5

(olAIdunbouuded that e* is periodic Mliu (1 H wi puv ( 'T JQ vuogioung [vQX Jo Huqjm) Ui JOJ VIHUIIOJ 049 91018 0 " Fol JO Wivua xoidunon 04 pUsAXD Hoqio...

Question

(olAIdunbouuded that e* is periodic Mliu (1 H wi puv ( 'T JQ vuogioung [vQX Jo Huqjm) Ui JOJ VIHUIIOJ 049 91018 0 " Fol JO Wivua xoidunon 04 pUsAXD Hoqioh QaOqV 04) 1w41 Suqutnssu Derlve Mule' fotmla "iM Wviiv Chm Hericn lor (" Maclaurin State (22 polnte) Hoiyhon10

(ol AIdunbouuded that e* is periodic Mliu (1 H wi puv ( 'T JQ vuogioung [vQX Jo Huqjm) Ui JOJ VIHUIIOJ 049 91018 0 " Fol JO Wivua xoidunon 04 pUsAXD Hoqioh QaOqV 04) 1w41 Suqutnssu Derlve Mule' fotmla "iM Wviiv Chm Hericn lor (" Maclaurin State (22 polnte) Hoiyhon10



Answers

Show that $\Phi$($\phi$) = $e$$^{im_l}$$^\phi$ = $\Phi$($\phi$ + 2$\pi$) (that is, show that $\Phi$ ($\phi$) is periodic with period 2$\pi$) if and only if m$_l$ is restricted to the values 0, $\pm$1, $\pm$2,.... ($Hint$: Euler's formula states that $e$$^i$$^\phi$ = cos $\phi$ + $i$ sin $\phi$.)

Okay, so in this problem, hey is really mathematical. One is more mathematical than physical, but there is a leader of physics involved. So we just need to show that the function off five, which is exponential off I am five is going to be equal the same function off. Let me see five plus to find. Okay, so this is something that we need to prove. So let's begin with the definition here. This left equation here. So the definition off the function off five this exponential off I am off I and we can use the oilers formula. Let's see, the Oilers formula is going to be for this particular case because signed off em five the US I sigh off m fire. Okay, so this is the definition off our function, but who is the the left side here? So the right side. Sorry. So who is this right side equation? So the right side, we have the function five plus 25 And this is going to be basically exponential off. I am five plus two pi. Okay. And using again the Arliss formula. Well, we have for this particular case. Sorry, actually, let's simplify. Let's make a smart movement before using the oil. This formula, this is exponential off the some off to constants. So let's separate the exponential us exponential off. I am fine. That multiplies exponential off. I I am two pi. Why we are doing this Because this exponential here we all right? You know, trown the first equation. So we just need to find out who is this second exponential here. So let's do this. We have exponential off. I I am five the multiplies the Oilers for the second term, which is course sign off em two pi. Plus I sign off em two pi. And since the problem arguments that if ups if am is a natural number, which means actually not natural integral. So it's just zero lost minus one los minus two and so on. Well, that means that the sign off to buy em is always zero. And this is the call sign off to pie. Am is always one. Therefore, we proof that the function or five is going to be equal function off five plus two y, with both of them being equal to exponential off. I, um Fine. Okay. So this is the answer to this problem. Thanks for watching

In this problem. We're looking at the third ah, function in the separated Schrodinger's equation. Uh, which will look like this sometimes with are as a function of little our theater as a function of little data. Oops, that's not right. I meant to write, uh, capital fee here, and capital fee is a function of little feet. So here we're just looking at a function representing fee as a mouthful angle. And what we want to show is that if we have this function and we give it an argument little fee, this is always going to be equal to, ah, the function. If we had give it feet plus two pi. So basically, what we're trying to show is that capital fee is a function that is two pi periodic. Um, So how do we do this? We're gonna have to remember Oilers formula, which is that e to the I theta is equal to co sign of data, plus I sine of data. So if we use this starting with what I'll call the left hand side over here, the original argument Well, we're gonna have that, um e so the M fee or I am fee is gonna be equal to co sign times m fee, Plus I sine of EMC. So this is our left hand side. So all we need to do now is sure that the right hand side is actually equal to this. Um, so this is our left hand side. Okay, So what about our right hand side if you go ahead and plug in feet plus two pi instead of just fee on what we're gonna get, um, I'll just write the whole thing out here is that feet plus two pi. This is going to be equal to e to the I i m times fee plus two pi and then this using oilers. Um, well, actually, let's expand this. First, we can say that using the properties of exponents, we have I and fee times e to the I to pi. OK, so using oilers on both of these terms here, what we're gonna get is actually co sign of m V plus I sine off and see. Um and this could be multiplied by the second term. Um, which is going to be co sign of too high? Oh, I forgotten m here. I've forgotten m here because this M is actually out here. It's one affected out. I had to Ah, distribute. Okay, so we have actually two pi em plus i sine to hi m. And one thing you'll notice is, um, any for any value of em. The critical realization here is that m has to be an integer. So this is just saying that M must be in the set of all integers from zero to plus minus. One plus one is two and so on. So this must be true. That is a defining property of our magnetic quantum number. M. So that is something we need to realize here. And if this is true, then for any value of m ah co sign of two pi times, some integer is always going to be equal toe one. It's always going to be equal to one. And similarly, the sign of two pi times. Any riel integer here is going to be zero. If you just think about how a sign looks anytime you hit some multiple of two pi, it's going to be zero. So that this leaves us with the right hand side, um, of this whole thing, which is the same is our left hand side times one. So that, um, basically concludes our work here, we can say that the left hand side is equal to the right hand side, which is saying that, um, big fee, given a function of fee, is the same as big feat, given a function of feet plus two pi.

In discussion condemned the power series under each of the X and Y equals you. The submission of the expo and in equatorial from zero to infinity. And now in discussion we need to find a migrant stories after function F X. Yes The fight is the integral Ramsar two x. Evil Manistee square minus Kwan 80 first. I'm going to find a powers risk. I'm just in here so we will have the even Manistee square. But lying the powers reason here we should get equity resubmission from here to infinity. When is the square about end up in Pretoria? Are we simply french again down from search infinity minus one power and and then we have to to an environment and pictorial and then we can try to uh minus one So a minus the square minus one which we didn't think of virtue. So it will use any co 20 which again the first time will be one. So I don't understand from one start because I noticed that we can write this 19 1 class Submission from 1 to Infinity -1 Bar and deeper to n november and natalia. So therefore -1. So we would have removed the first time here. So we have no standing from one to infinity -1 About an tell me about you and remember and natalia right And after that we went to do the integral on both sides. So we have the 50. This one moving at 50. And if we do so once again we can switch beyond the under integral and the submission We have the -1. About an inventory real integral on the T. About you. And and it's only to be friends with your ex. Okay Thanks. Promise that your ex. And this will be T. T. And we can signify the end up from 1 to infinity. And this will be -1 other. And uh I'm in vittorio. So this one we should get echo two entirely. Agree with you on this one. Echo to the T two, N plus one define invite you and plus one in varying it from there to X. And that's where we would have called you that so much you want to infinite eight -1 About and infantile. We put the acts inside so we will have the exposure to M plus one and dividing by the Children plus one. So this will be the power series on the function we're looking for.

So we're told this is a function of fi here. Capital fires a function of little fine, which is e to the i m l If I were oilers formula E to the I FIEs coastline five plus I sine phi So its periodic So we can replace that we find that capital fi which is a function of little fine This angle oh is equal to So this is co sign Oh oh, you have in Seville Sigh Plus I sine him civil If I Okay, so we know that, um for period Isett e sign of some value times a Nangle or co sign of some value times Nagel is two pi over that angle. So we know that period is city is two pi over ml well, and no is allowed to have the values. So this is hell. Use the same notation here, Okay, of plus or minus one in all other integer values, all the way up to plus or minus l. So in order for this to always be a multiple of two pi, so we can say sign of K X must equal the sign of two pi over K. Oh, Therefore, for poor period is city to be true. Sign of five plus to fuck plus two pi equals Oh, oh, oh, which is what we were asked to show.


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