First up is public comment. This is comment on anything that's not on the agenda. Hi, training is Robin Goodall on behalf of the library board, just giving to give the weekly update, the monthly update, I guess. We wanted we wanted the board to be aware that the library has closed for, I mean, I suspect you all notice that we have closed for in person, most in person. All electronic actions, business is still happening. We are meeting as a board of trustees weekly with Amy to make sure that we are keeping people safe, both the employees and the public. And we have two employees that are comfortable being in the library, Amy being one of them. And so, able to request materials and pick them up with outside pick up. Right. So, and there are lots of resources, the newsletter is sending out lots of resources and the Facebook page has lots of resources for community members that are at home, especially those that are also doing homeless things. And Amy, if I missed anything, you can feel free to chime in there. Thanks, Robin. Okay. Does the board have any questions about the way the library is proceeding through this very strange time? Sounds good. Thank you. Take care, everybody. You too. Thanks, Robin. We had somebody else join the call. I'm here, Larry. Okay. Got that Shannon. Just helping Shannon keep her note straight. Okay. Moving on to approval of the agenda. I'm sorry to interrupt you. Did have a question. Okay. I wanted to get an update on the status of the climate solutions resolution letter that the select board is supposed to be sending to our state representatives. Trini, I could answer that. This is Adolfo John. The select board discussed the crafting of the letter two meetings ago. Unfortunately, with everything that's happened since then, I have not had a chance to draft it with just issues coming up. But the board has given me direction on what to include in the letter to then bring to the board for their approval before sending to our legislators. Does the select board want any assistance in drafting that letter? If I need any, I'll certainly ask. I think too many cooks in the kitchen would delay the process. It's already been delayed for a while, so. Uh, yeah, but we're also dealing with a global pandemic. And I don't mean to be rude, but there are other issues at hand. Oh, I understand. Hence my offer for providing assistance, understanding that you are busy with other issues. Okay. Do we have any other out there for public comment? Yeah. So this is Amy and I'm here as a representative for the Randolph Area Mutual Aid Network. I just want to give a quick briefing about the purpose and goals of ramen. This is a community-wide network, I guess we could call it, that is working to maximize the services that each organization can offer to support people during the COVID outbreak, also to keep one another informed and to leverage the resources that we share to help support our communities. We serve Randolph, Braintree, Brookfield, and East Granville residents. There are three teams at this point that are active. There's a communications team, a fundraising team, and a volunteer team. The communications team right now is working on saturating as much as we can the communities with opportunities to know what services are available to them. And one of the activities that we're working on right now is to put together an every door direct mailing. A lot of ramen activities right now are conducted the robust web page, which I hope you've had a chance to look at. Also, postings on front porch form and so on. But that does no good for folks who don't live in the digital world. So we're working to get a literature into the hands of every resident so that they know that there are resources available to them if they have a need that needs and they need support. So the fundraising committee, I think that one is pretty self-explanatory. People will be making contributions to support their community members and they're working on a mechanism to make sure that folks are getting the help they need. And the volunteer committee, which is working to take in information from folks who have needs, everything from needing deliveries of groceries to help moving to with the organizations and volunteers who have the capacity to help. So that's what we're working on right now. Do you all have any questions? It's like a lot of coordination and collaboration. It is. And Linda Anderson with Capstone Community Action has been amazing with keeping us on track and bringing together the organizations and agencies that can really support the community at this point. So hats off to her. Absolutely. Amy, this is, and everyone, this is Tom. I've been participating in the weekly Zoom meetings of the Raman group and Ken only second part of Amy has said they've been doing some great work. It's perhaps best for me to bring this up during the other business portion of the meeting, but the Raman group has asked for the towns of Brookfield, Randolph and Brain part or founding members of this network. And I would recommend that for consideration. All that would mean right now is that we would have a official representative, if you will, at the table when we meet each week to represent the towns of the three towns. So I'll just put that out there for everyone's consideration. Also, and Josh has also been participating in the meetings as their time allowed. Seems like the town kind of was a founding member just wasn't official, at least the town. Yeah, we had a discussion among the founding members, especially the people on the communications team, which includes Amy and myself. Over the course of the last day or two as to whether we should officially ask the town through their select boards to be sort of sponsoring members of a group. We didn't want to be so presumptuous as to say in my capacity as the president elect of the Randolph Rotary Club, which is one of the participating entities in this process. But the participation by Josh and Adolph on myself was never really formally in an official town capacity as I understand it. Would we deal with that under new business? We would. We'll just need to understand what founding member means. Like, what is it? Are you looking for the town to be at the table to help with coordination efforts and whatnot? Or is there what's the ask? I think that would be part of the expectation. Right now, all of the participating organizations and I can I can list those later in the meeting if you want or I can list them now. There are 16 participating organizations mostly made up of area churches, social, the Kimball library, the two Rotary Club and the Orange County Sheriff's Office, a whole handful of groups. And really, they're all just participating in the discussion to coordinate different activities. At some point further down the line, there may be a modest financial ask or contribution of some of these organizations. But that's that's something that could come further down the line. It's not an expectation of participation. All right. Do we have anything else under public comment? Hi, Trini. This is Heidi and like to give a brief report on our park. As you know, we closed down our park area. So the tennis courts remain to be closed. We have had a lot of high traffic activity as well in the park and the playground, as well as in the courts, basketball courts. So I have put multiple signs up closed as well as the playground. And that goes along with all the parks and playgrounds across Vermont. And Monday, Sunday night, we had during the afternoon, a couple of kids broke into a shack behind the skate park and a skate park in the ice rink. They pulled out a few old ramps that were there. I found them Monday and I had a afternoon. The police had been notified and we talked to several of the kids. And we are trying to educate them like we get that you're bored and there's not much to do. But you know, breaking into a shack is not one of the things to do. So I'm for this group of bikers and kids in the park is getting bigger. But everything else is cleaned up on ice rink. And all the programs have been suspended. Baseball right now is the only one that we have and that is not, we're waiting for guidance from the leagues and the governor. And hopefully we'll have an answer by next week if the baseball season will take off or if he canceled officially. So that's all from the night. We are still hoping for summer camps. So that's what we've been working on summer and getting that staffing kind of ready. But that's also all pending. And hopefully we'll have better clear guidance within the next couple of weeks. So that's all for the record. I would tell you wherever you are tomorrow at 11 o'clock to tune in to the governor's press conference. We'll do. Yes. It'll be live on TV and it'll also be on Facebook on his thing, usually, and CAX lots of times live streams it on their website. Yes. And we've been, I've been meeting with via Zoom with all the Vermont parks and rec directors. And we're trying to stay all on the same page. And so it's been really great about summer and the pool. Great. Any questions for Heidi? Thank you. Any other public comment? Hearing none, approval of the agenda. I was going to ask if we could add under other business the Raman and other town would be an official founding member. We already added that under new business. We did. Okay. Yep. I'll second. Approval of the agenda. Okay. All those in favor? Hi. Hi. Opposed? Who should carries? Consent calendar. This is meeting minutes and warrants. In your packet you should have the meeting minutes for both the 12th and the emergency meeting on the 24th. Sorry, a move that we approve the consent. All those in favor? Hi. Hi. Opposed? Carries. New business. The VTRANS mileage certificate. This is an annual process with TRANS. They ask us to certify our highway mileage. The areas in your packet say document that included a background material to confirm that we have the necessary items that show that we are up to date with our highway network inventory. So if the board were to approve this, that would just need a motion and we could use our minutes and wall signature as proof that the board confirmed as opposed to all five signatures. Any questions? A motion? I move to certify the town and the roads. I guess the roads, this is the roads, right? It is the roads. Town roads and bridge standards. Is that part of it? It would just be the highway certificate title of document. Yes, certificate of compliance for town roads and bridge standards. Okay, I move to do that. Opposed in favor? Hi. Hi. Opposed? Do we have one opposing? No, no, it was a late I. Just helping Shannon. Next up is the water wastewater payment process. The water wastewater committee met recently to discuss what options they have available to them to rate payers in the water district and the sewer district. Concluding that meeting, our finance director reviewed some of the material that the committee discussed and pulled together an action item sheet for the board to consider and the recommended motion is included. To add more information, this would essentially mimic what the board had previously done with tax payments instead of it being a three-month delay on tax liens or delinquent water and sewer bill penalties. It would only do it for 90 days because 90 days more closely resembles when bills are mailed out for water and sewer payments. Any questions on this? Have we got any feedback that this will be a problem or curious why the committee was discussing that? There may be a requirement for being to place a ballot or an article on the ballot that asks rate payers for permission to retroactively approve this. That is a hold back. The challenge there is that the rate payers can say no and this year it would be roughly between three and five thousand dollars where the rate payers would have to then reimburse the town as a whole if the town voters were to decline this. This doesn't have anything to do with the town wide. This is just the water district. It would just be the water district for delaying of the penalties in the interest. We may still have to go to your right, Trini. We would have to have to go to the water district. That's right. Larry, when this was being discussed in committee, did they see any problem with that passing by the voters? No, the sense of the committee is that it was a good idea. We didn't have the authority as a committee to make this decision, but we did talk about it and that it would be pretty nice for at least a small group of people and thought it was worth doing. I think as it also points out, the amount of money involved is not so large that we really thought that no matter which way it would go, whether the district ultimately approves it or doesn't, it didn't seem to be a significant factor. So when we do our bill for water and wastewater, are we going to be able to, if somebody's delinquent multiple quarters, will we be able to break out that this one quarter doesn't incur the additional fees or are we waiving it for their entire bill? That's one of the things we ran into with the property taxes. Yeah, I'd have to re-confirm with Cliff. My understanding was that it would be easy enough for us to break out this 90-day period for fees and interests, but I would have to double-check with Cliff. Any other questions? That was my impression of the system. If I remember correctly, when we were looking at some of the issues with folks that hadn't paid, it was broken down by line item according to when the billings were made. Dolpho and Trini, this is Cliff. Hey Cliff. So on the billing system, Trini, can you repeat your question again? When we were looking at the, at waiving the penalty for property taxes, it was going to be real challenging to break out for those that had delinquent taxes prior to the period we were looking at. So my concern was that we didn't back you into a scenario where these people, somebody hadn't paid their prior invoice and you had to try to decide how to break out the interests or penalties on just this, this span of time. So I, I think we can do it. Um, because of the way the billing system works, impose interest balances from a certain date back. And, and I believe the motion is written so that the, the board would be putting a similar 90-day moratorium on those bills so that it would be essentially one billing cycle. So if it's the cycle, just so we're clear with me about giving some of the questions we're going to with the property tax piece, if I didn't pay my last bill and I'm already collecting fees and penalties, is that amount that I owed before this billing cycle going to collect tax and fees during this period? Or does it, does the motion need to be done so that there's no tax and interest in penalties accrued during this 90-day period? No, we can, we can continue to accrue interest on the old balances. Okay. That's what we did with taxes, right? That is correct, Larry. Sorry, Pat. That's what I meant to say. Just want to make sure we give you the motion so that you can implement it easy. Yep. And all your spare time right now out doing your photography this morning. That was done last night. That was not done this morning. Well, I can tell you the load of stuff went north today. So A&R was on site and shut them down. That's what I heard. There was a lot of emails going around this morning, too, so. Well, by the time it got to me, we already had A&R on site, so multiple paths. Anyway, different issue. I need just a few more. Okay. So any other questions on this? If not, we'll entertain a motion. I move that we use the suggested motion that it also has in our paperwork. I'll second that. Okay. We have a motion in the second. All those in favor? Aye. Aye. Opposed? Motion carries. Thanks, everyone. Okay. Thank you. Thank you. Next, we have the proposal for swimming pool, sanding, and painting. A draft RFP that Heidi and I have reviewed and pulled together. It has a lot of detail in here that provided by Heidi on what is still needed at the swimming pool. And much of that has been discussed by the board in the past, and it's mostly just the sanding, taking away of the old paint, and resurfacing and applying new waterproof paint. And this is just approval to send the RFP out? Approval to send out the RFP, and then also we would need a motion to allow us to essentially use the $40,000 in the Recreation Equipment Reserve Fund. The bids would come back to the board if we have multiple bids. The challenges, and Heidi's much more familiar with this because she's been dealing with the issue more directly, but the challenge is that there are very few people in our general area that could perform this work. So we may have to do a single source option at that point. The board wouldn't have to review the bid again, but we can bring it back, absolutely. I'm here. Sorry, do you have any questions on it? Well, I think even if you only have one bidder, it needs to come back to the board for approval. Just given the refresher course I've recently taken on our procurement policy. So what you're asking for right now is permission to send the RFP out or RFPs should go out with language in it. It allows us to not accept any of them, so I don't see any harm in sending the RFP out. Okay. Anybody have any questions on that or other thoughts? I might make a move to send out the RFP within the budget amount. I'll second that. I'll second. The motion in the second. All those in favor? Aye. Aye. Aye. She carries. The culvert on the Beansville Road. This is a project that's been ongoing for a number of years.  We are still working or I'm still working with Two Rivers and V-Trans to craft more appropriate. And we're also hoping that the town won't have to pay for this project if funds are made available for infrastructure repair through federal bills. But for now, I would like to ask the board for permission to, once this proposal is complete and has Two Rivers and V-Trans edits, I'd be allowed to send it out if there are some responses or I could also delay sending it out and wait to see what federal legislation does. The reason I say the last part, the latter is because I learned today from V-Trans, our representative, Chris Bump, that they would be more than willing to extend our existing grant form and expire in 2021, which gives us a whole other year to work with. Yeah, I think the only concern I have with sending this out now, the way it's drafted, is it has none of the pieces in it that will be required for a federal grant. Yeah. And there are coming, this is a prime project for us to get 100% funding on. It's awesome that we're poised and ready because a lot of towns aren't in that boat, which is going to make us a lot better positioned to compete for these funds. But I think we need to take this bid document and my recommendation would be to hang on to it a little bit and by next month, whether we're going to be able to go for federal funding for this or not. I like the idea of it being able to be paid for by 100% federal funding, so there's no cost to the town. And I also like the idea of maybe potentially being able to package it with some other projects at the same time, whether it's Maple Street or some of that, if we can get all of them funded. But that would be my... I agree, Trini. Yeah, if it's okay with you and the board, I'd like to just ask the board to skip this for now. I'm agreeable with that. I am as well. Sounds good. Do we need a motion to table or... No, we just skip over it. We'll see you again on another agenda. You might have a little fancier name on it though. I have a question on possible funding. Do we have any idea what applications would be due on? No, they can't even agree on how to put the bill together in Washington. So there's no... You know, the guidance isn't out to the town level yet, but the conversations are taking place, some of what their intent is. I've been in on conversations of that and what types of projects they're looking at. We've got some projects in Randolph that we would like to do that are like expanding water and sewer down the Beanville Road that don't really bring in a lot of long-term jobs, but will allow us to be poised to bring in additional businesses or to connect some of the folks up on Route 12. We wouldn't compete well in the normal funding because they're usually looking for sustainable jobs. Some of this money is going to be looking at jobs that you can get people into right now to do. So they're looking at how many people are going to be employed while that job is taking place versus what the end benefit of the jobs is. The Beanville Road culvert, Maple Street, redoing Maple Street, the water wastewater, connecting the town reservoir, those projects all will compete very well for that type of funding. And they may have already been packaged up and put a bow on and put some big numbers on to try and just give some examples of types of community projects that would benefit from this type of stimulus money that don't compete in normal funding pots. So we'll see what happens. Do you or Delpho have a list of what you're thinking so that the rest of us could be looking at that too? Well, those are the projects that we have going on anyway. So it's connecting the wells, the new wells to the water system, which is a project that's out there that we've been hammering on looking for funding to deal with the Manganese issue. So that's one of the projects that we have. Maple Street and our challenges there. You know, while we're looking at Maple Street and the, you know, you've got the issue of power poles and nobody wants them on their side of the street. So we might still just increase the size of the project and bury all those power lines while we're at it, right?


Hey Lisa, my first value this week is a guy who went nuts on Monday night and that's Pat Fryermuth. He's been awesome of late, he's been a top 10 tight end in each of his last three games over this stretch, he's averaging a target share per game over 20% and over his last two games, he's second in the entire NFL and targets from within his opponent's ten yard line. So he's seeing volume within his offense and that volume is also coming close to the end zone. He's a great value at $5,100. There's also D. Ernest Johnson, the Cleveland backfield could be out this week because of COVID, leaving D. Ernest Johnson with a big workload and the nice part is, is that we've seen Johnson with that backfield to himself back in week seven against Denver that week, he had 79% of the teams running back rushes, he had a 6% target share, he had a top five fantasy football performance. So if everything aligns and he's able to be starter this weekend, he's an absolute steal at $5,400. And then finally there's James Connor, Chase Edmonds is out for multiple weeks with a high ankle sprain that makes Connor a potential bell cow in this Cardinals backfield. Last week Connor saw 68% of the teams running back rushes and he had a 19% target share. We should expect that target share to drop a little bit and I also expect Eno Benjamin, their other running back to be involved in sort of a Chase Edmonds light kind of role, but Connor's getting a bump in work and he's already seeing goal line work in that offense. He's a great, great value at $7,000.


In the last class we started with a rigorous description of NMR, a quantum mechanical description which is called as density matrix description of NMR. We try to derive an expression for the density operator starting from the basic principles of quantum mechanics that how we make a measurement and what is the meaning of measurement in quantum mechanical terms that is when you measure the expectation value calculate the expectation value it represents a measurement and we also said that we have to take an ensemble average to represent the measurement. So as in the end we came up with this expression for the so called density operator rho is equal to 1 by N 1 minus this is the Hamiltonian divided by KT and this is an operator here. So rho is an operator and this represents the total number of states the Eigen states and that depends upon the spin system what we might have whether it is one spin system or two spin system or three spin systems and so on and what is the value of the I for most purposes we will be dealing with I is equal to half systems and we also said we will represent the Hamiltonian by the interactions and by and large it is represented by this simple expression here this is minus gamma h cross h naught and this h naught is the field and I z this is the z component of the angular momentum this is the I z operator. Now since the density operator is intimately connected with the angular momentum operator so we have to look at the properties of these operators in some detail. Now here I would like to mention that since this chapter is going to be quite intensive with regard to the operators and matrices, matrix representations, angular momentum and things like that it may be a good idea to take a small digression and spend some time on the properties of the spin space for the benefit of those who did not have sufficient exposure with quantum mechanics operators and so on and so forth. Therefore we will go slow here and try and calculate this individual we will introduce this concepts clearly and explain from the first principles how one can calculate these various elements. So now this is the same equation which we wrote earlier in the first class that any state is represented by wave function the wave function is written as superposition of the various states these are so called basis states they are form a basis set all the UMS are the eigen states or the form a basis set and this is the superposition of all the basis sets and you have the CMTs these are the coefficients which contribute to the superposition. In a simpler notation we simply write it as summation CMM and the summation goes over all the states of the spin system. So if as I said this goes from M goes from 1 to N the spin states satisfy this identity this identity also we use last time however let us come back to this identity and try and show how this comes and try and prove this. So we use this identity in the previous class and this M is the one of the eigen states or the element of the basis set and with this summation runs over all these M's. Let us take one of these element this one which is written here I take this here and I will show you that this is actually an operator. So this if I operate on the my wave function psi of t then I will get it like this this operator as it is and then for the psi of t I write this now it just instead of M here as the index of summation I use N it does not matter this just runs over all the states whatever index I use here it runs over the all the states. So summation CNN okay now this CNS are numbers okay so this CNS are numbers these are not operators therefore I can take this entire part outside of this operator term and then I put the summation here instead of putting it here then I say then CN comes here then I put the same thing back this M this is the ket this is the bra and this takes an element with the another ket here N. Now these are the sum in the expression gets simplified into this. Now you remember we also said that these states the elements of the basis set these are orthonormal. So what is the meaning of orthonormal that this bracket here this element is either 0 or 1 depending upon whether M is equal to N or M is not equal to N we write that as a chronicle delta. So this one is therefore is written as chronicle delta NM this delta NM is equal to 1 if N is equal to M or it is equal to 0 if N is not equal to M. Therefore this summation once again running on various N's here CNM delta NM this will simply become equal to CMM because for all other N's which are not equal to M this is 0. So therefore only one element of this survives and therefore we have here CMM. Now I take instead of this only one particular element which I took last time I take the whole summation I take the whole summation of all this. So therefore you notice you notice here that this is essentially projecting out one particular state here from the entire wave function. This was psi of t here and eventually I have got here only one state with its coefficient CMM. Therefore this operator this is now you understand this as an operator this operates on the wave function psi of t and gives me only one particular state therefore this is called as a projection operator this projects out the particular state M out of this psi of t. So now I am taking the entire summation of all these operators M and M all the M going from 1 to N. So this is the same here and this operator giving on psi of t give me CMM and this summation is basically psi of t again we have seen that this is in the definition. Therefore this equation if you see if I take out psi of t from both sides here then it simply reduces to this equation this is the identity which we actually used earlier and we have proved it explicitly here for the benefit of those who are not familiar with this principles. But these are generally used routinely in all various kinds of calculations in quantum mechanics and quantum chemistry and therefore this is very important to understand how these things come. Now let us do explicit calculations with regard to the angular momentum operators we said the density matrix is very intimately connected with angular momentum operator IZ and we will also deal with other angular momentum operators in due course and therefore it is important to describe the spin states and the representation matrix representations of the angular momentum operators. Let us start with one spin with I is equal to half in one spin how many states do we have in the presence of the magnetic field well there are anyway two states in the presence of the magnetic field they are non degenerate they have separate energy levels in the absence of the magnetic field they will be degenerate but nonetheless the two states are present and because we said the IZ is a quantized state and we have only two possibilities there therefore we have two states alpha and beta and these are the spin states and these are also the Eigen functions of the IZ operator as we see here. Now what are the properties of these states that is integrated here in this matrix element alpha beta goes to 0 whenever I say alpha beta actually in the conventional nomenclature those were used quantum mechanics the integral forms actually this ket this is called the ket this is actually the wave function psi and this side which is there which is the bra this is the complex conjugate. So therefore when I write like this it is essentially the integral psi star psi okay so therefore in the case of the spin states we simply write this as the bra here and the ket here and therefore this is orthogonal and therefore this is 0 all these basis elements of the basis set are orthogonal in nature and if I do that with the similar states alpha and alpha here this is equal to 1 beta alpha is equal to 0 beta beta is equal to 1. Therefore we say these states are orthonormal because this is normalized to unity and these individual matrix elements these are 1 here and the cross terms are 0 and therefore these are orthonormal basis set there are only 2 states right these 2 states are orthonormal. Now we also saw earlier that I z which is operating on the state alpha if it is m and this is the m value is this we defined in the very early stages of this course that this is the m value this gives me half I z operating on alpha gives me half alpha I z operating on beta gives me minus half alpha because these are the m values for beta the m value is minus half and for the alpha the m value is plus half therefore this gives me half and minus half and you see the same state is back here therefore these are the Eigen value equations and alpha is the Eigen state of the I z operator and beta is also an Eigen state of the I z operator with the Eigen values half and minus half. Now this I x alpha this we can take it for granted although this can be proved later this I x alpha is now giving you half beta. So I x alpha gives me half beta I x beta gives me half alpha notice here the alpha I x operator converts the alpha state into the beta state and it converts the beta state into the alpha state in fact this is what was used when we described the selection rules for energy absorption by the RF. The RF was applied along the x axis and the I x operator was coming as a perturbation by interaction of the RF field with the magnetic moment and then the I x or the I y operator is coming there and because of the change in the state here then the delta m is equal to plus minus 1 selection rule was derived as a result of this. So and that becomes conspicuous again when we will actually try to calculate this matrix representation of this I x and I y operator. Likewise if I do I y on alpha it gives me I by 2 beta and I y on beta gives me minus I by 2 alpha. Once again this alpha state is converted into the beta state here and the beta state is converted into the alpha state and therefore these are not Eigen value equations because the Eigen function is not retained. There is a change of state and but with a certain kind of a result here as a coefficient. So now I calculate the matrix elements of the I z operator. The matrix elements of the I z operator meaning what? So I have this sort of calculations I have to perform. I z in the middle alpha here, alpha here, alpha here, beta here, beta here, alpha there, beta here and beta here. So these four elements represent a 2 by 2 matrix of the I z operator. So this is called as the matrix representation of the I z operator in the basis set of the two states alpha and beta. You have the two states alpha and beta here, these form the ket individually and these form the bra individually once alpha and once beta and here the ket once alpha, once beta. Therefore I have a 2 by 2 matrix here. Now you notice I z alpha gives me half alpha. So the half alpha and alpha, this alpha alpha goes to 1. This is what we saw in the previous slide and therefore this results gives me half. Now I z beta gives me minus half beta as we saw in the previous slide. So if I take minus half out then I have the alpha beta here and that alpha beta has to be 0 because these are orthonormal. So I z alpha gives me half alpha. Therefore the matrix element will be half beta alpha and that will be 0. So and here I z beta gives me minus half beta. So therefore I will have minus half outside then I will have the beta beta and the beta beta is equal to 1 therefore this is equal to minus half. So if I put these four elements in the matrix then I will get here half 0 0 minus half. But I take the half out then I have half here 1 0 0 minus 1. So basically this will become the matrix representation of the I z operator for a single spin I is equal to half. Similarly we can do that for I x operator. Now I x operator if you remember I x converts alpha into the beta it gives me half beta. So the half beta means I take away half out then I have the alpha beta and that will be 0. So therefore I know if we see I x beta so I x beta gives me half alpha then alpha alpha is equal to 1 therefore I have here half. This was exactly the same thing what we used when we were actually calculating the selection rules with the perturbation magnetic moment interaction with the RF field and the operator part was I x there. So you see the difference of delta m between these two is plus minus 1. So therefore that is how we get a selection rule of plus minus 1 for RF induced transitions. Basically it comes from this sort of a calculation. Beta I x alpha if I do the I x alpha converts alpha into the beta state I get a half beta. So beta beta is equal to 1 so I get a half here. So here I x beta goes you half alpha and therefore beta alpha gives me 0. So therefore if I write it in the same manner here so alpha beta here alpha beta here these are the two kets and these are the two bras. So I x is in the middle so I will have four elements here. So I will have half here sorry 0 here half here half here and 0 here. Therefore this and if I take out the half half 0 1 1 0. So therefore this becomes the matrix representation of the I x operator. So similarly if I do for I y I y alpha I y beta and these ones now notice here I have a I coming here these are imaginary. So I have 0 minus I by 2 I by 2 0 and therefore I have here half 0 minus I I and 0. So this will be the matrix representation of the I y operator for a single spin half. Now if we eliminate this half we remove this half the matrices which are present for the I z I x and I y. So they are simply 2 by 2 matrices in I z you have 1 minus 1 0 0 for I x you have 0 1 1 0 for I y you will have 0 minus I I 0 and these 3 2 by 2 matrices are called as Pauli spin matrices for single spin and these are the ones which you actually calculated and these are called as Pauli spin matrices. Now let us turn to 2 spins. So 2 spins consider a and x I a is equal to half I x is equal to half. Now the individual spin states of these 2 spins are alpha a beta a and alpha x beta x. Now for generating the basis set for the 2 states we take the products of this individual spin states. This was done in the second chapter when analysis of spectra was described when you wrote the Hamiltonian you try to calculate NMR spectra using the quantum mechanical principles this also was done in a very generalized way. So for the 2 spins we calculate the product states for the 2 spins these will be alpha a alpha x alpha a beta x beta a alpha x and beta a beta x these are the 4 states which constitute the basis set. So 2 by 2 by 2 2 states for this and 2 states for this. So therefore I will have here 4 states. I represent these 4 states in the energy level diagram like this I have I label this alpha a alpha x a state number 1 alpha a beta x a state number 2 beta a alpha x a state number 3 beta a beta x a state number 4. And now the angular momentum operator a consolidated angular momentum operator for the 2 spin system with the I z or the I q or I x or the I y will be the sum of individual operators. So I z here for 2 spins will be I z a plus I z x I x for the 2 spins will be I x a plus I x x and I I y for the 2 spins will be I y a plus I y x. So therefore in abbreviated notation we write like this I q is equal to I q a plus I q x where q goes from x changes from x to x y and z. Now what are the properties of this spin states? So this will follow from the general properties of the alpha and the beta states if I label this now I will label the my 4 states as 1 to 4 right. So here therefore I take the products of the individual states I j then this is equal to delta I j the I j can go from 1 to 4 right. So the 4 states which are there so these are orthonormal states of the 2 spin system I j is equal to delta I j. So therefore how many matrix elements will be there for the 2 spin states since there are 4 states. So 4 ket states and 4 brass states and therefore this will be 4 by 4 matrix for the I z operator and likewise for the I x and the I y operators. So it will consist of these elements I z 1 1 I z 1 2 1 3 1 4 2 1 2 2 2 3 2 4 3 1 3 2 3 3 3 4 4 1 4 2 4 3 4 4 and these are the matrix elements we have to calculate. Let us illustrate this how these can be calculated. So let us calculate I z 1 1 I z 1 1 is 1 1 the 1 state is alpha a alpha x right.  Notice each of these operator operates only on its own spin state and it does not operate on the other spin state. For example I z a will operate only on the alpha a it will not operate on alpha x. So therefore if I am calculating the matrix element with respect to I z a I can take out the alpha x state. So this is what we shall do now if I have this 2 states here now alpha x I can take it out if I am calculating I z a here so I separate out these 2 into separate calculations. So in one case I have the I z a in the other case I have the I z x here. So if I have I z a here I keep this alpha a here then the I z a and keep this alpha a here and this alpha x this alpha x come out here as a separate entity alpha x alpha x. Similarly plus alpha x I z x alpha x that is if I am looking at this operator then alpha x alpha x these 2 are taken here and these alpha a alpha a come out separately here. Now because of the orthonormality of this this is equal to 1 and likewise this is also equal to 1 and what does I z a give me operating on alpha a it gives me half alpha a. Now therefore half alpha a half we take it out alpha a alpha a that gives me 1 again and therefore I have a half here coming from this term this is 1 and this gives me half. Similarly this gives me half and this gives me 1 therefore half plus half this gives me 1. Let us do for I z 1 2 I z 1 2 is alpha a alpha x here this I z remains the same and here I have the state number 2 which is alpha a beta x. So I do the same as I did here before so I have here alpha a I z a alpha a those are these states are taken alpha states and alpha x and beta x come out here that because I z a operated does not operate on the x spin states therefore these alpha x beta x come here. For the second one when I am putting I z x here I have to use the x spin states only therefore I have here I x alpha x and then I have the beta x here and the 2 alpha a alpha a come out. Now notice here because of the orthogonality this term gives me 0 here alpha x beta x is 0 whereas this gives me half on the other hand here this gives me 1 and this gives me 0 again because this is half minus half beta x and alpha x beta x gives me 0. Therefore both these terms actually become 0 and we vanish ok and likewise I z 1 3 and I z 1 4 will be 0. So let us do the same thing for I z 2 2 I z 2 2 is now we have the 2 state here and the 2 state here alpha a beta x alpha a beta x and I z a plus I z x. So this is alpha a I z a alpha a which is easily followed from here then I have this beta x beta x here beta x I z x beta x and alpha a alpha a. Now what do I get from here I z a on alpha a gives me half and therefore and this is 1 alpha a alpha a is 1 and therefore I get a half here. Now what do I get here I z x operating on beta s gives me minus half beta x and therefore minus half beta s beta x is 1 alpha a alpha a is 1. Therefore I get a minus half these two terms will then cancel I will have a 0 here. So therefore this the second term 2 2 will be 0. Now similarly you can do this calculation for all the other elements of the matrix and you will find that in the second row I z 2 1 2 3 2 4 and in the third row 3 1 3 2 3 3 3 4 that means all the four terms and in the fourth row 4 1 4 2 4 3 these will all be 0. So therefore in the first row also we said I z 1 1 was non-zero and other three terms were 0 so earlier. Now I z 4 4 also we can calculate and this is in the same manner when you calculate it this is beta a beta x beta a beta x here and I z a I z x in this terms. So this will turn out to be minus half minus half and this will be minus 1 because I z a on beta gives me minus half and I z x and beta x also gives me minus half. So therefore both will be minus half minus half and this will be minus 1. So totally therefore if I see list all of these elements here you can see that in this I z matrix I have only these two elements which are non-zero and all the other elements are 0. Therefore for the two spin state the two individual I is equal to half spins the I z matrix representation will be simply this. Now for the I x we can do the same thing I x also we will have this four elements here and now let us try and calculate for some of those. So if I can I x 1 1 this is alpha a alpha x I x a plus I x x and alpha a alpha x here we do the same trick as before.  Now notice here I x operating on alpha gives me beta that is this here whereas this remains the same and I x operating on alpha gives me beta. So therefore I have alpha x and beta x and notice these ones are 0 this is 0 and therefore this will be 0. So similarly I x 1 2 if I calculate I have here alpha a alpha x for the 1 and alpha a beta x for the 2. So here I have therefore the same expansion I x a alpha a alpha x beta x I x alpha x beta x alpha a alpha a. Notice here this fellow actually gives me 0 alpha x beta x gives me 0 right and what about this I z x operating on this is equal to 1 I x operating on beta x gives me alpha a half alpha x therefore alpha x alpha x will be equal to 1 and then I get a half therefore I will get 0 plus half and that is equal to half. So similarly you can calculate for these elements I x 1 3 I x 2 1 I x 2 4 3 1 3 4 4 2 4 3 all these will be equal to half and the remaining elements will be 0. So thus your I x representation will look like this all the 4 elements will have a half here this 1 1 here 1 1 here 1 1 here and 1 1 here and all the other elements are 0. You can do the same exercise with I y and then it will turn out that you have half here it will be very similar to this except that instead of 1 you have this I elements. So you will have minus I minus I minus I minus I here in the upper triangle you will have the minus I's and in the lower triangle you have I I and I I here. So this will be the matrix representation of the I y operator for the two spin states. I think you will stop here thank you will continue with this discussions in the next class.


Hi guys, it's Alex and welcome back to my channel. If you guys are new here, make sure to subscribe and also give this video a big thumbs up for what's in my purse videos. I haven't filmed one of these videos in forever and I have been loving my new Louis Vuitton favorite. I did a whole unboxing of this bag so if you wanna go check that out, I will link it down below but this is the Louis Vuitton favorite M.M. So it's the medium size and it is in the Damier Azzur print. I think this bag is the cutest when it's in this print and before we get into what is in my bag, this is a magnetic flap but it is very secure. So if I were to tip this over like this, as you guys can see, nothing is falling out right now so it's a pretty secure magnet. I've never really had a problem with the magnet so I just wanted to kind of say that and also the strap does kind of scare me because it's Vachetta and it's like untreated leather but you know, I bought a purse to use it so let's just get onto what is in the bag. So opening this on up, it just looks like this inside. You guys can kind of get a little peek. I just wanted to kind of show you guys how I organize my bag. Starting off the little back pocket, there is a little back pocket here. So basically I just have like gift cards, receipts, gas card, target, gift cards I got from Christmas. Not too fancy but that is what I keep in the back pocket. When I get my new business cards, they'll also probably go in the little back pocket too along with all of these cards. Okay so I'm just gonna go front to back so the thing that I keep in the front is my phone. So this is the iPhone XS Max that kind of gives you an idea on how big this little bad boy is. It totally fits in here, plenty of room. I love this bag because it fits so much but yet it's so compact. And then moving to the side, this is my vlog camera. It is the G7X so this fits perfectly just like in the side here. It just fits amazingly. Like I don't know, this bag was like meant for me because I love the easy access and the reason why I wanted this bag is because of the magnetic flap. I can like open it up, whip out the vlog camera and just start vlogging. So this is also in there, almost all the time. This is probably the one thing that sold me on this bag. I was able to fit a Ray-Ban's case in it. So basically this fits perfectly inside. I can keep my glasses in there so these are just the ones that Spencer got me for Christmas. I don't really know the name of them. I always love to keep my sunglasses in a case kind of like this so they're protected. Okay, and then I have my wallet. This is just a little Kate Spade card wallet. Very compact. I love these little things because they're perfect to like carry in your pocket or carry in your purse. And basically they're all the cards that I need. And then I just have some miscellaneous stuff in here. I have my keys. I normally lay my keys on top right over or next to my wallet. Because I know my cousin Perry watches this. Yes, I still have the overly priced key chain on it. I told her that I would probably always use this key chain until it died because I paid way too much money for it. But I just thought it was so cute because it was like a little tie.  And I also have some Tylenol in here. And then I have a pen. I also have a lipstick. And being the vlogger self I am, I have a camera battery and I have two pennies that I found. So that is everything in my purse. As you can see.  Oh, and I have a dollar. I forgot that. But that my friends is everything in my bag. That's pretty much everything that I keep in here. If you guys are interested in a review on this bag, I will be happy to do it. So let me know in the comments down below if you would like to see that. I think that's pretty much it for this video. Make sure to give this video a big thumbs up. Don't forget to subscribe to my channel and I'll see you guys in my next video. Bye guys.