a back on so you can see my facial expression so i'm not trying to hide here it was actually frozen on on a yeah yeah i sat down for a while while it was frozen it was kind of a nice break um so uh sarah this was proposed two years ago also this exact same proposal and it was withdrawn any comments as to why it was withdrawn before what were the concerns that caused it because it was never actually presented to the select board i don't it was removed from the select board's agenda two years ago i i did not go back and do that research andy so i my recollection was that it was presented to the select board um and the trustees because lauren and i made budgets with this change and then we had to scramble and make them without the change um but i think that there were concerns about ownership of equipment and that was the first um thing that came up then and then also let's i i can't remember the other any of the other concerns but i mean we could go we could dig it back up i uh i did just update lauren's memo so i we could uh go pull up the old minutes and see what the discussion was yeah i recall that it was it was included in the package but then it was at the meeting it was we were told it wasn't going to be discussed so i guess i guess it was my sense that the trustees backed away from it so i was just wondering if those concerns were still there but the trustees aren't here to speak for it so i i guess maybe it's an appropriate task i seem to recall that there were there were other funding things that they needed to the trustees needed to address regarding a increased transfer to their capital budget at the time that made them decide not to go forward with right right yeah yeah okay i'm just jarring some memories that there was there were other what they considered more important things in their uh whatever whatever was needed outside of this question okay yeah i remember that now you're right okay thank you pat anything i don't think so i mean i think initially last year um i wasn't really thrilled about the idea of eliminating the town highway tax um but in you know conjunction with everything else it seems like this is a much more sort of equitable solution maybe just kind of a random question but if the town maybe should say to be highway tax is eliminated would that be the last the to be gets exclusively levied against them if i think that's correct the only outstanding item correct so that's correct that's the only tax rate that's levied only on the town outside the village so i think there's a lot to like about this just from a the perspective of sort of bringing everything in line um you know and obviously simplifying it from your end so um i'm all for it okay and sarah just to sort of very simplistically repeat it the transfer is is basically a press we've already had this precedent because we transferred the village's public works budget into the town budget a couple years ago that has gone smoothly and if the village needs to the select board will have the ability to reject it if it goes over a certain percentage the increase in the budget when the village sets it on their annual basis correct okay correct well per the public works mo you if it's going to be more than a six percent increase it has to come to this left board for approval right and so the public works mo you is the document that we jointly reviewed two summers ago i believe and moving doing this move that we're talking about today was a recommendation of that review of the mo you so this is just a continuation of an existing agreement that we have with the town in the village and it's not necessarily something significant and new that we have to like do a whole bunch of additional work on this is like ready to go and we can take care of it and eliminate the tov highway tax at the same time correct okay well um and so in order to accomplish this sarah you need us to do two things you need us to approve the elimination of the highway tax in one motion and then you need a second motion for us to approve the village's highway budget increase above six percent correct i have no problem with them being the same motion but i do just want it on record that we are we are committing now to doing the six percent and the elimination of the highway tax because this only has that that's that's minimal impact if we do both things right and then the third aspect of it is the trustees have to agree to do their end right i tomorrow night i'll ask the trustees if they're okay with this and then wednesday's village budget day so here we go andy thank you for waiting for me to finish my questions go ahead um this is a budget question so i have reservations about committing to this without seeing the entire budget i'm i'm i'm fully in support of doing this don't get me wrong um as you said i've been asking for this to get rid of this highway tax for for quite a while and i and i i agree this is a uh clean and slick way to do it but i i um without approving it in the context of the entire f y 22 budget i'm you know i'm i'm i'm fine with giving you a consensus that yeah we're happy with it but i don't know that i'm i don't think that we should commit to it without seeing the entire budget that's that's a i can understand that i think that's you know i can understand where you come from coming from um i'm gonna be maybe a little more frank than i normally am if you're if you're not okay with with the concept outside of the budget then this isn't the year to do it i think i think we we just have you have one of me a lot of moving parts um and and i direction either way is really what i need in order to move forward wednesday the village is going to look at their budget and then it's it's time to compile the town budget i'll have that for you before christmas i'm hoping to also have you know fund balance from the auditors but i have a list of a couple dozen things that they're asking me for as well and and other looming deadlines like i just it's it's hard to make all the timing line up with the with the moving pieces and sarah the village needs us to act as well you know so that they can know that if they do it for their but they'll they need to know this from us for their budget day which is this week right right their budget day is wednesday now they ultimately vote on their budget after the town votes on their budget so we could go back in and change it in january after town budget day you know there are a lot of special requests of me and this would just be another one of them that would have to be prioritized and then and i would say for those in sarah close your ears for a minute uh there are dozens of decisions to be made in the budget that if you want to hold off on some of those they may have a little more uh flexibility and elasticity than this this is kind of a decision you make and it sets a framework for the budget and some other things and then there's the department budgets that if we nipped and tucked a bit here and there we could get to a place where um you're more comfortable but you won't know that until you go through it but this is kind of the first big domino that needs to be toppled to get to the end um and i wanted to remind people just just it's been a long night please remember we did a zero rate increase this fiscal year knowing that it was covid knowing that it was going to be hard times we are also going to be digging back out of a zero rate increase so there there's there's some gymnastics that go on with this and sarah's bending and twisting as best as she can thank you but this is the first domino because it sets some direction not only for the town but for the village budget i will say it is a lot easier with the new program because i have multiple scenarios set up than with all the spreadsheets but it is still i would still ask for direction one way or the other and in and i don't whatever way it is i'm fine and as we've mentioned before i mean i've been here almost three years this has been on your goals um for well before that i believe this tax was upwards of almost eight cents seven seven does that sound right it was it was fully eight it was fully eight it's fully eight we are down to point one uh one this this would this is something that's been on the wish list we're this close um so all right um if there aren't any other uh board questions or andy yeah i just i'm just going to say we do plenty of things based on select board consensus and uh again i mean last year's increase was over seven percent so it's been if this is be two years in a row where we've had a bigger than six percent increase and and i'm not saying that that's a problem i just want to i just want to look at the entire budget sure um and i i'd i'd i'd like to go forward with just consensus and not make a commitment here because again i haven't we we don't have the context of the entire budget what definitely i'm comfortable with that i will yell at ebbon if it all gets changed at the last minute and he already knows that wait wait wait wait wait why is don smiling at that why are you smiling that she's going to yell at me you gotta figure out how to make your camera for you so she can't i'm sorry i forgot to get the camera off for i did that put the cat back you put the cat back on your screen okay so andy you're requesting that this is not an actual vote but rather a consensus that would be my vast preference given that i don't have the context the entire budget correct our board members okay with that i am fine with that i'm fine with that um there's also public with their hands up a lane right right i just want to double check and make sure as long as it is sufficient for sarah to do the work that she needs to do and as long as it is sufficient for to allow for decision making at the villages level because they have it's dependent upon that then consensus works for me as well okay so before we move on from this business item are there members of the public who wish to talk about this particular item greg i don't know if you can like clear all the hands or i'm i i can't even make sense of how many hands are up right now and whether they're left over or not i'm sure if i can i this is seven i see ken and ireen and that's he done has her hand up that she done can we confirm those before we start and then also that's all i see and tracy delphi okay ken saying yes okay so ken ireen betsy uh betsy tracy harlan okay go ahead ken thank you thank you madam chair just a little bit of trivia for you on the highway tax was actually a whole dollar from 1959 through 1966 those years prior to that it was on point 75 and then point five since 1972 when our charter passed we've gathered up a whole 20 million dollars in highway tax and dollars at that time not a corrected for inflation i'd like to suggest that this is another um we lost you ken we don't have your audio ken okay let's let's move on to ireen and then we'll try ken again ireen are you able to speak she's muted at least on my screen greg you with us i'm here i don't have an option to unmute ireen though i seem to have an option myself if that helps anyone ireen if you i think you can unmute yourself maybe ken are you back hi there up there's ireen go ahead ireen they're just to refresh people's memory um i was on the select board two years ago three years ago when um the village first thought about transferring its rolling stock and the issue that was the stacking point was that if the town was going to pay for it the labels on the trucks would have to change and that was where the trustees at that joint meeting drew the line and said you know what we'll find that 200 000 dollars ourselves thanks okay ken can you try again we're gonna drop down to betsey i'll aim my net go ahead betsey you're muted i'm unmuted i can hear you now go ahead my hand is down i don't have it up i'm looking at my screen on my hand set up oh okay okay never mind carry on uh tracy oh go ahead wait i'm sorry ken is back can give it a shot uh yes the wonders of technology there we go so where was i um i'm a little bit uncomfortable with the whole mou concept because it ends up that part of the town ends up contributing to another municipalities operating expenses and now capital without gaining any control over that budgetary item i can't vote on a village budget yet i'm going to be effectively paying part of their operating and capital expenditures so i know that i sometimes come up with somewhat radical ideas that um that very entertaining such as the zip code concept so here's a more radical idea for you let's get it over with let's do with public works what we've done with police let's roll up the village public works department into a town public works department that takes care of the whole town to make life simple for everybody village won't have to have a budgetary sequence for public works they'll come to it doesn't sound so good huh so the village would come to the but the the town budgetary meeting and lobby for expenditures that they would like we would get that those economies of scale i keep on hearing that we're going to be getting but don't ever seem to see and i think that would be a better approach so just to underline i'm just a little bit uncomfortable with this concept of part of the town paying for another municipalities expenditures without having a vote on it that's it for me thank you thanks ken um tracy she's trying i think there we go there she is uh this is what we in the it field like to call job security um so sort of along uh ken's line of thinking my my immediate thought when i read the packet was is this a precursor to merging departments okay i mean i think you know along now those same lines both the town inside the village of the town outside the village obviously we all have a need for public works we all have a need for rolling stock if we can leverage those savings by only having one i think that's something that we should do or at least look into after merger if it goes through we're going to have to do that anyway so this would kind of give us a leg up on that my other question is in the packet the recommendations number three talks about village highway transfer to rolling stock into the f y 22 village highway budget but the table um the second to last row talks about the effect of adding village highway rolling stock to town budget um so is which one is correct i guess both of them are tracy so one of the challenges with things that are so complicated is that they can see you know is is that we have all we have all of these moving pieces and so in the village budget what has to happen is that there are different segments of each of the municipalities budget and in the village they have a segment just for transfers and miscellaneous and that's currently where this transfer exists and this is where it's budgeted year over year and that segment is raised entirely on village only tax levy and then there's the village highway operating budget which is a different segment in the budget just like finance or administration manager it would be in the town budget for example so the in that segment of the budget when we look at it in this entirety is then moved the net the sum total of the expenditures are moved into the town budget as a single line which then gets funded through the town budget so the trustees have to approve the change to the structure of their budget which moves that transfer into the town into I'm sorry which moves from the transfer segment into the village highway operating budget so that we don't have pieces that are funded differently in different places so that kind of meet ends up tidies up the village budget and then that all gets funded through the town budget does that help answer your question it does thank you um and just one last comment that I have in dealing with MOUs when I put my professional hat on I get nervous whenever you know if a binding boat is taken to oh let's agree to increase it without a maximum so I would just echo Andy's line of thinking as far as waiting to see what those actual numbers are so that you can put you know a maximum number on that if that makes sense and thank you thank you okay I think I see so I have Harlan to go still and now I see Lorraine and Greg am I missing anyone else that is all that I see right now okay thank you go ahead Harlan hi can you hear me yes we can actually I did not have my hand up but since you called on me and there's been a conversation going on um to simplify or at least simplify it in my head Sarah um basically the village used to pay for all of their capital on the rolling stock and the town through the highway tax used to pay for their capital on the rolling stock and now we're just going to combine them all together and everybody's going to pay for everybody's rolling stock sad short and simple and correct you're muted Sarah no I am I'm just trying to uh think about where where this the super simple version might miss something and I don't think that it has so it it just like all taxpayers pay the same effective rate for highway operating now all taxpayers are going to pay the also pay the same effective tax rate for um all highway that's funded through transfers to capital out of the operating budget so yes does that then I just say what you just said we're all paying for all of the equipment that services both the TID and the TOV on the capital end of things that's basically what we're doing is balancing it out so that all of the taxpayers are paying for all of the equipment right that's yes for all the highway rolling stock so this does not include like this doesn't move into fire right yep no no totally get it totally understand that piece this is all we're all talking highway here yeah okay um thanks for the opportunity I didn't have my hand raised but as the conversation went on it was like wait a minute so I'm good thanks thank you okay Lorraine and then we're going to bring it back to the board hi thanks Elaine um Sarah I I can watch you all night long I wish I had you in college or my calculus classes so it's just I know um but I uh one of the questions is because I too like Ken I would love just to see the whole budget merged and into like the police and just make it one and I think that would make it easier for you as well in the staff member in terms of chewing up labor too um but with that said and now that I'm living outside the village and no longer in the village um if we because the village in a way because they're part of the town they are currently paying for a rolling stock on top of that or does that village uh I mean that highway tax cover the whole rolling stock for the TOV area so it without pulling up the town capital budget which is funded through the two cent capital tax that's levied on all taxpayers the amount of funds that are raised by the town outside the village highway tax annually approximate the operating transfer to capital out of the town highway budget which funds so that transfer then gets transferred into the capital fund and then it funds the town rolling highway rolling stock plan over time so like some years we spend more some years we spend a lot every year we put an equal amount in that's the same way that the village does things um if there is on the town side if there is not enough money in that fund from the operating transfer it live it exists as part of the larger capital fund and there's the two cent capital tax that can be that can be leaned on and that would happen during the budget process if that were the case the last few years that I'm familiar most familiar with since I've been here we have not needed to do that but I wouldn't go as far to say that it's it's never going to happen or has right because you can't predict that right or maybe you can so so do that um and then does the the rolling stock for the town does that service both the village and the town well generally no however the town and the village highway departments work together often to support each other and if the town has a piece of equipment that's down that the village has they'll they'll share and and they work their open lines of communication they're often you know because of the public works mo you because of the way that the public works departments have been working together more and more we have more of this blending so strictly you know I think that the village equipment primarily services the village the town outside the village equipment primarily services the town outside the village but just like we have a town engineer who provides services back and forth and you know this there's collaboration oh that's yeah there's a lot of collaboration between the two departments that starts to blend that line so shared services basically so that the village rolling stock also at times has serviced the town as well and so because Dennis serviced both the village and the town um in a way even though we don't have control as to veers for the village rolling stock in a way you kind of do ish because because of the because he serves both the village in the town outside the village am I getting that right I'm just trying to understand this yeah I think so also well because the thing that concern me was legally is just it's weird to then pay for something that you don't own you know like in terms of not having the representation and not be able to vote on whether you add that rolling stock or not and whether that eventually increases my taxes without being able to vote on it so that that's where I get concerned so I just want to make sure I'm understanding this right it would be awesome if this could all be one budget and all so simple and one vote and one rate I wish I would be offered just doing one town I mean really in terms of merging stuff I it seems to work for the police I echo what Ken says I would rather just see it one department so but thank you for your time so would we awesome we're all on the same page I think one more person has a question Brian Sheldon and then we need to bring it back up okay never mind he's withdrawn his question okay so back to the board we have a consensus that we are okay with the elimination of the town highway tax and the highway budget increase above the village highway budget increase above six percent but we do still will need to vote to approve that I'm assuming however we could do that as part of the overall approval of the budget that is correct I've got to think about the timing yes yes because when you guys the select board approve the town budget to be warned then there is still enough time for me to make any final changes to the village budget for it to be warned the challenge becomes if there are changes made to the town budget on the floor that affect the village budget there's not enough time to then get them into the village budget before village budget day so yes so sorry to answer your question Elaine yes there's no floor changes to the budget this year right right because it's going to be australian ballot yep well Sarah all of that said if there's anything you need from us that you need official decisions on don't hesitate to ask all right and mad madam chair if I can just one quick thing since I'm the manager of both the village and the town and the rolling stock budget has to come through me before the board see it and our goal of alignment and consolidation of service we are trying very hard and being successful not purchasing duplicate equipment we are trying to make sure that we have just enough equipment we need so yeah we try not to buy two end loaders if we don't need two end loaders we try not to buy two you know so one time the town will buy one type of piece of equipment and share it with the village and the village buys another piece of equipment and shares it with the town so we're not both buying very expensive equipment that sits in the shop for three days waiting for somebody to use it they're being used a lot so there you have it thanks seven okay so thank you everyone for the discussion and the questions and Sarah thank you as always for the really clear explanations you really help us make some complicated stuff understandable and I hope that we are doing just a little bit to make your job slightly less complicated all right business item five f which is what we moved from the consent agenda earlier in the evening the minutes of November 16th don you had some edits you'd like to make I do aligned 150 to 152 should say that Mr. Signorello said that he developed a chart going back to 1950 and village residents held the majority of the select board seats for 20 years the way it should read that's what I wrote anyway okay and I'm line 158 where it says the other six six should be replaced with five Kathy do you have that confirmed oh no that's not Kathy Kathy can you just confirm that you've gotten that change okay she's got it great thank you Kathy no that was Tracy I was just telling you that Kathy can't unmute because she's an attendee right now I understand she also said that she got it so thank you um okay so then we need to approve those minutes separately as amended would anyone like to move that don't move don't move thank you dawn is there a second to approve the amended fence any further discussion all those in favor of approving the minutes of November 16th please say aye aye opposed okay thank you now we do have some executive session items but we're going to skip down to the consent agenda first before we handle those would anyone like to move the consent agenda I'll move the consent agenda thank you Vince is there a second thank you dawn all any further discussion all those in favor please say aye aye opposed okay and now we are at the reading file um board members do you have any questions or comments regarding the reading file or anything else at this time okay I do really quickly very fast I know we're late but kudos to our police department for the recent updates and the apprehension of the vehicle fires we've had sort of a rash I would say in the community of you know some break-ins and the vehicle fires recently so I'm very pleased to see that we're getting resolution on those cases um big thanks to Jim Berniger who has stepped down as a chair of the EDC over this last week he did quite a lot of work in that role and just wanted to express my thanks to him for serving as long as he did I think uh one last thing just to note that I think there was a mention earlier in the evening in regards to the merger financials since that is in the reading file there was some talk that we never got any information about separation financials that was part of Sarah's presentation back in September so people are curious about that it's just something that they can go and look at and we'd be happy to find the links if someone wants to get that to us okay thanks Pat any other board members with comments on the reading file okay and we had Irene at the beginning of the meeting asking if she could make a comment on the reading file go ahead Irene thank you I'd like to comment on the staff response to the 45 residents requesting more robust and up-to-date financial numbers about merger I appreciate that Sarah gave us an immediate response to that memo and I would hope she'll take the time needed to generate complete answers to our questions also the staff memo uh refers to 37 insiders and that should be updated to 45 thank you thank you okay um if and I'd also like to echo Pat's um thanks to Jim Berniger serving on the EDC he was an excellent member and we're going to miss him and wish him the best okay we have two executive sessions that we need to adjourn into so we have a couple different motions to make and I do believe that we will be exiting this meeting and entering a separate meeting for the executive session and do you need us to come back Greg for any other business if we're going to point somebody we do yeah we might have to have you come back we'll see how the discussions go and like you said for the appointment thank you Greg I appreciate that if there's an appointment so we'll leave this meeting open we're going to hang out in the other room do our executive session and come back if we need to okay so um Andy would you like to make the motions to enter into executive session please I certainly will I move that the select board enter into executive session to discuss the proposed public official appointments in accordance with one VSA section 313 a3 and to include the unified manager and deputy manager is there a second second thank you Vince all those in favor please say aye hi hi hi next Andy okay I move that the select board make the specific finding that premature general public knowledge of the town's position concerning the proposed contract discussion would place the town at a substantial disadvantage is there a second second thanks Pat all those in favor hi hi hi host thank you next Andy I move that the select board enter into executive session to discuss potential contracts pursuant to one VSA 313 a1 a to include the unified manager deputy manager and community development director thanks is there a second second thanks Pat all those in favor hi hi the post that was just three right Andy yep okay thank you so we're entering executive session at 952 and we'll be back if we need to do any business afterwards we'll just wait for Don to arrive and we can finish okay I think Don's here I just want to confirm that Don can connect I see her but I don't hear her Greg can you unmute Don please okay try that that's better perfect thank you okay would anybody like to make a motion I make the motion that we appoint Annie Cooper to a term on the economic development committee thank you is there a second I'll second that thank you Vince any further discussion just to be clear it's economic development commission sorry thank you all those in favor please say hi hi hi host all right congratulations Andy welcome to the EPC congratulations I am so grateful and I really appreciate the respect and honestly I take this very seriously and I appreciate it more than I can really articulate and I will serve you well and wise and graciously from all angles I promise thank you for the trust we know you will I am really quite moved actually to be honest with you I I'm so grateful thank you thank you okay that concludes our business for the evening would anyone like to adjourn vote to make a motion that we adjourn is there a second second thank you Vince all those in favor hi hi host all right thank you all very much and have a good evening


So now what we're going to do is essentially extend that kind of analysis representing a function in terms of its oscillatory components to the continuous case where we have the oscillatory components are functions of a continuous variable and the interval in principle can go from minus infinity to infinity. All right, so there's no prescribed range over which the function has got a repeating oscillation. All right, now the Fourier transform is something that's very, very important and it's probably something that you already have done and may not have even known about it so anytime you turn on your MP3 player or your CD player you're doing that. If you've done FTIR spectroscopy in organic chemistry you've done FT stands for Fourier transform, et cetera, et cetera. So it shows up everywhere. It's a very important technique. Whenever you have a signal that depends on time it's convenient to be able to represent that in terms of a related function that is in terms of frequency. Okay, so I'm going to just write down the formulas so we can see what they look like and we are going to use them to evaluate some Fourier transforms for a simple function and that will hopefully help us to appreciate what exactly the transform is doing even if we ignore going into all the mathematical details. Okay, so I'm going to suppose that I have a function F of X, okay, and here X is, can be for example X could be position of a particle or most commonly in Fourier transform analysis X would be, it would be a time, okay, and what I want to do now is transform this function to another space where it's expressed in terms of another function whose variable is related to X in this sense of this transform, this Fourier transform, okay, and I'm going to call that variable K. Now the way the Fourier transform works is that we multiply F of X by a complex exponential which involves both K, this other space variable, and X, okay, and then we're going to integrate this function, this product from minus infinity to infinity, all right. So what we do is we get rid of the dependence on X and we're left with a function of K which is called, we'll call it F of K with capital, okay, and this complex exponential contains oscillatory contribution, sines and cosines thanks to the famous Euler relation which says that E to the I times a variable say theta is equal to cosine theta plus I sine theta, okay, so this is an oscillatory function and as we're going to see the process of doing this transform, the integration and multiplying by this oscillatory function, it's going to actually pick out the oscillatory components of X and express them in terms of the variable K and just to remind you, if X is equal to time, then K would be equal to frequency, so for example, hertz or second to minus one, or if X is equal to position, it turns out that this would give K is equal to momentum, all right, so time frequency transformations are very useful in signal processing, in image processing, in GPS, in spectroscopy, and position momentum transforms are the basis of being able to determine the structures of molecules in solids, so crystallography, by measuring the diffraction of waves such as X-rays or electrons or neutrons, so that structure determination relies on the transformation because you measure the quantity in momentum space and you want to transform it to a function in position space, in other words, the molecular structure, the coordinates in Cartesian space, and oftentimes we measure a signal in time and we want to express it in terms of its components and frequency, so that would be an example of that would be NMR, modern NMR experiments, they measure a time varying magnetization and then a Fourier transform turns it into the NMR spectrum, which you've learned is relatively straightforward to understand if you understand the basic principles of NMR, say, for organic molecules, okay, but the time signal may actually look very, very messy compared to the frequency spectrum, okay, so this process here would take, if X was equal to time, would take a function of time and turn it into a function of frequency, all right, and this is what we call the forward Fourier transform and it's actually invertible, so if you know F of K, you can recover F of X, or if you know F of X, you can recover F of K, okay? And the reverse just looks like this, F of X is equal to minus infinity to infinity, F of K, E to the plus 2 pi I KX times D K, okay? Now it's, like I say, it's probably not obvious from looking at these formulas what is actually going on, what are you actually doing when you're doing these transforms? And I think that at the level of this course, the best way to see how it works is actually to go ahead and play around with it, and even if we don't care about the intimate mathematical details, we can at least get a feeling for what's going on, and I think that should be the goal of what we do here. All right, so what we're actually going to do then is we're going to construct a time signal that we know a lot about, and then we're going to look at its Fourier transform to see what actually we get when we do this process, okay? And the particular form of the signal that I'm going to look at is as follows. We're going to define a function S of t, S for signal, okay? And let me just make sure I have consistent notation, so I'm going to grab my notes here. It's going to be equal to e to the i omega t, where omega is now going to be an angular frequency. Omega equals 2 pi times the frequency in hertz, and then it's going to be plus some quantity which I'll call alpha times i, all right? And this, so this is, well, let's go ahead and consider a particular case that we'll do first, so, all right? So first we're going to do nu is equal to 5 hertz, and we'll do alpha equal 1, okay? So when we put that in to this form and use this formula for omega, then what we get is S of t is equal to e to the 10 pi i t, and then times e to the minus t, okay? All right, so this part here, this is the oscillatory part of the signal, all right? Because it's got a cosine, the real part is proportional to cosine, and the imaginary part's proportional to sine. And then this part, which comes from this term here, this is just an exponential decay, all right? So what we expect to see is that the real part will look like a cosine that's decaying in time, and the imaginary part will look like this sine decaying in time, and then we'll play around with these parameters and see how it affects the transform. But the bottom line here is that for the first example we know that we do, we know that the frequency is 5 hertz, and we'll be able to see that very clearly when we actually do the Fourier transform or actually compute the frequency spectrum of this function. Okay, so let's go ahead and get started. We'll type that in, and then what we're going to do initially, so there's lots of ways that you can do Fourier analysis in Mathematica, and we'll see a couple of them. So what we'll do initially is we'll just do the integral that's written on the board, we'll just do it numerically, okay? And then later we'll see how to use existing tools in Mathematica to do Fourier transforms. All right, so I'm going to define omega as written down, so I'll go over and get myself an omega, and that's going to be equal to 2 times pi, and I'm going to put in nu is equal to 5, so there it is. And then plus alpha times i, alpha here I'm just going to say is 1, so I'll just put in i, and i is capital i in Mathematica, or you could go up here on the pallet and put in this i here, okay? And then I'll define s of t underscore colon equals exp of the capital, and then I'll put in i times omega times t, all right? Now, just for fun, let's have a look at what this thing looks like, okay? So the first thing I'll do is plot the real part, so this should be the cosine part, and the way we get the real part of a complex object in Mathematica is we say capital R, little e, and then bracket, all right? So what I want is the real part of s of t, and then I'll just plot that from t goes from 0 to 5 seconds, all right? So we enter that, and let's put in plot range arrow all, okay? So there you have what I already described in words. It starts at 1, it's a cosine, and then it's got this decaying amplitude, that decaying amplitude is coming from the exponential e to the minus t, okay? And another thing you can notice if you count how many times it oscillates before it gets to 1, you can see it's 5, and so you see that the frequency of that signal is 5 hertz, okay? So we know all about that signal. We can also plot the imaginary part which is going to be proportional to sine, and so what you see is that it's, we have to put in imaginary part, I m is imaginary part, and so now you see it's the same as the real part except it's out of face because it's a sine, okay? It starts at 0 instead of 1, okay? Now the whole idea behind the Fourier transform is that it's going to give us a function that has a peak in frequency where that frequency matches that of our signal, okay? So in this signal we have a single frequency. So when we do the Fourier transform we should get a single peak and that peak should be at 5 hertz. Okay? And that'll be, actually that's in the real part of the Fourier transform. Okay, so here's how we're going to do the Fourier transform. So we're going to say n integrate with a capital I, and then we're going to multiply s of t times exp minus, is it 2 times pi times I, capital I times f, f now is going to be my definition of frequency, and then times t, okay? So that's the same as what's in the formula over there except I'm calling k, f, and x, t for frequency and time. All right? Now when I do n integrate I have to say how far I'm integrating. And I can see from this, the behavior of this function that if I go out to about x equals 10 I'll be in a place where it's completely decayed out and so I'm going to do the integration for t goes from 0 to 10, okay? All right, so if I just did that I would get the Fourier transform. Now what I want to do though is actually plot that as a function of frequency. So I'm going to say plot, and because it's complex I want to look separately at the real and imaginary parts. So let's go ahead and look at the real part first. So say plot re and then put another bracket here, okay? And now I have to say over what range of frequency do I want to plot it? And we'll do it from, I expect my action to be at 5 hertz so if I go from 0 to 10 I should see something. So f goes from 0 to 10, okay? All right, so let's go ahead and let that go. And let's go ahead and put a plot range arrow all, okay? So there you have it. So notice what we've done. We've taken our, well with the real, we've taken our function, our complex function, we've done the integration corresponding to the formula for the Fourier transform and we've generated a complex output and the real part looks like a frequency spectrum. We have a single peak here. Remember this is frequency now in hertz and it's at 5 hertz which is exactly where the oscillation in the signal is. So the Fourier transform has pulled that out for us. Okay, now there's one other aspect of this that I wanted to show you. So let's go up here and let's change alpha, okay? So if I put in say a factor of, let's see what I want to do here. Let's go ahead and just put in 2. Actually what I want to do, let's go ahead and do a new one. So let's grab this because we want to compare. All right, so we'll have that and let's go ahead and put in a factor of 2 on this thing which is called the damping. Now what this is going to do is it's essentially going to turn the decaying exponential instead of e to the minus t it's going to be e to the minus 2t, okay? So it's going to decay more quickly. It's going to be damped more strongly and we can see that by just entering this command. You can compare this function here. It has the same frequency. We didn't change that. But notice this one takes much longer to die out than this one. So increasing the damping has, if you like, localized our oscillating signal. All right, now let's see what happens to the spectrum. So we should still get a peak at 5 hertz because we didn't change the frequency. So if you do that and I'll let you scroll up and take a look, what do you notice that's different between these two spectra? So the peak is at exactly the same place, it's at 5 hertz. What's changed? Compare the widths of these two spectra. Which one is wider? The second one, okay? So that's kind of interesting and this is, if you like, a manifestation of the uncertainty principle. All right, it turns out that these two spaces, time and frequency or as you're taught in G chem energy because energy is related to frequency, they're related by an uncertainty principle. All right, so notice a function that is more delocalized in the time domain becomes more localized, if you like, the narrower, this one's narrower in the frequency domain, right? When I go down here and I damp this out more and make this oscillation more localized, it produces a broader frequency spectrum, okay? Now let's have a look at the imaginary part. This is something worth looking at. This form here, the real part, which looks like an ordinary spectrum that you might see, say, in a peak that you might see in, say, an NMR spectrum. This is what's called the absorptive line shape, all right? And it's the real part of the spectrum. The imaginary part contains similar information but it has a different shape, which is called the dispersive line shape, so let's have a look at the imaginary part, okay? So this is the imaginary part of the same transform and notice that it has this funny shape, the dispersive line shape and where it goes through zero is actually where the frequency was in our signal, 5 hertz. And if you think about it a little bit, it turns out that these two are related, okay? So in principle, they contain exactly the same information and the relation is, and I think you can see this if you look at the behavior of this guy and the slope of it, this turns out to be the derivative of this because sine, which appears in the sine of frequency, which appears in the imaginary part is the derivative of cosine, which is the real part, okay? So this is what's called the dispersive line shape. It contains the same information. It's just usually we're more interested in identifying frequencies by where peaks occur in the spectrum, so you're often more interested in the absorptive line shape, okay? Now let's add another frequency component. So here, in this example, we didn't need to do a Fourier transform to see that there was an oscillation in the time signal at 5 hertz, right? You can just count it off the graph. So we didn't need to go through the trouble to see this here. But let's make it just a little bit more complicated and then I think you'll be able to see where the real power of the Fourier transform comes in. So what I'll do is I'll grab all this stuff and I'll go ahead and put the damping here back to 1.  And I'll make its frequency 3 hertz and then I'll copy it, put it back, and now I'll define a second frequency, omega 2, which oscillates, which is 6 hertz, okay? And now I'm just going to add the two together to produce my signal, all right? So we have two contributions, okay? So now let's make a plot of the real part of that superposition of two oscillating signals. I'm sorry? I believe you want the 7 hertz. Oh, right, yes, exactly, thank you. Very good, okay. Okay, so now look at what you get. This is a little more complicated, okay? And it may not be completely obvious what the frequencies are. You can play around with it even more. You could say like 3 and say 5. And it starts now to become difficult to just look at the graph and see what are the underlying oscillations. And this is just with two frequencies. What if you have many as you may have in an audio signal or even an NMR signal, okay? So now we'll see that we can take this thing which already looks fairly complicated and we can make it look very, very simple by doing the Fourier transform, all right? So let's go up here and get this. We'll do the real part, the absorptive part first. So what should we get? Anybody? What should we get? We should get a frequency spectrum with two peaks, correct? One at 3 hertz and one at 5 hertz. And low and behold, you have one at 3 hertz and one at 5 hertz. Okay, so you can see through these simple examples how it is that this somewhat intimidating looking expressions over here actually do something really nice. They take your time signal, convert it into a frequency spectrum and pull out the power if you like contained in the oscillations at the different frequencies. What happens if you play around with how much goes into each? So let's say we multiply the second one by 0.5, all right? So now it looks kind of weird. And then do the transform, notice, there's less of a contribution from the second oscillation and that shows up in terms of the size of the second peak. So the size of the peak is the measure of how much power if you like there is in a given oscillation. And then of course we can play around with the damping. So let's put in a smaller damping for the second one. So let's say 0.5 instead of 1 and then redo it. What do you think is going to happen now when we do the transform? I made the damping, the exponentially decaying and time part more rapidly decaying for the second component which has the frequency of 5 hertz. It should narrow the second peak, right? Hmm, it's taking a while. Okay, there you go, all right? And it grew because the area is a measure of the strength of that contribution to the signal, okay? And then let's look just at one more thing here. I think you'll get the idea. Let's have a look at the dispersive part, the imaginary part. What do you think we'll get here? Probably going to have to wait a while. All right, so there you have it. So you have your two dispersive line shapes and they tend to overlap, see? So this can be a problem sometimes because if they overlap a lot, the positions of the zeros may actually be shifted by the shape of the neighboring guy, okay? But in this case, they're reasonably, well, they're enough separated that we go through 0, right at 3 and right at 5. And notice that this one is narrower, sharper feature than the lower frequency one. Okay, so now you've had your brief introduction into how Fourier transforms work. And the next thing I would like to do is to introduce you to the symbolic Fourier transform that's available in Mathematica, okay? So many functions, not all functions, but many functions have Fourier transforms that have, you know, analytic forms, meaning you can write down an equation for them, okay? And where those exist, Mathematica will find them. And so I want to show you a couple of the more well-known ones that appear very frequently in chemistry applications, okay? All right, so the first function we're going to look at is it's going to be a Gaussian function, all right? And I've told you many times that Gaussian functions have many wonderful properties. And one of those properties is that the Fourier transform of the Gaussian is particularly convenient, and we'll see that in just a second, okay? So I'm going to define now a Gaussian F of X colon equals, and I'm going to say it's exp to the minus a constant times X squared, okay? So this is just a generic bell curve, all right? So we can enter that, and we can go ahead and plot it. X goes from, actually let's use a replacement rule to put in a value for C. Let's just put in 1. We'll have X go from minus 3 to 3, okay? So there you see you get a nice bell curve. Gaussian function centered at 0, okay?  So it's actually going to try to find the closed form Fourier transform of this function, and it will find it. And so here's how that works. What I say is Fourier transform, and then I say, what is the function that I want to transform? So F of X, and then the variable that I'm transforming from, and then I say what's the variable I'm transforming to? So I'm going to say K, so it's going to be just like what's written on the board there, okay? And I'm going to assign that to a function of K. So I'll say G of K underscore colon equals, all right? So let's go ahead and let that rip. And now let's ask what is G of K? And here's the part that I think is interesting, okay? So we take F of X here as a Gaussian, all right? E to the minus constant times X squared. And notice what we get. We get a function back that's also a Gaussian. E to the minus K squared, because this is a function of K, all right? So that's one kind of useful thing to know right off the bat is the Fourier transform of a Gaussian is a Gaussian. So that's one of these really nice things about Gaussians that turns out to be very convenient. Now, what's also very interesting is the relationship between the parameters of these two Gaussians, and in particular their widths, okay? So let's have a look at that. Oh wait, let me show you one more thing. Remember, we have a way to go back. So when we do a Fourier transform, we get a function in the other space, in this case it's K, and previously it was frequency. We can apply the inverse Fourier transform to go back to the original function. So I just want to show you that that works, okay? And so what we can do is we can ask for Fourier transform of G of K, K goes to X. And notice, aside from these square root factors that cancel and give one, we get back exactly the same function, okay? So that's an illustration for the case of a Gaussian of the forward and the inverse Fourier transforms. So now what I want to do is let's have a look at these functions because I want to reiterate a point that I made earlier. So what we're going to do is we'll go ahead and make plots of the Gaussian in real space here. And I'm going to do it for two different values of C, which is going to determine the width of the Gaussian. All right, so I'll first plot for C equals 1, and then we'll add to that plot the same function but for C equals 10, okay? So here we're just plotting the Gaussian as a function of X, just for two values of C, all right? And what you see is that C equals 1 gives a fatter Gaussian and C equals 10 gives a narrower Gaussian, okay? Because the parameter C is inversely proportional to the width, all right? Now let's have a look at the plots of the Gaussian in the K space, in the Fourier space, okay? So here what we're going to do is we'll plot G of K now with C equal to 1 and G of K with C equal 10, all right? Something happened here. Oh, I have to plot as a function of K, all right? Okay, so now we're going to notice again a manifestation of the uncertainty principle, all right? So notice here that actually let's go ahead and say plot range arrow all here to make it on the same scale. So notice when I transformed the narrow Gaussian, I got a fat Gaussian in the opposite space, in the Fourier space, okay? And I'm still not getting the complete range of this guy. All right, plot range arrow 0 to 1, okay? So notice the blue one gets a little fatter but the purple one gets a lot fatter, okay? So the more localized we are in X space, the more spread out we get in K space. If you like, the uncertainties are connected, okay? So there's your introduction to Fourier transforms, a little taste of the symbolic version and then we'll see another example later and then we numerically integrated the formulas ourselves to see how it actually works, okay? So now we've concluded our treatment of some series expansions and now the Fourier transforms. So what I'd like to do is go over a few examples from chemistry that involve these various things, okay? Now the first thing that we're going to do is relevant to statistical mechanics which is something that you'll learn about in Chem 131c and we already talked a little bit about the central object in equilibrium statistical mechanics which is called the partition function and the partition functions can be written as a sum and so what we're going to do, I'm going to show you a couple of examples and then we'll actually evaluate the sums in Mathematica, okay? So the way the partition function is calculated is we suppose that we have a quantum mechanical system for which we know how to calculate the energy levels, okay?  They may or may not be degenerate. Let's just say there's a few levels there and energy is increasing, all right? So we can say that this one is 1, 2, 3, 4, et cetera and we can also introduce, so this will be, say, a label for the quantum number. So that's N or let me go ahead and call it I because that's what's written in the notes and we can also define the degeneracy of each level, all right? So I'll call that G sub I and degeneracy means how many energy states do I have that have the same energy, okay? So here there's only one in the I equals 1 level but here there's 2, so I put a 1 here and a 2 here and then 1, 1, okay? So that's the degeneracy. Now the formula for the partition function is it's usually denoted Q. What that is is it's a sum over the I's and in principle it can go to infinity if there's an infinite number of levels and then you take the degeneracy and then times E to the minus epsilon sub I where epsilon sub I is the energy value at which those levels occur divided by KT where K is Boltzmann's constant. Another way to write this would be sum over I GI E to the minus beta epsilon sub I where beta is equal to 1 over KT, all right? So what's the significance of this? Well as you will learn when you get your introduction to statistical mechanics, if you know this function then you have the connection to all the thermodynamic properties of whatever system it is that you are describing, okay? And in order to be able to calculate this function you need to know the energy levels, okay? And those come from a quantum mechanical calculation. So for example, those of you who are in Chem 131A now, you've seen what the energy levels are for particle in a box, you've seen them for harmonic oscillator, you've seen them for the rigid rotor for rotational motion and you've seen them for the hydrogen atom I suppose by now, okay? So the point is for all of those systems we know the levels and therefore we can in principle evaluate the sum, calculate the partition function and then ultimately thermodynamic properties of the system. And this is the connection then between the properties of microscopic objects. So individual molecules for example and thermodynamic properties is through this connection. Quantum mechanics gives us the energies. Statistical mechanics tells us how to calculate the partition function and then turn that into macroscopic properties, okay? So you'll see how that works later. But for us right now it's just going to be a sum that we're going to evaluate using Mathematica, okay? And we're going to do this for two systems. The first one is going to be the harmonic oscillator. Does anybody remember what the energy levels for the harmonic oscillator are? Okay, I'm going to use a slightly different notation. I'm going to call the quantum number for the harmonic oscillator N, okay? So I would be replaced by N.  Okay, so it could be H bar omega, it could be H nu. All right, nu zero now is the characteristic frequency of the oscillator, okay? And for the harmonic oscillator the quantum number goes from zero to infinity, okay? And another thing about the harmonic oscillator, does it have any degeneracies in its energy levels? No, so all the G N are equal to 1. Okay, now the next example that we'll do, so I'll go ahead and write it down now. Okay, so in this case then Q would be equal to sum N equals zero to infinity E to the minus beta H nu zero N plus 1 half, okay? So that's the sum that we need to evaluate if we want to calculate the partition function for the harmonic oscillator. Okay, so then the next one that we're going to do is going to be the rigid rotor. So this is a quantum mechanical model for describing the rotational motion of diatomic molecule. Does anybody remember the energy levels for that one? Mm-hmm. And it's usually expressed in terms of a quantum number called J. J times J plus 1, where do I put that? And then, okay, so then there's some numbers that go in front to give it energy units. And I'm going to use the notation H C B where B is something called the rotational constant. I don't know if you saw that in your formula. But in any case, the book that I used has that notation. Okay. And what is the, what are the allowed values of the quantum number? It starts at zero. And what about the degeneracy? 2 J plus 1. Great. Thank you. Okay. All right. And we're going to do this one for a particular case where this, I'm going to write this in a funny form just because it's the way that I looked up the constants, the way they were given to me. Okay. So I'm going to call, sorry, let me get rid of this thing here. What I'm going to need to evaluate the rigid rotor partition function is I'm going to need to know beta times H times C times B. Okay. So I'm just going to write down the value that I'm going to use for that. Beta times H C B is going to be equal to 10.591 divided by 207.224. Okay. And this happens to be for hydrogen 35 Cl. So we'll actually evaluate the part, the rotational contribution to the partition function for HCl. And it will involve using these numbers. Okay. So let's go ahead and work on these using the sum command. Okay. Now, what's nice about the harmonic oscillator, one of the many reasons why it makes such a nice model system, is because as we're about to see in just a minute, the sum that we need to evaluate, which is a sum over an infinite number of terms, it actually has a closed form. Okay. So let's try that out and see what we get. Okay. So what I'm going to do now is let's go ahead and make it fancy by using the palette. So I'm going to say sum N, whoops, not B, N, tab, zero, tab, infinity. All right. And now what do I need to do? I need to put in the degeneracy, which is one for all levels, so I don't have to put anything. And then I'll put in E and make that a superscript. And so now I need to put in minus beta, space H, space, and then I'm going to have new zero. So where's new here? And then tab zero. Okay. And then space times N times N plus one, half. Okay. Now, sometimes you get lucky and sometimes you don't. Hmm. That doesn't look good, does it? So something is wrong with my formula. Just make sure I do all this stuff, right? Anybody see what I did wrong? We should get a nice formula. Capital letter E. Okay. So let's, yeah. I thought that one should work, but maybe not. Oh, yeah. Thank you. Yeah, I used the superscript. That's right. Darn it. See, I don't like the palette. Here we go. All right. E minus beta times H times, yeah, there's something wrong there, but I don't want to waste your time debugging it. So let's just try it a little more straightforward way here. Times N times N plus a half. Oh, I know what the problem is. Dummy. I'm talking to myself here. Let's try that one. Ah, there we go. Okay. Gave up too early. There shouldn't have been an N in the, in front of the N plus one, half. I was thinking of something else. Okay. Now, so what, after all that to do, what's the significance of this? Well, this is one of the few lucky cases where the sum that you need to evaluate for the partition function actually has a closed form. Meaning, notice, there's no sum here. All right. So we could just plug some numbers in and we get a number for what's written originally as an infinite sum. So that's a very fortunate case. And this is another example of one of the very nice aspects of the harmonic oscillator, why the harmonic oscillator is such a useful starting point for talking about molecular vibrations. Okay. All right. Now, the next one is the rigid rotor. Okay. So what I'm going to do here is I'm going to define the sum end as a function of J. So I'm going to say F of J colon equals. And then I'm going to put in the degeneracy. So that's two times J plus one. And then times exp minus A. Okay. So that's what I wrote down over there. Beta HCB, I'm going to call A here. And then times J times J plus one. Okay. So this is the thing that goes in the sum to calculate the partition function for the rigid rotor. All right.  Let's go ahead and enter that. And let's see if we can actually get a closed form for this one. So let's say sum F of J, curly, J goes from zero to infinity. And we're not going to get lucky this time. Okay. So we got the thing spit back at us exactly as we put it in, which means no luck. Okay. So then here's an example of a case where we actually want to calculate a number for this but we have to actually do the sum explicitly. And then this brings up an issue that we have to confront when we do sums initially, which is how many terms do we have to include to get a good answer. Okay. And you can kind of see that this thing is going to, if you like, converge at some point. Because notice, as we add higher and higher values of J, the thing that we're putting in the negative exponent here is going to get bigger and bigger. So as we increase J, the contribution coming from the higher J terms diminishes in size. And at some point it becomes insignificant. Where significant depends a little bit on, you know, how accurate we want to be. Well, it depends a lot on how accurate we want to be. Okay. So one useful thing to do in such a situation where you need to approximate an infinite sum with the finite number of terms is to take a look at how the value of the sum depends on how many terms you put in. Okay. So that's all we'll do now. We'll look at it a couple of different ways. All right. So to do that, I'm going to put in an explicit value of A and that's the ratio over there that I wrote on the board. So that's going to be to the end point 591 divided by 207.224. That's for HCl, okay. And now what I'm going to do is make a table of the contributions to the sum as a function of J. So we can kind of see the behavior of the things that we're adding up to get the sum. All right. And I'm going to do make that table for values of J up to a number that I'm going to call J max and I'm going to set that equal to 15, all right. And I'm going to call the table data. So here's what I'm tabulating. Table, curly, J, and then now the individual term. So that's F of J. And then I'm going to do that for J goes from 0 to J max, okay. So what is this table going to give me? It's not giving me a sum. It's giving me all the values of this thing, including the degeneracy for each value of J going from 0 to J max, which in this case is 15, okay. And now what we'll do is we'll list plot those so we can see the behavior of the individual contributions to the sum. So let's go ahead and enter that and then we'll plot it. But here's what I wanted to show you, okay. So notice that as you go to higher and higher Js, the number gets smaller and smaller, right? See how it's behaving here? So let's have a look at the plot. So the plot is going to be a list plot because it's discrete data, all right. So we just list plot data and here you can see the behavior of the individual terms. Okay, so the first one is 1 because we put in zeros for J and then it quickly rises up to a pretty large contribution, rises up more and more and then at some point it starts going down, down, down, down and you can see that once we get out to around J max the contributions are very small. So we might safely conclude then that if we include only 15 terms in this particular sum that we may get a well-converged result. So let's check that. So now what we'll do is we'll actually create another table where instead of looking at the individual terms we'll look at the value of the sum itself as a function of the number of terms that we include, okay. So for that what I need to do is I'm going to do sum of F and now I'm going to put in a dummy variable K and then K is going to go, I have to put a curly bracket here. K is going to go from zero to J max, curly, square bracket and then J max is going to go from zero to 20. You see what I'm doing here? I'm making a table of this, well let's wait, I got something wrong here. This should be a table with respect to J max, okay. So what I'm doing now is I'm making a table where J max is being incremented from zero to 20, okay. So the first entry in the table is going to be J max equals zero, okay. It's only including one term, the first term that's going to give me one in the sum. And then as I increment J max I'll be actually evaluating the sum here up to J max in each case until finally J max equals 20, okay. So we're basically going to see how does the sum itself depend on the maximum value of J that we include in it, all right. So let's go ahead and enter that, okay. So here's only one term, J max equals zero, we get one as expected. And then we add more and more terms and what you see is that after 15 here to the precision, the default precision in Mathematica the number doesn't change, okay. So we would say then that to this number of decimal places, this sum converges after 15 terms. So let's have a look at the convergence if you like. So what we can do is just do list plot data and now you can see if you only include one term you're way off and then you get closer and closer as you include more and more terms and then after, you know, around 15 it's converged. Okay, so if we wanted to quote a value then for the partition function of HCL and this is, I forgot to tell you that this corresponds to room temperature 298 Kelvin the value that we would say it's equal to is 19.9028, okay. So the main point of this exercise here though is to show you how when you are not able to generate a closed form for some, how it is that you can look at the behavior of the individual terms and then figure out how far out you have to go to get a given level of accuracy and how you can do this analysis and plot the results yourself quite easily using Mathematica. This would be a real pain to do with the calculator. Okay, now there's one other thing that I wanted to show you now that we actually have a converged partition function for HCL, there's another quantity that we can calculate and this will be familiar to those, well maybe not familiar to those of you yet in Chem 131. Did you guys talk about the populations of the levels? Okay, all right, well this is an interesting thing that you'll see later on in when you do the statistical mechanics. All right, so the next thing we're going to do is we're going to calculate the population of the levels, okay. And those come from what's called the Boltzmann distribution, I'll just write down the formula here, I need my pen. Okay, so the Boltzmann distribution tells us that the probability of being in the ith level is equal to the degeneracy of that level times e to the minus beta epsilon of i divided by the partition function, okay. So now what we'll do is we'll look for the first so many levels, I don't recall exactly how many, I guess we'll do the first 12 levels, 12 rotational levels, what's the probability of being in that level? We've already seen how to calculate q, we know how many terms we need to get a good value of q, so we'll just now evaluate this guy for j goes from 0 to 12 and see how that looks, okay. So now what I'm going to do is I'll just go ahead and grab this thing, put it down here, all right, so I'll say I'm going to set this equal to q, so this is going to give me a number for q and I've just seen that I only need to go out to how many terms? Fifteen, so I'll evaluate this out to 15 terms and that'll be my single number, converged number for my partition function, all right. And now I'll make a table that's going to hold the populations of the first 12 levels, rotational levels, okay, so that's going to be, I'll call it data again. So now what I'm going to do is table j and then I need e, well actually we already have the function defined, right, we've defined it here, it's fj is this exponential factor that we need, so we can go ahead and put that in, f of j and then we have to divide by q, all right, and then we'll go, j goes from 0 to 12, all right, so let's go ahead and we can enter that and then lastly what I'll do is plot the populations as a function of j, so list plot, plot data. So now what this is, this is the probability of finding hcl in the jth level, okay, so you can see for j equals 0 at room temperature, you expect to have 0.05 population and notice that it increases and the most probable rotational state of finding hcl at room temperature is, what is it, is this 3, looks like it's 3. 0, 1, 2, 3, okay, so if there was, you know, hcl vapor at room temperature, the majority, well the most probable rotational level would be j equals 3 and a consequence of this, these populations is, is that when you do vibrational spectra of hcl, you find that the shape, that the heights of the peaks are actually proportional to the populations of the levels when there's rotational find structure in the spectrum and the function, the envelope of those peaks actually has this shape, so you can actually see that in a spectrum. Okay, so there's your little introduction to using some, to do some statistical mechanical calculations, partition functions and what we're going to do next time, which is going to be next week is we will look at some examples of Fourier transforms in chemistry. All right, so if you haven't already done so, please register to use sci finder scholar following the instructions that I sent you yesterday because you're going to need to be registered if you want to follow along during the library resources presentation by Dr. Mitchell Brown on Thursday and as I think I already mentioned a couple of times, next week's homework assignment is going to be to use what you learn on Thursday to find data in the chemistry literature and you will be able to do that from any computer from which you have access to UCI library resources. Okay, so you don't need to come to the lab, so we're not going to have lab sessions next week. And in addition, you can do the homework at home if you want. The proviso is if you don't know how to establish a VPN connection, you're going to need to learn how to do that because to get access to the library resources you need to be able to use the VPN. All right, so we'll see you on Thursday.


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I think that the government should not pay for everyone's college education. Okay, very interesting. I kind of didn't expect you to say that. But with that being said, check out my other videos right here. I made them just for you. Go ahead, gently tap that like button, hit the subscribe button, ring the notification bell and comment down below any thoughts, comments, criticisms, etc. that you have on the video and I will see you next time.