like I said first of all, they didn't communicate this to people until the week of so for people who are Flying in from someplace or traveling from a far place We're kind of getting screwed because you had to show up super early in order to get in the autograph line and some people's Itinerary may not allow that So really bad on there and in my opinion for not getting the communication out there earlier But I will say maybe it was a situation where they didn't know this until then I don't know I I just I don't know. I'm not sure but then once they did know it They actually didn't do anything to other than the ticketing system, which actually was a really good idea They didn't think think further. They didn't think after the ticketing system So it became a situation where it was just like you get your ticket, which I did get a ticket It was $50 for the ticket that guarantees you a Bruce Campbell autograph that day, which is nice You know as soon as you got it as soon as I got it I was like thank goodness because who knows when he's coming around again. He's very popular So I want to be able to check him off the list and just meet him because he seems like a cool guy plus I like a lot of stuff. He's done so When you got that ticket they were just I asked I'm like they weren't even telling people They're just giving them the ticket So I proactively asked and I was like, you know, what do we do? We just come back here at 12 o'clock like how is this working and the lady was just like Yeah, you know at 12 o'clock you just come back here because that's when he's gonna start signing I was like, okay, so that makes me assume you come there and you get in line and as you get in line That's the order that you know, the line will go although they had told us ahead of time Everyone who pays for the VIP passes, which I did not do I'd never do the VIP because it's $150 and that just gets you to skip a line once which this time they did two different types of VIP They did VIP specifically for Bruce Campbell and they did regular VIP that you could use for anyone So the whole idea was starting at 12 o'clock They would work through all the VIP people first which makes sense because they paid for that Then they would work through everyone else who got a ticket So the VIPs the good thing about that paying for VIP for them is that they could just show up You know kind of whatever and then just go to the front line for Bruce Campbell So if you have the money to spend that way That's a really good way to go if you don't care about throwing an extra hundred and fifty dollars Basically, if you don't mind paying two hundred dollars for a Bruce Campbell autograph, which I was like I can't justify that I'll just do the fifty dollars and I'll wait in line because it's gonna suck when I'm waiting in line But later, you know like a month later. I'll look back and be like yeah, I'm glad I did that for sure actually kind of immediately I was like yeah glad so Based on what I've been told about just show up at 12 I my buddy and I were like oh man, you know Maybe we should show up earlier because if they're doing it where you just show up and get in line We'd rather be in the front as opposed to the back of that line because then we can make it into his first time slot of signing of 12 to 2 o'clock because then from 2 to 3 I think he was doing more photo ops or taking a break or doing both I don't remember what and then he was coming back from and doing a 3 to 5 slot So we were like we'd much rather be in the 12 to 2 and get the autograph Then so we don't have to wait even longer in that line until 3 o'clock So we actually decided to show up even earlier. We went there about 1045 ish To get in line because they're like let's get further up now We were standing in line for Matthew Lillard because I really wanted to meet Matthew Lillard I've seen a lot of his stuff. I really like what he's done. He's amazing in SLC punk Obviously a lot of people love him from Scream. I love him most from the movie hackers I have this weird obsession with the movie have hackers. I know it's not a good movie, but I just I love it It's fun. It's a great time and his character and hackers is my favorite So I wanted to meet him because of that. So we're standing in line for him. His line was crazy slow It wasn't super long but it was crazy slow because he really takes his time with fans and that's kind of a double-edged sword where You hate it when you're in line But once you get up there you're you're glad that he takes the time with you and it's very meaningful and special So, you know, some celebrities are just like that Robert England is notoriously like that You wait in line for a long time because he talks and he tells people all sorts of things and answers questions And you know, he takes times with with people which is cool so Waiting in line for Matthew Lillard and we ended up jumping out after standing in it for a while because we were like We'd rather make sure that we get a good spot for Bruce Campbell So we got there we started waiting in a line that they said that We asked people are like you're in line for Bruce Campbell, right? And they're like, yeah So we waited in line and then we heard people talking about the photo op because there's a Bruce Campbell photo op at 11 o'clock And then he started signing at 12 and then I was like, we better figure this out So my buddy Rich went and asked a few people and they verified Oh, the line you're standing in is just for the Bruce Campbell photo op at 11. They're like, oh, okay Where's the line for the autographs? So then they pointed to another line for the autographs now the issue with this type of confusion is that they could have solved it by having Signs saying Autograph line photo op line. That's all you need and it only needs to be a piece of paper Like you don't have to pay for professional signs Like they just they have a hard time thinking these things through and it really seems like at least the way Things developed while we were there It seemed like they had a basic idea of we'll just sell the tickets and didn't think past that And we're just kind of like if anything else comes out after that will wing it That's not a good idea it's really not a good idea you should think think ahead This is the 44th monster mania that's been done 44 you should have an idea of how to project ahead and problem solve and anticipate what's gonna go on Especially with a celebrity who's as popular as Bruce Campbell is you have to anticipate these things Because you're a professional Convention you got to do this. I'm sorry. You've been doing it 40. This is your 44th time. You got to make this happen so like I'm saying just simple just have a sign it's easy so They did end up putting up a photo op sign at one point, but it was in a place where no one was actually Lining up for photo ops so that didn't even make sense So then after we got out of the line for the photo ops and we got into the actual autograph line We're like, okay, we're in a pretty decent spot here looks good And then they on the other side of the kind of hallway, but we're outside Archway, I guess they were lining up all the VIPs for the autograph So we would like look over and be like how long does that look? That's pretty long We might not make it in the 12 to 2 slot figures cross So then they came by and started saying hey, I think we're gonna organize everyone by their numbers on their tickets And I was like, okay I mean like that makes sense because if you showed up early and you were in a particular slot in line Like that can dictate it But the problem is why didn't you say this to people when they were buying the tickets? Because then people could have stayed in line for Matthew Lillard like me and met him and had his autograph Instead of showing up an hour and 15 minutes earlier than I needed to for a line that I was just gonna have a place in already This is what I'm talking about about the communication and it does kind of piss me off because you have to anticipate this stuff and if you're gonna make these types of Decisions on the fly that's what your Monster Mania app on my phone is for because you promote it saying Download this app. It's got all the news you need and if things are going on we'll communicate with you through the app I kept checking the app. No news about these changes bad communication so When they said they're gonna do this number putting people in order by number situation I was like that can get really messy and so they would have to like push the line backwards and be like, okay You know these people with these numbers come forward and they're manually trying to put people in the right order Which I thought was gonna be really really hard to do But the volunteer staff actually did a pretty good job pulling that off so good on them um and So it wasn't smooth But it was probably as smooth as it could have gone with making that decision on the fly to be honest So props to the volunteer staff for making that happen um So yeah, so we we actually ended up being in about the same spot that we were But I'm just really mad that I missed out on that time when I could have gotten Matthew Lillard's Autograph within that time like the whole idea of the ticketing system was so people wouldn't have to waste their time And we didn't have to waste some of our time But we wasted a lot because they didn't communicate that you would be in a number order that needed to happen They need to say that so um the lines when they actually when the autograph started actually ended up moving pretty well And the main reason is because bruce cambell's personal assistant came out And he was on top of things this guy knows how to run stuff He was very professional and he was like hey guys he yelled to all the lines He's like you gotta listen up. He's like help me help you basically we want to get through everyone We want to make this as seamless and quick as possible Um purchase if you're gonna purchase this new book come up and do it now Make a line to purchase this book before the autographs start If you're gonna get a photo sign that we have on the table over here come up now and grab that now So you have it while you're in line If you have anything that needs to be prepped Like posters rolled up in tubes have that out before you even hit the door To the room where bruce cambell is for the autographs because we want it to be as soon as you get to that table You can just hand them the thing with your ticket and he can start writing And we can just get things moving and I was like Okay, I like the way this guy works very organized Wants to get it taken care of so We actually made it in the 12 to 2 time slot. I was impressed Bruce Campbell signs fast, but he actually talks to the fans um My interaction with him was pretty cool. He um, it was weird. It was very unexpected, but it was pretty cool He was a very nice guy shook his hand. He was like, hey, how's it going?  Like well, originally I'm from Ohio and he was like, we're in Ohio. It's like napoleon It's a really small town and it's kind of like one of those dying towns now in the midwest and he's like, oh, yeah There are actually a lot of those at this point. He's like actually I'm really I'm reading this really interesting book right now historical book on Ohio and how the midwest was settled called pioneers and I immediately was like I said Bruce my my mom is reading that book right now and she's Just a few weeks ago. She was just hounding me about how much I need to read this book because it's so interesting And he was like it is it's great and he starts telling me about this book and he's just like How hard it was for these people and like they had their wagon caravans and In order to get through mountains and large hill areas They had to like actually manually take all the stuff apart and carry the pieces to the other side And then rebuild it to keep on going and he was excited about this book. So The interaction wasn't wasn't how I expected it to be But uh, it was cool. It was very unique and um, I'll show you what I got signed by him Uh, nothing crazy special. My friend actually had like a figure That he got signed which was cool. Let's see if I can get this up here right without too much glare There you go Looks good. Yeah, he just wrote. Hey carlin put his autograph and put ash on it That's all I really need. I I just want to make sure it's personalized. I always like when they're personalized. So You know And it looks good. It's a good photo Um, so it's nice to have gotten his autograph and when we got done. We were just like We did it. That was our main objective. Now. Let's go see what else is going on So I looked at Matthew Lillard's Lillards line after that and it ended up just being too much to handle to be honest I got back in line for Matthew Lillard It was much longer than it was before and it was still moving very slow And I waited in line for probably about another 45 minutes to half to an hour And then someone came out and was like, hey, just letting everyone know we are um, We're doing a we're having a break for Matthew Lillard right now.  So he's not gonna be back for like another hour and 45 minutes And then I started calculating in my head and I was looking at how long the line was adding an hour and 45 minutes Plus whatever I think the line is going to take based off how long it how how slow it's been going And I was like, I would end up being here for another four hours And I'm not gonna do that and I'm just gonna take my chances Not meet Matthew Lillard now and hope that he hits another monster mania in the future Which is possible. These things can definitely happen.  So It'll happen I'll just keep my fingers crossed because I would really like to meet them. But at that point I was just like, I'm really tired I haven't eaten enough today. Um, I was standing in the sun because the line was going outside and I was just like, uh But to their credit they did say We're gonna get everyone a ticket So you have a number and then we can reform the line so you have to physically wait here for an hour and 45 minutes I was like, oh, that's cool I was like, but I don't want to do it because I just don't want to be here that much longer So skipped out on that kind of sucks But I went and hit one of the other autographs That I really really wanted to get which was Danny Lloyd who played The kid in the shining the original Stanley Kubrick shining now this guy He's just a normal dude like he's a very nice guy and part of the reason for that I think is that he's You know, he was a kid actor He was five years old when he was in the shining and that's what he said and he was just like And I don't remember a ton is what he told me I don't remember a ton from the film He's because it was so long ago and now he's like very much detached from it just because it's been so many years but um Super nice guy really easy to talk to willing to to chat And I got a really cool Autograph from him. You just said two carlin best wishes red rum Danny Lloyd, I like they put the red rum on there and this was cool Thanks rich for thanks for my buddy rich for steering me towards this one because I was considering some other ones And as rich pointed out to me, this does a good job of showing a range of acting emotions that Danny used So, yeah, that's cool. Also, they had the the twins From the shining were there as as well as the woman from room 237 the young version They were there signing but I didn't really feel that's like it was necessary for me to get their autographs I wanted Danny Lloyd because he was a Pivotal part of the movie and he had a lot of screen time And he was actually like doing a lot of acting with like the twins I mean, they were just kind of like in it briefly as well as the lady in room 237 just kind of in it briefly so That's why I didn't feel the need and you know it adds up for anyone who goes to conventions You know the money adds up super super fast. So I really like this I like this a lot and I really loved my interaction with Danny I used super cool when I was talking to him. I was just kind of saying I was asking, you know What's your favorite memory from the set of the shining and he was like to be honest since I was so young it was just like getting the Opportunity to ride a big wheel around so much as they were trying to figure out How to track the shot of of like following behind him as he was Riding his big wheel through all the hallways in the hotel He's like, that was a lot of fun So, um, then I was talking to him about, you know, just watching the film and how it's one of my favorite films And how I was lucky enough to to a bunch of years ago I have seen it on the big screen for a fathom events and he was like, oh really he's like I didn't know that they did that. I haven't seen like the full version of the film in on the big screen And I was like it's kind of game changing as far as the film goes because it makes The hotel feel like even more of an imposing character because everything's much bigger You feel more immersed in it and so you can kind of like feel the presence and that And the the evil kind of of the hotel itself and it's really amazing It does make a difference So I was telling him if you ever get the opportunity you definitely should check it out that way And he was like I would love to and then he told me how Because I asked him I was like, when did you first see the film? And he said that he saw like a kids safe version of it Not long after it came out. They did like a special cut So it was safe for him to watch because one so it wasn't so intense And he and his mother were allowed to go to this theater and it was a private screening Just the two of them watching this film He's like so it was kind of weird to like see this movie with just my mom in this big theater He's like but it was cool at the same time Uh, and then I was like, well at what age did you see the full film? And he said well like age of like 11 or 12. So Yeah, but a really cool interaction with him. I really loved it Nice guy if anyone gets the opportunity and you're a big shining fan, it's definitely worth it And chat with him man chat with him. See what's up Uh, so as far as like vendors went they always have really good vendors there. Oh, um, I will say real quick I didn't I purchased something I'll get to that last but I Spoke with the people at vinegar syndrome This will be very interesting for people who are into shudder and the joe bob joe bob stuff I was speaking to the guy at the vinegar syndrome booth Looking at a bunch of their films and noticing that a bunch of the films joe bob has done Episodes on of his last drive in on shudder and they were kind of point the guy was pointing He's like, yeah, this was done on boat joe bob. This was done He said actually for his upcoming stuff. He signed a six Film contract with us which has been big for us And I was like really and he said yes So He's gonna have six vinegar syndrome films coming up and I was like, huh I was like, can you say are you allowed to say which six films they are? And he was like, honestly, I couldn't I actually don't know what they are But he pointed to their gigantic booth of like all these tv d's and blue raisin was like But they're here somewhere most likely And he was like if I had to guess I would guess this one I would guess this one this one this one He goes basically our craziest ones like all the ones that are just like the most nutty and like out there and crazy I was like, yeah, it will be cool to see so that's interesting to know that there will be six Vinegar syndrome films that joe bob covers. So that's awesome also right next to the vinegar syndrome folks was Matt from horror movie podcast Horror movie night podcast. Hold on. Let me make sure I get that right because when I was talking to him I Downloaded one just to make sure Yeah, horror movie night is what the podcast is called So he just kind of like comedically talks about horror movies on these Episodes and I was like, I wasn't aware of this podcast. This sounds up my alley. I'm going to check it out Talking to him was really cool. He's a really nice cool fun guy We kind of just like geeked out about different movie titles and he and I kind of connected with Um Because sorry, I got a little loud there with podcasting We geeked out about because I do a craft beer podcast that's been going for almost seven years now And he's been doing his podcast for a long time as well Since I think he said 2004. So that's 15 years. That's crazy Um, so we were just kind of sharing some stories about this is what we learned with podcasting You got to do this you got to do that and never do this never do that So he was a joy to talk to but as far as vendors go when I went into the convention I had one vendor I wanted to hit in mind and that's what I did and that was the people from mixtape massacre Now if you watch my review for I think it was monster mania 42 I um I had Purchased the first board game the main board game for mixtape massacre at that time and I talked about it on that video So I wanted to grab they have an expansion pack and two booster packs for the game because My wife and I have been playing the game and we both really like it and confession I can't win. She always beats me. I've never won in this game We've played it a bunch of times and I've never won. I I got to figure that out But anyway, so I got the expansion which is the black mask expansion Which is pretty cool um Haven't played it yet because I was just at monster mania yesterday when I'm recording this Uh, then we have the booster pack of bad dad Which looks like a good time and scarlet Which kind of looks inspired by uh return of the living dead So uh excited to get those popped open to start playing if you have interest in mixtape massacre Just go to their website. They actually have playthrough videos that are very uh educational to kind of get you going when you get it Or just you can see how it works So you know if you want to buy it because you can order online and they actually have a new version of the game That's kind of from the opposite perspective that's coming out in december that I think i'm going to get Uh because the game is like you're one of the killers going after a bunch of people to kill And the new version coming out is the opposite where you're like trying to survive and the killers are coming for you So i'm going to buy that one. I'm probably going to go out and pre-order it. Um pretty soon maybe even today Um But yeah, so that's kind of my overall feelings that that's what I did and that's kind of my review of it overall convention was good They always have really good celebrities. That's one of the best draws the venue I like the hunt valley venue more than I like the cherry hill venue because it feels like the the hotel Is in much better shape to be honest like they've renovated that uh since two Years ago, I think when I was there and it's it's works better also the layout for convention I think works better because they can kind of spread things out a little more So it feels like you have more space. So I like the hunt valley one I Am okay with the cherry hill one, but there are some problems there with the actual venue itself. So um Overall, I had a really good time. I really liked it. I think they do a good job with a lot of things Obviously, I had some harsh criticisms about how things went with the Bruce Campbell situation, but You know that that's the truth of it and I'm not gonna Not tell you what I think about it because I really Because I love monster mania because I do I think they do an excellent job overall But I'm gonna point out the problems when they happen and it's up to them to hear the criticism internalize it and not get you know defensive about it, but instead say Hear what you're saying and we'll see what we can do because Me as a person one of the ways I operate whenever there's an actual problem or someone messes something up I don't want excuses. I never want excuses. I just want to hear I hear what you're saying and We'll see what we can do basically or we'll make it right That's what I want to hear. I don't want to hear excuses.  It's the worst So, um, yeah Hopefully they get that fixed if they have another big celebrity come through But like I said, if anyone from monster mania is watching this know that I don't hate you guys I actually love you guys. I think you do an excellent job, but the criticism just happens like it's it's a part of life So I'm sorry. It's a little hard to hear at times, but it's essential But anyway, thanks everyone for checking this out. Um, did you go to monster mania 44? If you did I want to hear in the comments what you did and what autographs you got and specifically Your interactions with the celebrities because that stuff is the most interesting to me Um, so yeah, put some comments down there. Let's geek out about it. Hit that subscribe If you don't mind it would really help me out. Oh and real quick. I gotta give a shout out to john I can't remember his last name. I think it was like laundauer or landauer or something like that I'm sorry, john But he's a guy we met in line when we were waiting for Bruce Campbell and we ended up just hanging out with him for like A lot of the convention because he was just a really cool nice guy And that's one of the other great things about these conventions is there's All these like-minded people in the same place who are there for the same reason and everyone Can talk while you're in line and geek out and be nice to each other and just have fun Because what else are you gonna do? You have to entertain each other when you're waiting in lines And that's one of the things that makes it okay, you know waiting in line Isn't as bad if you have cool people around you to talk to and I experience that a lot at these conventions And I love the horror community. That's just one reason I love the horror community, but Yeah, so thanks everyone for checking this out hit that subscribe Please I would really appreciate that and until next time keep it brutal


Greetings, we have come a long way in relativistic quantum mechanics, the subject is vast and there is only so much one can do, but I would like to introduce you to the solutions to the coulomb problem, that is the hydrogen atom problem. The reason is that when you read any literature in relativistic atomic physics, you are going to deal with atomic wave functions and their probability densities, charge densities and then various atomic processes such as collisions from atoms or you know photo absorption by an atom and so on. The first thing that is going to hit the I is the atomic wave function and the atomic wave function will be described by a certain set of what you call as good quantum numbers.  So, you need to recognize what the good quantum numbers are and the subject being vast, I have planned to at least give you a basic introduction to this, so that you recognize what the quantum numbers are and what how to process you know information about relativistic wave functions. So, let me remind you that our strategy to deal with the hydrogen atom is to set up the spherically symmetric Dirac Hamiltonian. So, v r is a spherical, this has got spherical symmetry, this is the Dirac Hamiltonian and now you need to separate the radial and the angular part and for relativistic case, this is not at all a straight forward task, it is not trivial at all, it is not difficult, but certainly far from trivial and you will see why it is complicated because this term alpha dot p has to be handled very carefully and we are going to do that in today's class. So, in the few minutes we had before the last class ended, I introduced the sigma matrix which is the Dirac sigma, so I will quickly remind you about our notation. So, your spin is h cross by 2 sigma and then depending on how you interpret this sigma or this sigma, this is just a matter of notation and some books use this, some books use this, but depending on the context, you know whether it is a poly sigma or a Dirac sigma, Dirac sigma is the 4 by 4 matrix operator, poly sigma is the 2 by 2, so that is the essential difference.  So, now let us see how to deal with this term sigma dot p and sigma dot p, I have prefixed this by a unit operator because omega dot r is the projection of sigma along some direction and this direction this is only a direction because this is the position vector divided by the length of the vector which is r over r will give you the unit vector and there are 2 of these, there is an r position vector here and another over here. So, sigma dot r sigma dot r over r square is the square of the projection of sigma along some arbitrary direction and that sigma square along any direction is equal to 1 as we know very well from our experience with the poly sigma. So, now this term can also be treated very easily using the poly relation which we have used very extensively and this is therefore, r dot p plus i sigma dot r cross p and now from the r cross p you have got the orbital angular momentum vector here and then from the r dot p you can express the momentum operator which is the gradient operator explicitly in spherical polar coordinates and then you are left with only this component because you will take the partial derivative of some function with respect to r and then take the dot product with this E r. So, only the E r dot E r term will contribute. So, that is the reason only that one is of significance you get the radial momentum which is not just minus i h cross del by del r, but this is the same result as a non relativistic quantum mechanics.   Now, pay very special attention to this operator here h cross plus sigma dot l and this is coming up for very special consideration. It is it has got a structure which you had not made before and it is going to play an extremely important role in relativistic quantum mechanics in as a matter of fact it will be connected to a quantum number that we will get for relativistic quantum wave functions. So, you have got this h cross plus sigma dot l coming here and now you have alpha dot p written in terms of some radial features, but then there also are some angular features here. So, it is not completely free from angle dependence, but sigma dot p is now this alpha dot p now has some radial features which will come in handy when we want to separate the radial part from the angular part which I have alerted you is not a trivial thing and you will then be able to insert this in the expression for the alpha dot p in your Dirac Hamiltonian. So, this is where we introduce a new operator and this is h cross plus sigma dot l which is here, but it is pre multiplied by beta which is what gives the new operator k and if you pre multiply k by beta you will get beta square which is equal to 1 and that is what will give you this h cross plus sigma dot e sigma dot l which is the operator you have seen here. So, this is not k itself, but it is beta k.  So, this is now your alpha dot p and now you can insert this in your Dirac Hamiltonian and find a beta k over here. So, this is what we have got the beta k appears in the Dirac Hamiltonian you can write p r in terms of the derivative operators and you know what the radial momentum operator is. So, you have two terms del over del r and then h cross over r. So, you can write it in some equivalent forms and you have the beta k.  So, the k operator is going to give us a new quantum number which is called as a kappa quantum number. So, this is not the lower case k it is written as kappa that is the notation you will find in most of the literature and those of you who are reading about atomic wave functions would have met the kappa quantum number.  Now, these relations we have used earlier as well.   So, l square is now the square of sigma dot l plus this is the minus sign here. So, it will come with the plus sign here h cross sigma dot l. So, you can factor you can pull out the appropriate factors this h cross sigma l dot sigma is j square minus l square, but l square now has got these two terms and then you have got this minus s square. So, now you notice that j square if you write this expression for j square take all these terms to the other side then j square you will have a square in sigma dot l then you have two terms in h cross sigma dot l which is twice h cross sigma dot l. Then you have h cross square three fourth because s square is half into half plus 1 that is three fourth. So, you get three fourth h cross square and you immediately recognize that this is a whole square minus h cross square by four. So, this is what you have got and that suggests to you because you do know that j square is conserved it commutes with the Dirac Hamiltonian. What you have on the right side will also commute and anything that commutes with the Dirac Hamiltonian is a conserved of motion. So, it offers itself as a candidate to give you a good quantum number. So, you immediately see that possibility and this is what you have got. So, j square plus h cross square one fourth is we define this operator as beta k if you remember this is the operator beta k. So, this is beta k beta k beta is this one minus one along the diagonal. So, this is beta k and again a beta k. So, that will give you k square times the unit matrix. So, the Eigen values of k square will be the same as the Eigen values of j square plus h cross square one fourth and you know the Eigen values of this side of the left hand side. So, what are the Eigen values of the left hand side? The Eigen values of the left hand side are j into j plus 1 times h cross square plus h cross square over 4 and that gives you the Eigen value of k square and we extract the dimensions in h cross square. So, that kappa is the dimensionless number and you get kappa square which is defined by this relationship and now you can strike out h cross square from the two sides and you get kappa square equal to the whole square of j plus half. What it means is that kappa is either plus or minus of j plus half and then you see that j will be depending on j being l minus half or l plus half you pick the appropriate sign over here that is how it goes together. So, for j equal to l minus half kappa is plus j plus half and for j equal to l plus half kappa will be minus of j plus half, but you will see this develop further. So, now we catalog the good quantum numbers in the Dirac scheme. So, where do we get them from? We get them from k. So, kappa will emerge as a good quantum number then we get it from j because j square j commutes. So, j will be a good quantum number and then what also commutes with the Dirac Hamiltonian is the parity, but parity has to be defined very carefully in relativistic quantum mechanics because beta is an operator which is not the diagonal unit matrix, but it has got the 1 0 0 minus 1 structure and because of this very special feature of beta parity has to be defined in different ways. So, this is called sometimes as the Dirac parity p written with a subscript d for Dirac parity or sometimes you write it just as p for parity, but just to rub it in I am using the subscript d because the Dirac parity is different from ordinary parity it has to be pre multiplied by the beta matrix. So, this will become the 4 by 4 structure will come from p 0 0 minus p and you will see how it gives you the appropriate quantum number. So, these are the quantum numbers that you get for the Dirac state. The quantum numbers are n kappa and m these are the quantum numbers kappa contains information about parity. So, there is an omega which is introduced which is defined to be equal to plus 1 when j is equal to l minus half and it is equal to minus 1 when j is equal to l plus half. This is what gives kappa equal to plus or minus j plus half when j is equal to l minus or plus half this is what takes care of it and omega as you will find it has got information about the parity. So, kappa has got information both about j and omega because it is given by j plus half times omega and if you now look at the quantum numbers for various atomic orbitals look at the first column these are the atomic orbitals that you work with s half p half p 3 half d 3 half d 5 half f 5 half and so on. They have the orbital angular momentum quantum number. So, l is equal to 0 for s 1 for p again 1 for p 2 for these 2 d's 3 for f and so on.  So, in this case the omega quantum number gives you that information the j quantum number is here which is half it is half for this state 3 half for this state and so on and kappa will be integers because kappa will be plus or minus j plus half depending on j being l minus or plus half.  So, these are the quantum numbers that you are going to see in relativistic atomic physics you will see the kappa quantum number and they will have this structure and it is important to keep track of this that the s half p half p 3 half d 3 half d 5 half etcetera kappa quantum number correspondingly will go as minus 1 plus 1 then minus 2 plus 2 then minus 3 plus 3 then minus 4 plus 4 and so on. So, there is a systematic manner in which these quantum numbers will show up. Now, what we need to do is to separate the radial and the angular part and that is the non-trivial part that I had mentioned. So, let us work with the Dirac Hamiltonian and we have the beta k operator. So, this is the Hamiltonian this operating on an Eigen function which is now designated by the quantum numbers n kappa m these are the quantum numbers that we have recognized. So, the state is now described by these quantum numbers n kappa m. So, this is an Eigen value equation this is the Dirac Schrodinger like equation if you like, but this is of course the Dirac equation and this is the Dirac equation for the hydrogen atom V r being the Coulomb potential. Now, k operating on u n kappa m will give you the kappa quantum number because that is an Eigen state of k. So, you get I c h cross kappa over r and then the rest of the terms have been written just as they are and then this whole equation I operate upon by beta. So, beta matrix is what multiplies this entire matrix equation. So, beta pre multiplies this left hand side and then on the right hand side beta pre multiplies u which is a 4 by 1 unit wave function vector. So, this is what you have got, but now you know that beta alpha r is minus alpha r beta. So, if you want to change the order of this beta and alpha r you have to take care of this sign it will be the same over here, but when you interchange the position of beta and alpha r over here you will get beta square which will be equal to 1. So, you will get some simplification coming from those terms likewise in the third term you will have beta square m c square and beta square is equal to 1. So, you will get good bit of simplification. So, here I have beta moved to the right of alpha. So, I have a minus sign here the beta square in this term goes to 1 same thing over here then you have beta v r and beta u. Now, these two equations are what I call as equation a and equation b and we are going to handle them as a pair of equations. What I will do is to take half the sum of equation a and b. So, you sum the two equations sum all the terms and take half of it. What do you get? You get half of these two terms. So, this does not have a beta this has got a beta with a minus sign. So, you get c alpha r p r by 2 and 1 minus beta it is very easy to see and likewise if you take the next two terms which has got i c h cross kappa over r which is what you have here as well, but then this is with a plus sign and this is with a minus sign and this has got a beta this one does not. So, if you put the two terms together you get minus i over 2 c h cross kappa over r alpha r 1 minus beta. Are you do the same with the remaining two terms? So, now you see terms in 1 plus beta and 1 minus beta coming in and this we have to see how this function u is going to respond to 1 minus beta and 1 plus beta. That is the first thing that is going to happen because alpha will only operate on the result of that. So, alpha is of course, would operate kappa has already operated and given you the Eigen value kappa the k has already done its task and now we have to see how 1 minus beta and 1 plus beta would operate on the function u. So, the wave function u we write in terms of the spherical harmonic spiners or spinners and this is 1 over r p omega i q omega where these omegas are the spherical harmonic spiners. So, this is written as in a short notation as u plus and u minus. So, p and q are the radial functions which we hope we will be able to separate. It is not that we have done it yet, but in anticipation of a prospective separation of the radial part of the angular part we begin to use that notation and then we ask ourselves what p should be like and what should be the nature of q of the function q. So, that is something which is yet to emerge it will emerge as we carry out this analysis further. So, this is sometimes you use some other notations as well which I will mention toward the end of the class and then now you ask what will 1 minus beta do to this wave function which is written as u plus u minus for the for brevity and 1 minus beta will give you this 1 diagonal matrix minus beta which is 1 minus 1 and this matrix when it pre multiplies u plus and u minus you get twice 0 u minus that is very easy to see that is the advantage of having these block diagonal structure. So, it is very straight forward and then likewise when 1 plus beta pre multiplies the wave function you get twice u plus 0.  So, those are the terms that you get you also have a 1 half factor here you have got that is coming from the fact that you took half the sum of a and b, but then you get a factor of 2 and then there is a similar half factor here and the 2 over here. So, the half and the 2 cancel everywhere in all the terms because they are present in every single terms.  So, minus i c h cross kappa over r times this alpha r and these are the common factors of which pre multiply the matrix u 0 u minus which is here and then you have similar 2 terms which you can combine for u plus 0 that is good. Now, we know that the radial momentum operator the p r commutes with alpha r and you can therefore, interchange these positions.  Alpha r you remember this was the matrix rho alpha was rho sigma this is the radial component of that. So, you get rho which is 0 1 1 0 sigma is sigma 0 0 sigma you take the radial component dot r. So, you get sigma dot r and essentially it is sigma r along these 2 of diagonal positions. So, actually you notice that this sigma r and this sigma this upper case sigma they play the same role in 2 dimensional space and in 4 dimensional space that is what you find and using this feature.  So, you do the same thing over here and now you are in a position to carry out this matrix multiplication because you have got a 4 by 4 matrix which looks like a 2 by 2 because each element is 2 by 2 and then you have got a 4 by 1 column wave function which looks like a 2 by 1 because each element has got 2 rows. So, that is the structure of these equations. So, let us look at this expression it is the same one which was at the bottom of the previous slide and now you have got half diagonal elements over here and the diagonal elements are 0 and they must pre multiply through matrix multiplication this 0 u minus.  So, that is what you get from these terms here you get m c square plus v r that is multiplied by the unit 4 by 1 4 by 4 unit matrix and essentially you get a relationship which is free from all odd operators. Now, there is no alpha here you have gotten rid of odd operators and there is nothing that is angle dependent. So, as a matter of fact we have achieved the separation of the radial and angular part, but this is not the whole story but this is the essential idea because we have to do a little more algebra with this to figure out how exactly the complete Dirac equation is handled. So, I will show you how to do that, but now you have a result which is free from odd operators it is also free from angle dependent operators and we have also succeeded in having a 4 by 4 structure reduced 4 by 1 structure reduced to a 2 by 1 structure. Of course, there is another it is not that the 2 by the remaining 2 rows are lost can you see them what do you get from the remaining 2 rows here you have 4 right. So, here also 2 of those rows are here what do you get from the other 2 rows you get 0 on the left hand side and 0 on the right hand side. So, you get 0 equal to 0 and there is no physics in it. So, which is why I have not written that, but essentially you get complete consistency you are not thrown off anything from the remaining 2 rows you get 0 equal to 0 and you are not losing anything in that as you knew it even before you were born right and you have an essential 2 by 1 structure. Now, let us go back to the equation a which I had used earlier you remember we had taken half the sum of equation a and equation b. So, it is the same equation a which I have written again, but this time I am going to do something different with it. What we will do is instead of operating upon equation a by beta we will operate this time by alpha r the same kind of technique, but now we take the other operator which was there in the equation of motion. So, we operate now by alpha r. So, you have alpha r pre multiplying this whole equation and then you have alpha r pre multiplying this wave function which is the 4 component wave function. So, now you have got alpha r square which will give you unity in the first term. So, you get C P r from the first term again from alpha r square you get unity then you have got alpha r beta right and then alpha r v. Now, on this result you pre multiply by beta these are some nice tricks that you play, but it also tells you that when you are dealing with this kind of mathematics you know what is it that you can do and what you stand to gain by doing. So, you pre multiply this by beta and now you get beta C P r then beta will come here every term you have to pre multiply by beta and take advantage of beta square being equal to 1 and now what I will do is I will call this result as equation x and this result as equation y. So, I name these equations as x and y and then take half of x plus y. So, for our convenience I have rewritten the equation x and y at the top of this slide. So, that we can keep track of each term and now we take half the equation x plus half the equation y. So, you get the factor half C P r then you have to take the sum of these two terms.  Now, we have done this earlier. So, you can very easily see how these terms fall in place and once again we can make use of the fact that we already know what 1 plus beta does to you and what 1 minus beta does to you. So, those results we can already use. So, now I have gotten rid of the two and the one half factor. The half factor goes everywhere and we have these results 1 minus beta operating on the wave function pre multiplying the wave function will give you twice 0 u minus and this 1 plus beta will give you twice u plus 0 where plus and minus are these components of the four component wave function. So, these are two component functions each has got two components.   So, you would get twice u plus 0 over here, but you are going to get two in all the remaining terms. So, they will cancel out. So, in anticipation of that they have already been removed and now you have over here alpha r which will pre multiply 0 u minus. So, alpha being odd you can once again do the same kind of matrix multiplication and you get a simple 2 by 1 reduction just the way we did in the previous step and then of course, you will still have two other rows which will give you 0 equal to 0. So, now I will spend a few minutes discussing the properties of these spherical harmonic spinors omega. So, this is the definition of the spherical harmonic spinors these are made of the regular spherical harmonics these are the spin functions. So, this is spin function spin up and spin down it could be 1 0 or for the spin up and 0 1 for spin down or up and down you know however, you know whatever notation you want to use, but so these are the spin Eigen functions. So, this is essentially a composition of angular momentum of these two angular momentum individual factor states one which are Eigen functions of the orbital angular momentum second which are the Eigen functions of the spin angular momentum and these are coupled using the clebsch-gordan coefficients and you have you are the world's experts on clebsch-gordan coefficient. So, you have a double sum now m s going from minus half to plus half and m l prime I have used which will go from minus l to plus l, but you know that the clebsch-gordan coefficient would vanish unless this m is equal to m l prime plus m s.  So, only the term in m l prime equal to m minus m s, so that pins down this spherical harmonic this quantum number m l prime must be m minus m s otherwise the clebsch-gordan coefficient is 0 we know that and now you have a very simple sum because this is a sum over m s going from minus half to plus half which means that you must sum over both of these spin Eigen states which are respectively 1 0 and 0 1. So, you have these two terms one corresponding to m s equal to minus half here and the other corresponding to m s equal to plus half and then you have the corresponding clebsch-gordan coefficient here. So, let us look at these two terms. So, these are the two terms now you have summed over m l's you have summed over m s you have got everything has been summed over now these spin functions this is m s equal to minus half. So, this is the 0 1 this is the spin down this is m s equal to plus half. So, this is the spin up which is the matrix 1 0. So, which means that you will multiply this term and this term and it will come in location in the first row but the element in the second row will be 0 over here the element in the first row will be 0 the element in the second row will be non 0 and then you sum the two terms what do you get you get a matrix of two rows and one column the first row comes from the lower term because this has got this the element in the first row to be non 0. So, this is what you get and from this term you get the lower row which is this which is the second row. Now, this is your simplified expression for the spherical harmonic spinner and now you need these Clebsch-Gordon coefficients you have to determine them and you can determine them quite easily from you know standard you know tables of Clebsch-Gordon coefficients and so on and their values actually depend on whether j is l plus half or l minus half. So, you have to handle these cases separately because the value of the Clebsch-Gordon coefficient will certainly depend on this and then if j is l plus half then this Clebsch-Gordon coefficient the upper one is root j plus m over 2 j the lower one is root j minus m over 2 j on the other hand if the j is equal to l minus half then the Clebsch-Gordon coefficients are given by d. So, this is something that you can extract from standard tables of Clebsch-Gordon coefficients. Now, you have got everything you plug in the corresponding Clebsch-Gordon coefficients and you will have two different sets one for j equal to l plus half and the other for j equal to l minus half because the corresponding Clebsch-Gordon coefficients are different. So, I have just compiled those relations over here for simplicity and now you have got all the elements that you need you have got the radial equations you have got the wave function which is written in terms of this u plus and u minus and you have got the spherical harmonic spinners which are now written in terms of the spherical harmonics and the Clebsch-Gordon coefficients. So, all the elements are there now all you have to do is to see what these equations simplify to. So, now this u plus u minus which we have used I used explicitly the functions p and q. So, this is i q this spherical harmonic and here I have got p omega. So, I have got two sets of equations if you remember I have used both of them and what we need is sigma r u plus u minus because that is still to be determined. So, you have got the sigma r over here do not forget that. So, this sigma r what it does to the spherical harmonic spinner and the wave function u plus u minus is something that is yet to be explored. So, this is something that you can work out that when sigma r pre multiplies u plus you get minus omega minus kappa m and this comes just from the definition of the spherical harmonic spinner because sigma r has got this matrix structure. So, you just use the matrix structure and the definition of the spherical harmonic and you will find these quantum numbers this omega kappa m when pre multiplied by sigma r will give you minus of omega times omega, but the quantum number kappa will be replaced by minus kappa and you have got a similar relationship over here and you can now use this result back in these two relations in both of these you need these two relations. So, plug it back and you get now the angular part is completely separated out. So, the partial derivatives with respect to r del by del r can be written as d by d r and you get two coupled equations for p and q and this is the completely radial equation and this really comes as a surprise because it was not at all obvious from the beginning that the Dirac equation can actually be separated into the radial part and the angular part because it had a very complicated structure in terms of the operators that we were working with. Now, you have these two operators two sets of couple differential equations for p and q you can you will find them in different forms in different you know books and papers because you can replace kappa by minus kappa and vice versa and you see them in a slightly different form, but it is completely equivalent it is just minus kappa instead of kappa that is it, but then you can also have these relations you multiply everything by minus 1 and again you see it in a slightly different form. So, you will see these equations in somewhat different forms in literature and sometimes instead of p and q you find these equations written in terms of g and f rather than p and q where g is usually defined as minus i p r and f as minus i q r. So, you will find couple equations for g and f rather than p and q. So, different authors use different notations, but that is just a matter of very minor detail and it is not going to worry you. So, these are the couple equations for p and q these are the radial equations you can solve them and depending on what source you are using whether you may be reading Bjorken and Drell or Masaya has got a very good chapter on Relativistic Quantum Mechanics in Volume 2. The article which I strongly recommend is an article by Grant in advances in physics 1970 and you will find these relations in Grant's paper as well Pratt Ron and Seng have a review in modern physics. Bjorken Gratt have got a nice paper and you will find equations radial equations of this kind or some equivalent forms either in terms of g and f or p and q or the same equation multiplied by minus 1 or kappa replaced by minus kappa. So, some small transformations you may have to do, but essentially what I was hoping to do in this class is to introduce you to the radial differential equations and then you can go ahead and solve it like any other system of couple differential equations. That is a technique by itself, but that is not new to you if you have come thus far you know how to do that. So, I am going to stop here for this class and with this I conclude unit 3, there is a whole lot of physics that you do with relativistic atomic wave functions. In fact, anything that you do deal with atomic structure and atomic processes requires you to use relativistic wave functions for the simple reason that nature is relativistic. There is no escape from it, there is no escape from the fact that the speed of light is finite and everything begins there. So, the speed of light is finite, laws of nature are quantum mechanical and therefore, there is no escape from relativistic quantum mechanics and we have a bare introduction. All we did was the hydrogen atom which is the smallest thing that we can think about. Once you take any atom which has got more than a single electron then it becomes that much complicated and then you want to do relativistic quantum mechanics with it, it becomes even more complicated. So, we first of all we need to learn how to deal with atoms with more than one electron and that is more than one is many. So, we will get into what is formally called as the many electron formalism of atomic structure. So, the next unit will be on the Hartree-Fock self-consistent field methods of dealing with many electron atoms, but if there are any questions on the relativistic hydrogen atom at least I hope that you have got a basic introduction to the topic. You have your handle on the techniques and these are the methods that you have to use means I am sure that you will be confronted with various problems which will require some effort, but you will not require new techniques, you have those techniques with you. If there is any question I will be happy to take otherwise, thank you for now and then we will meet for unit 4. Angular function right, I mean it is very normal. Yeah, but that has been separated out it does not come here, in this equation there is no spherical harmony. Now, even the y not y, why yeah. So, in the previous result where we use that the separation was not explicit, we use that to arrive at the original expression of course, has got the radial part and the angular part just like the Schrodinger equation has the radial part and the angular part. So, the angular parts are there, they are sitting in the spherical harmonics which sit in the spherical harmonic spinors, but then using this algebra of the direct matrix operators the sigmas in we dealt with the radial components of sigmas and then when we had the sigma, the radial component sigma r operating on the 4 component wave function, we use the result in the previous relation here and then all the common terms cancel out and what you are left with is only the radial equation. So, this will get give you a set of terms, which will cancel out completely. So, all the angular parts go away and you are left with only the radial part and this is a great wonder, which is why it is often said that the Dirac equation can be solved for a few problems for which you have exact analytical solutions, the hydrogen atom is one of them. It is not for you know many situations that you can really do, but then that is not much of a concern, because we have already reconciled with the fact that in most situations, the challenge for a physicist is not really to get the exact solutions, but to get the best approximation. Exact solutions do not exist, there are existance theorems, which tell you that an exact solution does not even exist. So, your challenge is not really so much in looking for an exact solution, because that could be a search for the impossible, but to get the best approximation and for the hydrogen atom, you do get exact solutions, but when you go beyond the hydrogen atom, you do not. Then your challenge is what is the best that you can do and the many electron theory, the Hartree-Fock formalism is a wonderful exercise in getting approximate solutions to a many electron problem, but of course there are ways of improving on the Hartree-Fock as well. So, we will talk about it in the next unit. Any other question? So, thank you very much and goodbye for now.


This video will talk about systems of nonlinear equations. So let's think about how many solutions we would even have if we're talking about nonlinear. We've talked about linear and realized that most of the time we're going to have one. But if I have a line in a circle, how many would I have? Well if I have a circle and I have a line that looks like this, then that would mean that I have no solutions. And if I have a line that's going to be tangent to my circle, that means that we have one solution. And then my only other possibility is if my line goes through my circle, and that will give me two solutions. So I have zero, one, two, or possible solutions. What about a circle and a parabola? We could have any combination.  We could even have a log and a parabola, all kinds of things. But we're just taking a couple into consideration here. So if I have a parabola that looks like this, and then I have a circle below it, say, well that would give me zero solutions. Could I have one? Well if I have a circle with a parabola that the vertex is tangent to that circle, I could have one. Could I possibly have two? Well let's think, a circle. And what if this vertex of my parabola was inside the circle so that only my legs went through the circle? Well that would give me two. Could I possibly have three? Sure. Shift that up. Remember these parabolas could be flipped upside down too and do these same kinds of things. I'm just doing one direction here. In fact I'll switch it just to be different. What if I had my vertex right there and then I went through my circle like this and I'd have three solutions. I could do four if my vertex was outside of my circle but it went all the way through it. So that would give me four. And five I think would be a little difficult since I only have two legs of my parabola. So when we're solving these things, we have multiple solutions but we're going to use the same methods that we did for linear equations. So let's try one. This says substitution. So I see that this is a y equal. So I want to put this one into this equation. So here's my y. Minus one is equal to ln x plus 12. And some of you are beginning to see that these definitely look different than the linear ones. So what is y equal to? It's equal to ln x squared and then plus one. Well if I drop my parentheses because there's nothing on the outside to distribute I can see that I'm going to have positive one and negative one that cancel each other out. So I'm left with ln x squared on this side and I'm equal to ln of x plus 12 on this side. Well I have that uniqueness property. Same log, same base of my log. So I can just say that x squared is equal to x plus 12 and we have the quadratic. So we can say that we have x squared minus x minus 12 bringing both those terms to the left hand side. And I'm going to use my quadratic formula on it. Find out that when I have a is one, b is negative one and c is negative 12 and it tells me that my x's are four and negative three. So I have two x's but remember a system is ordered pairs. So I know I'm going to have x equal four but I've got to find out what y goes with it. And then I also know that x is going to be equal to negative three and I need to know the y that goes with it. So let's go back here. Remember we can plug it back in to any one of our original equations. I'm going to go back to that top one. It looks a little bit nicer for me. And I'm going to say that y is equal to ln and then I've got my four that I have to square plus one. I go to my calculator and ln of four squared close the parenthesis plus one tells me that I have 3.77 approximately. So now I have negative three that I want to work with and I'm going to do the same thing. I'm going to say y is equal to ln of negative three squared but it's inside plus one. So this will be nine when I squared ln of and then I need another parenthesis because my base on that square is actually negative three and then squared and then close my argument plus one. And I find out that y in this case is going to be 3.197 so we'll say two zero. And those are my two answers. So then it does say that we should check by graphing and I already put these into my calculator. What I did was I said this was y is equal to ln x plus 12 but then I add the one to the other side. So here's my two equations. ln x squared plus one. That's the top equation. Then ln x plus 12 and then plus the one when I brought it over and I get this graph. So I can see that I have two intersecting points right there. So second trace five. And notice I don't have anything in my y right here so I need to move my cursor until I do and then I can just press enter, enter, enter for the first one. And there's my four and my 3.77 and then I do second trace five again and if you get this trick it's helpful but if you don't it's okay. If I start moving trying to get closer to there you see I'm moving along this bunny looking graph. And you see that that's the ln x squared plus one. If I toggle down and say my down arrow now I have y2 and I'm looking at the other graph so I'll be on this log graph and I can move to the left but I get there a lot quicker because I am just on the simpler curve and I press enter three times and I get my negative three and my 3.197 or 3.2. Elimination remember we said we had to line up like things and have opposite coefficients on them. Well this is a linear and this is a circle so what are we going to do there? Well let's look at it. I can solve this thing for y so y is going to be equal to 5 plus 2x and I can solve this one for y squared is equal to 85 minus x squared. Now if I tried to get to y here I'd have a square root and that makes things really difficult but I see this y and I could make it y squared. If I square both sides I could make this left hand side become y squared. So squaring this side I get y squared. Squaring this side remember it's 5 plus 2x times 5 plus 2x so 5 times 5 will be 25 to take the first term and square it and then I'm going to have 5 times 2x twice so that would be 10 times 2 or plus 20x and then I'm going to have my 2x times my 2x which would give me plus 4x squared so now I can say underneath this bottom equation that my new equation that's going to combine with it is going to be y squared is equal to 25 and I'm going to rearrange this so that things line up so 4x squared and then plus 20x I'm just going to kind of switch these two terms so things line up nicely and if I can get them to be opposites then nice things happen so I'm going to multiply through by a negative in here and when I multiply through in a negative that becomes a negative y squared and a negative 25 and a negative 4x squared and a negative 20 and now I can add so y squared plus negative y squared is 0 85 plus a negative 25 is going to be 60 negative 4x squared and a negative 4x squared will be minus 5x squared and then minus 20x because there's nothing to combine with that one and you can see that we have a quadratic if it helps you we could say negative 5x squared minus 20x plus 60 it's a positive 60 there and I'm just going to use my quadratic formula so that I don't care that that's a negative 5x squared so negative 5 is my a negative 20 is my b and 60 is my c and I find out that x is equal to negative 6 and x is equal to 2 but once again I have to find out what my y is that goes with those so I'm going to go back up to this linear equation and say that y is equal to 5 plus 2 times my negative 6 well that's negative 12 plus 5 so y is going to be equal to negative 7 so up here I'm going to write my ordered pair of negative 6 for x and negative 7 for y and then I also have an ordered pair that starts with 2 for the x and then I'll work my problem so y is equal to 5 plus 2 times my 2 which is x so y is equal to 9 so I also have the ordered pair 2, 9


I'm going to get a motion for approval of this. Second. Second. All in favor, please say aye. Aye. Aye. And it's approved. We just roll it into our presentation. So thank you, sir, for being here. Bring introduction. First, let me say, no mic, test, test, test. Good morning again. Good morning. Thank you for allowing us this opportunity to share with you our proposal for generating job opportunity for some of our most challenged citizens. Some of these challenges may include alcohol abuse, drug abuse, lack of self-esteem, and yet others may be a lack of education, or low work skills, or low confidence. This proposal represents a collaborative effort between St. John Community Development Corporation and the City of Columbia, led by our very innovative city mayor. I am Jackie Epps, and I chair of the St. John CDC Community Development Corporation. And a company of me is our career center director, Gwen Floyd. Our pastor of St. John Baptist Church is unavailable and not able to attend today, but rest assured, let me say emphatically, that he fully supports this initiative. And I say that with total confidence. It's good to know that we have a Vanna White with us today, so, I guess some of you, it's too young for this. Hopefully you don't have to spell anything. Our CDC mission is essentially to revitalize the Book of Washington Heights community with major emphasis on youth, young adults, and senior citizens. The Book of Washington Heights community has a very rich history of success, and has been experiencing significant high level of crime and violence recently. Our church is centrally embedded within the Book of Washington Heights community, and we were challenged by a county council member to get involved, to implement solutions and solve problems within the community. So we accepted the challenge to make a positive difference in our community. A major area has been on jobs and employment opportunities and workforce readiness, because we fundamentally believe that as employment increase, crime and violence will decrease proportionately. We think they're inversely related, which is why we organized and launched the Career Development Center. We opened the Career Center by offering soft-skilled workshops, such as resume writing, interviewing skills, rest or impress, thinking on your feet, interpersonal skills. And we later expanded our curriculum to include technology courses like PowerPoint, Excel, Word, keyboard, and et cetera. And now we're offering drone flight training classes to get FAA certification for good jobs. I met Rickerman, visited our Career Center a couple years ago, and thought our program could play a vital role in helping challenged youth and young adults adjust their lives and position themselves for today's workforce. So with that as a background and as a backdrop, let me now present to you the key elements of our Pathway to Employment proposal. This initiative represents a public-private partnership between the City of Columbia and St. John Community Development Corporation. Our primary focus is on young adults aged 18 to 25, albeit not limited to that age range, who are facing barriers and life challenges that may include substance abuse and other personal challenges. This program provides an opportunity for candidates to participate in a program designed to earn real income while operating on the old concept, yet still valid, a full day's work for a full day's pay. Hopefully, participants will gain a great appreciation for good, strong work ethics and value the significance of commitment and responsibility. Are we on slide three? We're on slide three. This initiative is planned around a 12-week program with a 40-hour work week that includes a combination of field work and professional development. This program has a goal for all the participants to evolve or transition into a full-time job offered by the city or some local business or some other local industry. A typical work week will be as follows. Monies and windies, wind days, will include a combination of a four-hour professional development session, which will focus on soft skills and also technology skills, followed by four hours of field work, focused on city beautification and sanitation work. Tuesday, Thursdays and Fridays will be eight hours of field work only. So those three days will be full on-the-job work experience. Monies and Wednesdays will be half-day work and half-day professional development. Participants will be observed and monitored by their supervisor and career center instructors, and we will assess the need for support from community resources. We will appoint those special resources as needed, like drug-addition counseling and other personal needs based on our conversation with them and based on our observation with them. But we, in some respect, we view our role as a personal coach to each person participating in this program. Slide five, transportation. Our participants, participating in the program, will be responsible for arriving at the church prior to 9 a.m. each day. A work transportation van will depart the work site, will depart for the work church at 9 a.m. for work and then return at 5 p.m. A city bus stop is located at Farrell Road in Douglas Street, which is just in front of our church, for those who need to commute by bus. We will have bus passes and tickets available for all those commuting by bus. The CDC will purchase a van from the city at a nominal cost for transporting the workers to and from the work site locations. At the end of this session, the end of the program, the vehicle will be returned to the city at the same cost as purchased. If it perches for $1, we will return sales for $1. Slide six, supplies. The city will provide all jobs, sites, supplies as well and any needed safety equipment, such as goggles and vests. The CDC supervisor will hand out supplies and equipment each morning and then collect them when they return in the evening and store them for the next work day. Next slide. By the way, feel free to interrupt me if you have any questions about anything. Session development. Our Career Center is well-prepared for this role as we are currently offering soft-skilled workshops and interviewing skills, resume writing, dress to impress, mock interviews, work ethics, et cetera. Additionally, we offer technology courses that include Word, Excel, PowerPoint, introduction to computer, keyboard, et cetera. And oh, by the way, we have a partnership with South Carolina Works. We are a hub for South Carolina Works. Any of our community members can come into our Career Center and search for jobs, as well as file for unemployment. And we also have a partnership with Richland One Adult Education Program if they're interested in earning a GED. So we're well connected. We have our outstanding partnership to enable these participants to better prepare themselves for the workforce. We will engage a banking industry consultant and a tax consultant to discuss personal finances because each participant will be issued or receive a 1099 at the end of the year. We'll view them as independent contractors. They will be responsible for their social security and paying other taxes. So at our first meeting, we will review all those responsibilities. And we have a bank person come in and talk about checking accounts and saving accounts so they can cast their check, open up accounts, cast their checks. Again, that will be on day one. We will start out with these kind of personal money management kind of questions. Any questions? Next slide. Pogethilp Investment. Now I'll open this session by indicating that this is a collaborative effort each partner have in skin in the game. We both have invested interest in a successful outcome. Each partner, each party, I should say, is investing significant resources to make this program a success. As you can see on our screen, there are many contributions from each party. And of course, there's a financial component to this program. So we're requesting from the city an annual, the annual cost of this project is the neighborhood of $275,000. We are requesting funding for a three-month allocation, three months to support a three-month pilot program in the amount of $64,215, you know, in my previous life while I was working at Kraft Foods, prior to launching any new product, we would always conduct a test market. And I view this pilot program as a test market where we can get immediate feedback on projected success. This three-month pilot program should provide a good projection of our return on our investment. Next slide. We fundamentally believe that this initiative represented great opportunity to make a positive difference in the lives of many challenged, unemployed citizens of our community. We're grateful for the effort to our innovative mayor and for his foresight and very strong leadership in this effort. If we aim to accelerate progress, we must implement these kinds of innovative and transformational programs. We believe that this initiative is one that, if successful, can be expanded broadly across the city and beyond. It has been a pleasure working with the mayor and his support team hammering out this proposal and we're very excited by the opportunity to participate in this pilot program. And we're ready to get to work. I thank you for your attention, I thank you for your interest and I'll respond to any questions that you may have. Mr. Epps, how many participants do you think you'd have during the pilot program? We budgeted for eight. We would say eight as a maximum. We see four as a goal. We would begin the program with four as a minimum. What do you think, if you're projecting eight, is there a maximum that you... Eight, maximum is eight. Maximum is eight. Maximum is eight. If the concept is full-time employment, do you have some business community partnerships lined up that might be the next landing place for these eight? The city would be one source of... But we will be actively engaged in identifying other business consultants, business partners. Okay. Currently, we don't have any on our list, but I think one of the things that I would like to do is we talk about we want the business community more involved and I think the business community talks about being more involved and we have a hard time connecting those two dots. I think as part of this program, I would like to help and connect those dots between this and the business community and if you could expand it, then you've got even more opportunities. Absolutely. If you had... Suppose you had 20 groups with eight people and then that's even more chance for the business community to be involved as part of the solution of these types of issues. So you already have four people lined up, though? Oh, no, no, until we're funded, we haven't committed to anyone yet. We have someone in mind, but we haven't committed to. Once we get the green light, then the process kicks in. We have some... Got you. So what would be the measure of success that the pilot was successful? How would you... I think we talked about that and we're saying in the 50% to 60% range of participant evolving into a full-time position at a meaningful work site, a job, business, industry, city, whatever. Anything north of there we think is a good result. I think we changed one life. Yeah. I think there's statistics to say, if somebody can make it in a new job for 60 days, the likelihood of success moving forward is pretty strong. So it makes me very... How do you... I guess I'm... My question is getting there by nine o'clock. How do you... Do you have any counseling or communication pathways lined up to really go that extra mile to make sure these talented workforce folks make it to the training every day? The first little hiccup, the first time they miss, might send them on a spiral where they can't finish the process. So I guess the ultimate goal was make it that 60, 90 days. Right. And then transition into a municipality job, a county job where there's wonderful workforce demand. So the question is, do we have a process for... Are you going to go above and beyond to make sure the eight candidates come see you every day? Yeah. First thing we're going to offer, we're on our bus passes. We're going to try to get bus pass, so they have transportation. Depending on what the special need is, I'm not sure if we'd be able to utilize the van to go pick up, but we're a part of them. We're going to make things as easy as we possibly can for them to succeed. So hopefully they'll be utilized to bus this and get there. We're going to transfer them from the church, they get to our church, to the site each morning. We talked about if they're late, what do we do, how long do we wait for them, those kinds of things. But we're going to kind of work through all that as we go. We've thought about it and we have some backup plans that kind of help them along the way. And I think especially some sort of counseling layer. Oh yeah. That would be daily. Yeah. That's part of that soft skills. You know, kind of knowing where their direction is going, I think it's very important. I hope you work that into your... Yeah. That's part of our work ethics and program that we're going to be sharing with them doing those soft skills. So are there eight candidates from the local zip code as a target? Where will you do your recruiting for the program? We'll start with the Book of Rights community, but we span it out. We have a relationship with the olive of gospel where they come from all over. So we have a network where we're going to try to recruit these participants. Yeah. Fantastic. And whenever the presentations ever, I'd love to hear from Robert Anderson about the vacancies right now and public works, maybe Clint can speak to what we're looking at at Columbia Water for these types of positions, which would just make us feel like we're going to get this great pipeline of talent. The natural candidate. Well, I think every municipality is struggling to find great workforce. And it sounds like y'all have tailored this as a pipeline for us to fill that. So I'd cue those to the mayor for that. But yeah, just would love to hear about our vacancies right now. Yeah.  And I was in the, you know, fund raising part of the business, but I had a relationship with every major corporation in South Carolina at that time. That was 10, 15 years ago. I don't have too much relation with corporate, but I have that kind of relationship building with industry. So I look for the opportunity to get out and talk to some of these businesses locally about hiring some of these, some of these participants. Good. I think that's a, I mean, I really, I really think this could be a neat program because it really is tailored to the individual and, and, and, you know, you, you said if you can change one life, but it's even better if we can change eight lives. Absolutely. So, are there any other questions? All right. Thank you, Mr. Epps. Thank you. Thank you. So thank you, Mr. Epps. Thank you for your presentation. I do want to point out that during our discussions with the church and the pathways to employment that there was a lot of talk about the signing of a, of a rules of conduct, I believe it was Mr. Epps that, you know, and I think that helps out with your question, Councilman, is, you know, this is what we expect and that's what the church expects. This is what the employer expects. So when we look at it from a vacancy standpoint, I think we look at it, not only speak for Columbia Water, but I think we all have rolling turnover, you know, where Samantha said she has five today. She may have 10 tomorrow. She may have two tomorrow too. So we may hire three people, but we continually have rolling vacancies, not only in public works, but throughout the city. I see her work at Job Fair. She puts it on well. She does great work. So, but I'm also seeing the, the, the other side of this where we have people that come in that have been on drugs for many years that come and do something different. So I also want to say that when we talk about our partnership, I don't know that we're training only for the city of Columbia. We've talked a lot about this with the church is we're training for Richland County, Lexington County, where we're training for Forest Acres, City of Casey, West Columbia, but we're also training for our vendors ourselves. You know, we have mowing contractors. We have cleaning contractors. You start a cleaning company right now. It's probably gold, you know, to get these people to do that. So it's not only just to teach them how to pick up litter, but it's it's teaching them, you know, how to plant a tree or, you know, how to dead head of flower or making sure that, you know, something looks better or, you know, how to wax a floor, maybe, or just teaching them the soft skills of life so they have a better opportunity to go out and do this. So I don't think there's a need in public works. I think there's a need to City of Columbia through Parks and Recreation, Public Works, Columbia Water, but I think there's a need through the contractor side too. And it's hopefully getting these people out to understand some of a couple of things we're working on, you know, CDL drivers are always, always, always a must. We're working on a partnership right now. We haven't even talked to the church about this. Through Midlands Tech. Midlands Tech did a machine which run cat, which we found out yesterday is actually has multimillion dollar machines that is teaching people how to operate trackers and motor graders and stuff. And we're going to bring those trainers soon to a conference we're having here in Columbia, but we've actually met with Midlands Tech about, you know, going out there, maybe sponsoring somebody to learn these, you know, all of our operators are sitting out here on 126. We need a few more operators. So I think there's a lot of opportunities for us to bring these people in and train them and hopefully teach them a better way. Well, and that's great information. And our previous work sessions about recruiting and retention, I think it's important for people to realize what a great future they could have as an employee with the city of Columbia. And that's really showing them if you retire in whatever 20, 30 years, here are the benefits. I think that needs to be played up in the off days when you're doing, you know, I'm not sure what programs, but just for them to really feel good about their path forward, I think it's very important. And that's something that our HR team can probably help out with showing the benefits of sticking it out. I think that brings us to a conversation we was having just before he walked in that, you know, I started when I was 25, I think Brian said he started when he was 26. I think Samantha started when she was 25, you know, they're looking at retirement and where it fits into their life, you know, now that they're fifties or so, I think. But a lot of people have that, but they're actually looking at their road to retirement, too. And I think when I came here, that's what the city sold itself on was a solid job in retirement.  I've always got to check with the city of Columbia forever. So, you know, and always got that through hard times. So I think I think what's really good about this program is the collaboration between the community, we want the business community, the city, and I think I think that's really what this council has been all about is collaboration and working together for solutions, solutions to the issues that we've had in the city. And I think this is just such a great time to be doing something like this. So my question is going to be, we have a three month pilot program. So would you like to present maybe halfway into the program, like in a 45 day mark, or, you know, to kind of give us an update? Yeah. And then we can talk about it at the 90 day. I think it'd be good to hear about it, you know, in the middle before, you know, just to see how things are going and to kind of see where we're leading and where this is going to head. So I will tell the committee currently we do have the funding for the pilot program. We've actually added money to our budget for the full program next year. I do think there's a couple issues that we need to work out with, maybe legal or something else, you know, maybe like a resolution or an ordinance or some kind of memorandum of understanding and we still got to work out a vehicle issue that, you know, how can we, can we sell a city vehicle for a dollar and get it back for a dollar? But, you know, we sell vehicles at Red Fleet all the time. So it may be something easy. It may not be. That sounded, that sounded too easy. Oh, it's got to be that easy. So, but so there, but once we get the council's blessing, then, you know, our committee moved this forward. And I want to thank Peyton for taking the lead on this and ACM Benjamin for setting in on the meetings and making sure this moves forward. Thank you all for, for getting us to where we are. I think this is an important thing. That's why I want to get a midpoint update because I think it's something that could, could go city wide, you know, perhaps and you can really make a difference with multiple different groups doing this. So you're saying, so to Robert, you have the three months currently budgeted and the current budget or on the next fiscal go round? Well, so we have the funding for the three months. We've actually requested the additional funding for next year in our budget. And this doesn't have to wait for funding to be set in place. This can start pretty, pretty quick. Yes. We have to get approval though. That's correct. All of the council. Right. Right. Yeah. Yeah. Yeah. Right. To extend those funds. Yeah. We'll get a contract. It'll come. Yeah. Okay. So yeah, to that point, I think we could start this fiscal year and then if y'all approve the the full year's funding and the budget request, continue on, you know, subsequent to the progress update and the project going well, which we all think it will. So. And then when's fiscal year start? So I do a lot first. A lot first. Yeah. So we would have to go ahead and put the line item for the full year in. Approved. Approved that in May, June. So pretty much kind of, we would need a kind of a halfway, month and a half update. Yeah. All right. And what we would do after today, even though it hasn't been a full council, is probably start working with legal and the church on a memorandum or something so we could, you know, hopefully just start it as a pilot program and move forward.  Are you really happy to see you? That was pretty cool. So, but we would start working on some of the. We're taking an agreement. I don't know how we're going to do the vehicle. So this will take a while to get on the agenda. So we could do a month out before. Yeah, I was thinking the same thing because our next meeting. So wouldn't it just be worth to do it early April? Just do the new fiscal year? Yeah, fine. Yeah, look, I mean, I can't. That gives me no time to. I mean, I'm in no hurry.  But as it does get kicked off, I think it's important for us to to stay in touch with how it's going until we can release a fleet vehicle, the dollar purchase or whatever monetary amount purchase. My thought was, is when you, if you sell the vehicle for a dollar, you've transferred the whole ownership of the vehicle and the liability that goes with that vehicle. Therefore, they're carrying the liability insurance and carrying of that. If we actually lease them a vehicle, we may still have some liabilities. And that was my early thoughts on this. Is it easier to give a vehicle than to sell a vehicle? It's just, you can't just sell something purchased with public funds. We have, there's a process you have to go through. Is it easier to give it away or harder? No, there's, it's just, it's been done. Let's just share it. And we currently, I will, I will tell the committee that, and you may hear about this, we actually worked on this program, a similar program to this about three years ago, four years ago. And so we have some draft resolutions, some draft stuff already prepared that we can move forward as we go. So not so many, who would actually drive the van, the bus to the, an employee or somebody under. We would hire a supervisor, a supervisor and a driver. Okay. Okay. So no city employee would be. We'd provide the work. Any more questions? Ms. Benjamin Payton, am I missing anything? We're good. I think Councilman Brown hit on the, just the big thing. So we'll, we're going to move this along, but we're also, you got, you got, you've got cleanup to do along the way before it can come to full council. So we would expect it to come to full council before July 1st. Yeah. Okay. Okay. All right. All right. Thank you, Mr. Epps. That was a good presentation. Appreciate you. All right. Anything else at this time? Motion to adjourn. All right. Motion to adjourn. Second. All in favor, please say. All right. Thanks everybody.