Below we have students questions which I have answered by email. The questions and my responses appear in order with the most recently (and probably most interesting to the class) answered questions first. =============================================================================== Fcc:8.04s96 Subject:Printing out notes... -------- A student wrote... |> thanks for typing up notes on scattering theory |> some questions... |> |> how can i print them out from my athena prompt? |> (my dialup computer doesn't run the web) From the athena prompt, the command to print from the printer names "printer" would be lpr -Pprinter /afs/athena.mit.edu/course/8/8.04/8.04s/www/lec_scatt/notes.ps , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Typeo in Prob. 2.5 of PS#8 -------- Dear Class, There is a typo in Prob. 2.5. Where it reads psi(x)=N/sqrt(E-V(x)) exp { i ... it should have double square roots like this... psi(x)=N/sqrt(sqrt(E-V(x))) exp { i ... Be advised that that the same error will appear in the problem set solutions as well. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Re: 8.04 In-reply-to: Your message of "Tue, 23 Apr 1996 16:52:41 EDT." <9604232052.AA08721@m4-167-2.MIT.EDU> -------- |> Hi. Is there a more clever, faster way to solve problem 1 of problem set 8 |> than the way you introduced in lecture today, using what we have learned in |> 8.04? Hi! No, none that I am aware of. :^( Believe it or not, this is the simplest model of molecular bonding that you can write down! The best way I can see is to match the jump conditions at the boudaries, writing even [odd] wavefunctions like Psi(x) = A exp(kx) x<0 Region I A cosh(k(x-R/2))/cosh(kR/2) [-sinh(k(x-R/2))/sinh(kR/2)] 0 -------- |> dear prof., |> everytime i tried printing from athena, the top line |> of every page is wiped out because the page allignment is about |> 3 cm too high. is there a way to fix this problem? |> i usually print the .ps file using: |> |> athena% attach postscript |> athena% /mit/postscript/decmipsbin/multips -l -r1 -c2 file.ps | lpr -Pprint |> ... er2 |> |> |> this saves paper, but i have the same problem even if printing using |> lpr file.ps |> Strange... I have never had this sort of problem. Maybe it is the printer you are using? I always use lpr -Phelios file.ps (Helios is in 16-034) and have never had a problem. Maybe you can try that printer. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Study Guide In-reply-to: Your message of "Mon, 15 Apr 1996 18:53:06 EDT." <9604152253.AA28853@m1-142-22.MIT.EDU> -------- Dear Class, Four announcements/reminders/responses: a) I will hand out the solutions to Quiz II, 1995 at the end of tomorrow's review session. After that time, they will also be available at the undergraduate office. b) Unfortunately, there are no solutions to last year's practice quiz, though I would be happy to discuss them during office hours. c) Remaining office hours before Quiz II: Tue 3:30p-6:30p Tue 8:00p-9:00p (Review session, actually) Wed 5:30-6:30p d) I have made the study guide on the web page (file:/afs/athena.mit.edu/course/8/8.04/8.04s/www/Final_prac/study.html) also available in postscript form at the bottom of that page. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To:evan1@mit.edu cc:muchomas@mit.edu, yeantebi@MIT.EDU, alexb@MIT.EDU, cjborges@MIT.EDU, eboyden3@MIT.EDU, burleigh@MIT.EDU, cmbuttz@MIT.EDU, jocave@MIT.EDU, tetrion@MIT.EDU, benjie@cyber.mit.edu, icicle@MIT.EDU, adc@MIT.EDU, dajulio@MIT.EDU, purple@MIT.EDU, esler@MIT.EDU, mgolinko@MIT.EDU, ggomez@MIT.EDU, govereau@MIT.EDU, mrgraham@MIT.EDU, evan1@MIT.EDU, hackett@MIT.EDU, angieh@MIT.EDU, sunflowr@MIT.EDU, awhoward@MIT.EDU, mijotz@MIT.EDU, jakek@MIT.EDU, jenkile@MIT.EDU, pking@MIT.EDU, funkybob@MIT.EDU, speed@MIT.EDU, lbt@MIT.EDU, lucylim@MIT.EDU, tliptay@MIT.EDU, mert@MIT.EDU, minkovam@MIT.EDU, klmitch@MIT.EDU, vmohta@MIT.EDU, wrawlind@MIT.EDU, rnoguchi@MIT.EDU, ouyang@MIT.EDU, dave@poobert-central.mit.edu, arun@MIT.EDU, jsreese@MIT.EDU, ereich@MIT.EDU, noraa@MIT.EDU, clindsey@MIT.EDU, spidey@MIT.EDU, enstrong@MIT.EDU, zct@MIT.EDU, karent@MIT.EDU, sgtist@MIT.edu, trenczer@MIT.EDU, aaront@MIT.EDU, jvande@MIT.EDU, watters@MIT.EDU, fatih@MIT.EDU, tanyaz@MIT.EDU Fcc:8.04s96 Subject:Extra Review Materials -------- Dear Evan, In preparing for Quiz II, there is also the practise quiz which I gave out last year, and last years problem sets. All are available in postscript form on the web! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== |> |> Dear Michael, |> |> We'll see in lecture TODAY, but, no, by the time you get your |> transcendental equation, all the constants B,B', etc. should all be |> eliminated. |> |> Your question about the wells is a very good question. It just dawned |> on me that I forgot to motivate our study of the square well for the |> class. The fact that you are wondering about that shows that you are |> definitely thinking like a scientists. Congrats. |> |> I'll address that question for you and the rest of the class in |> lecture. |> |> Also, I'll see about moving the lecture up one hour. If it helps more |> people that it hurts, then we'll do it. |> |> |> |> , |> Tomas Arias, PhD |> Assistant Professor |> Department of Physics |> Massachusetts Institute of Technology |> Cambridge, Massachusetts , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To:evan1@mit.edu Fcc:8.04s96 Subject:Normalize 2.1? Yes! -------- Dear Evan, I got your v-mail. Yes, for Prob. 2.1, please normalize your answer. I glad to see everyone getting an early start on this one. It is an important problem set. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Cult Scientist Fcc: 8.04s96 Subject: Alternate Derivations of the Schrodinger Equation In-reply-to: Your message of "Sun, 07 Apr 1996 15:13:29 EDT." <9604071913.AA29120@hayden-3.MIT.EDU> -------- Dear Beng, Yes, I have been very very busy this last week, so I hadn't had time to properly answer your questions. I don't have time to respond in detail yet. Perhaps you'll have to remind me in a few weeks to write back more. Basically, you are slightly off, but on the right track. In fact, if you follow your track to its logical conclusion, you will end up deriving the Dirac Equation, the equation for electrons which takes relativity into account, and from which Dirac was able to predict the existence of the positron, BEFORE THERE WAS ANY EXPERIMENTAL EVIDENCE for such a thing. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Details from PS#5 needed for PS#6; The amount of flow in PS#5 was -8/(3*pi), the time was to=(2/3) m L^2/pi/h-bar and the wavefunction was psi(x,t)=1/sqrt(L)[ sin(k1 x) exp(-i w0 t) - sin(k2 x) exp(-i w1 t) ] where k0=pi/L, k1=2*pi/L, w0=h-bar k0^2/(2m), w1=h-bar k1^2/(2m). You can get a new copy of PS#5 by executing athena% add 8.04 athena% lpr /mit/8.04/8.04s/www/ps5_96/ps_all.ps or simply view it from your machine with athena% add gnu athena% ghostview /mit/8.04/8.04s/www/ps5_96/ps_all.ps , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: "Paul Konigsberg test sig file." Fcc: 8.04s96 Subject: Reflecting Packets In-reply-to: Your message of "Tue, 02 Apr 1996 15:43:05 EST." <9604022043.AA12824@marinara.MIT.EDU> -------- |> |> |> I was wondering about one of the animations you showed, |> the one with the step function for the potential. |> After hitting the step why was the probability |> greater in the higher energy region? I thought the |> greatest probability would be to exist in the minimum potential |> energy area. |> |> thank you |> |> Paul |> Very good question, Paul! You would be right if we were talking about the ground state, some equilibrium situation where the state can adjust to the environment. In this case the particle just propagates through. The way to think about it is to realize the the relative amount of reflection is related to the size of the step in the potential energy. A large step will, clearly, result in most of the packet being reflected. In the limit of a very tiny step, things just look like free space and most of the packet keeps moving in the same direction. What we had in the film, then, was a case where the size of the step in the potential was significantly less than the average kinitic energy of the packet. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== |> Dear Prof. Arias, |> I have another question about 3. Are the m you state is an arbitrary |> constant the same m as in the TISE, or something different? |> From, Michael S. Golinko |> Dear Michael, It is a constant DIFFERENT than the "m" in the TISE. I agree that it is a little confusing, but unfortunately, this is the standard notation. Whichever "m" is meant at each point should be clear from context. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: mgolinko@MIT.EDU (Michael S. Golinko ) Fcc: 8.04s96 Subject: Re: physics In-reply-to: Your message of "Mon, 01 Apr 1996 15:19:50." <9604012019.AA00268@MIT.MIT.EDU> -------- |> Dear Dr. Arias, |> How was the conference? I hope everything went well! I have a question |> about 8.04. In 2.2.1, do you mean the negative sign to distribute through |> both terms in the numerator, i.e. -2hAB - Fm, or just the 2hAB term?, |> because when I did the work I got that the sign on Fm should be positive? |> Sincerely, |> Michael S. Golinko Dear Michael, The conference went very well, thank you. You can see our talks on the web at: "http://elrio.mit.edu/Talks/talks.html". No, I'm pretty sure about the sign. Your problem may be that A(t) is defined a little differently in this problem. Note that NOW we have psi=exp(A*x^2+B*x+C) NOT psi=exp(-A*x^2+B*x*C), like we did last time. Also, note that V(x)=-Fx with a minus sign. Probably one of these two is your problem. Good luck on the Pset! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: vmohta@MIT.EDU Fcc: 8.04s96 Subject: Re: Problem Set 4 grades In-reply-to: Your message of "Sat, 30 Mar 1996 21:39:04 EST." <9603310239.AA03904@bolognese.MIT.EDU> -------- Dear Vivek, In response to your zephyr: |> Prof. Arias, I have recently been reading about classical mechanics and |> a few questions seemed to arise |> naturally when I thought about |> translating to quantum. |> Is there way we can salvage the notion |> of phase space? |> And is there a resulting counterpart to |> Louiville's Theorem? That is a very good question. As you realize, the notion cannot be carried over directly because the notion of a "point" in phase space violated the HUP. If we blur our points out into little wave-packets, then Ehrenfest's Thm, which we will cover Tues, shows that those packets move like classical phase space points, and then Louiville will take over in the classical limit. In semiclassical quantization, we look in phase space an merely insist that our orbits enclose an integral number of multiples of h. So, there is still the notion of phase space here, but it is approximate. You will also learn that the equations of the time evolution of our operators look very much like the classical canonical equations of motion in form, and so one can produce something creative along the lines of Louiville's theorem. Then, instead of looking at the density of points in phase space, one looks at an alternate formulation of quantum mechanics in terms of objects called "density matrices" rather than wavefunctions. We probably won't get to discuss those in 8.04 this year. Great question!!! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Hints/Typos for PS#6, Prob. 2.2.2 -------- Dear Class, Please note the following for Prob. 2.2.2 of PS#6: ------------------------------------------------------------------------------- 1) Where the PS currently reads: Re[A(t)]=(-1/4)/(sigma^2+(h-bar t / 2 m sigma)^2) note that, technically, there is an integration constant t0 that makes the most general form Re[A(t)]=(-1/4)/(sigma^2+(h-bar [t-t0] / 2 m sigma)^2) YOU MAY TAKE THAT INEGRATION CONSTANT TO BE ZERO and keep the form given in the problem set, or carry the constant t0 through until the end of the problem if you wish. ------------------------------------------------------------------------------- 2) Where the PS currently reads: Re[B(t)]=(m sigma^2 t (Ft + 2 v0 m))/(4 sigma^4 m^2 + h-bar^2 t^2) it SHOULD read Re[B(t)]=(m sigma^2 t (F t + 2 v0 m + 2 x0 m/t))/(4 sigma^4 m^2 + h-bar^2 t^2) where x0 comes from an integration constant. (This expression will also have the replacement t -> (t-t0) if you choose to carry the t0 term through the problem.) ------------------------------------------------------------------------------- 3) HINTS: for finding the solutions: A) Each time you solve one of the Differential Equations (for A(t) or B(t)), you general integration constants which, in general, will have both real and imaginary parts. You may wish to write them them separate to keep proper track. B) Recall how to get the real part of a fraction: Re { (a+bi)/(c+di) } = Re { (a+bi)(c-di)/(c+di)(c-di) } = (ac+bd)/(c^2+d^2) C) To solve for B(t), assume that it has the form B(t)=beta(t) A(t), where beta(t) is unknown. Substituting this for B(t) will simplify considerably your Diff Eq for B. This is a standard "trick" from 18.03. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To:minkovam@MIT.EDU Fcc: 8.04s96 Subject: Error on problem 2.4 In-reply-to: Your message of "Sun, 24 Mar 1996 17:02:50 EST." <9603242202.AA22912@alfredo.MIT.EDU> -------- |> I seem to be stuck with problem 2.4. I can easily show that |> delta_x*delta_p >= 1/2|<[ksi, rho]>| |> >= 1/2|<{ksi, rho}>| |> And while |<[ksi, rho]>| = |<[x, p]>|, this is not so for anti-commutator {} |> <{x, p}> = 2

+ <{ksi, rho}> |> Getting the latter expression led me to think that in fact |> delta_x*delta_p >= 1/2|<{x, p}>| might not necessarily hold: |> what if we have a Gaussian wavepacket with finite uncertainty |> in x and p with delta_x*delta_p = h/4pi at some very large (-> infinity) |> position x. Then <{x, p}> = 2

+ <{ksi, rho}> is very large -> |> infinity while delta_x*delta_p is finite. I am sorry Mariya, I do owe you an apology. I really did miss the point of your question entirely. I was in error when I wrote on the problem set Delta_X Delta_P >= |<{x,p}>|/2. I should have spotted this, because there is clearly no a priori limit on . What I should have written was Delta_X Delta_P >= |<{ksi,rho}>|/2, which follows directly from the previous problem. There is an immediate way to get as what you said. For large and

, which we have in classical physics, we expect |<{x,p}>|/2 to approach the classical limit |<(xp+px)/2>|-->

, which clearly may be made arbitrarily large while keeping Delta_X Delta_P fixed. In sum: 1) The second part of 2.4 should read delta_x*delta_p >= 1/2|<{ksi, rho}>|. 2) Mariya very nicely spotted the error and was the first to bring it to my attention. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: icicle@MIT.EDU Fcc: 8.04s96 Subject: Typo in 2.4, PS#5. Math saving hints to 3.4 -------- |> |> i was doing 8.04 ps #5 |> i couldn't solve 2.4 and 3.4 |> i thinkn i'm missing a little trick in 2.4. |> how do you get rid of terms? It seems we have a typo; the second inequality shoud read, Dx Dp >= (1/2) | <{chi,rho}> |^2, NOT ``Dx Dp >= (1/2) | <{x,p}> |^2''. |> for 3.4, |> do we just do gaussian integrals to get the answer? |> i'm just missing out of little math tricks... |> would you be so kind to answer my questions? |> and leave those girls alone for five minutes? Yes, just use Gaussian integrals. Actually, if you just notice where P(x) is centered, by symmetry you can just read off . Similarly, using your Gaussian integral knowledge, you can get P~(k). From where that is centered, you can get without doing any integrations. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Re: ps#5 In-reply-to: Your message of "Tue, 26 Mar 1996 11:41:30 EST." <9603261641.AA02229@carbonara.MIT.EDU> -------- |> Prof. Arias: |> |> What's the xact meaning of the average of a quantum operator? |> We have a few of those in problem 2. |> Is < O^ > = < O^ psi > / psi = (psi, O^ psi) ? |> |> Often in quantum mechanics, physicists don't make the distinction between the average of an observable and the average of an operator, although technically they should. So, < O^ > = < O > = (psi, O^ psi). Your other expression, "/psi", has no meaning. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts <9603220653.AA07655@pick-gun.MIT.EDU> -------- ===============================================================================Fcc: 8.04s96 Subject: Good questions about Tuesday's Lecture!!! In-reply-to: Your message of "Fri, 22 Mar 1996 01:53:11 EST." Now, to your questions: |> * Parseval's Theorm. Is this is a result of the "niceness" of |> psi and phi, or of the inner product ( , ) or of both? Good question! Ultimately, Parseval's thrm is a consequence of the fact that we defined the transform just so that normalization was maintained (remember the funny game with the sqrt(2 pi)'s?). You can prove that if (psi,psi)=(psi~,psi~) for all psi and their alternate representations (transforms), then (psi,phi)=(psi~,phi~) for all psi and phi and their transforms psi~ and phi~. So, I would say that it comes from the definition of inner product and the physical constraint that we maintain normality. |> * Hermitianness. I presume from the notes that Hermitianness |> stems from the linearity and physicalness of operators. |> However, it is still not quite crystal clear. Is any |> special relationship between psi and phi required? |> (Where psi and phi are the two functions between which |> the operator may leap.) Very good!!! Yes, you are exactly right that Hermiticity stems from physicalness and linearity of operators. Note that no special condition between psi and phi is required. From class we showed by example (psi,O psi)=(O psi,psi). However, you can repreat those arguments and show that in general (psi, O phi)=(O psi, phi). You should try this as an exercise; you do it by first writing things down in the represention associated with the observable O and then seeing how it transforms into any other representation. |> * On to Dymanics! (This is where the majority of questiond reside) |> - Once you expressed psi in terms of U hat operating |> on psi, what motivated taking the derivative wrt t' ? Good one! Actually, you can formulate quantum mechanics just in terms of U^ ("U hat"). That takes you down the road to Ferynmann's diagrams... On the other hand, since you guys are more familiar with differential equations, we will learn the Schrodinger equation in 8.04. Here the idea is that since psi(t) determines psi for all later t' > t, then we expect that a first order differential equation in time will define the dynamics. (If we need more information like psi and psi' then we'd need a higher order diff eq.) |> - Okay, this is hairy, but your expression after taking the |> derivative wrt t' was: (note, my qustions will be in []'s) |> |> partial (wrt t') evaluated at t'=t : partial (wrt t) of psi(x,t) |> [why is this wrt t and not t'] ... at first t' was different than t, but since we are evaluating the derivative at t=t', after taking the derivatives, t' just becomes t. |> = partial (wrt t') U hat (t', t) psi(x,t) |> [why is U hat still a function of t'? Does the |> derivative not affect it? Or have you just not |> evaluated it yet?] Very good, we just haven't evaluated the derivative yet. |> * You wrote, if psi is pure wrt H, psi (underscore H) of t = (blah) |> Why does psi only depend on t? Here I am just emphasizing the time dependence of psi. I should have written something like psi(x,t) ~ exp(-iwt), or more explicitly, psi(x,t)=f(x) exp(-iwt). This tells us O^ psi_H(x,t)=-iw psi_H(x,t)= [ ... from de Broglie ... ] -i (H/h-bar) psi_H(x,t). Here, H is not an operatorm, it is just some value for the energy/Hamiltonian. Rewriting this a little more neatly gives, i h-bar partial_t psi_H(x,t) = H psi_H(x,t) = i h-bar O^ psi_H(x,t) |> * You also derived what H hat operating on psi of x = (different blah) |> But then, you turned around and used this operator on psi (x,t) |> Is this legal? At this point we know that there is a general operator O^ which always give the time derivative of psi(x,t) and we are playing a game to use the de Broglie hypothesis to tell us what that operator must be. We now know that i h-bar O^ acting on each and every pure state psi_H(x,t) just gives back the value H for that pure state times the pure state back again. But, that is just what the operator H^ does! Thus i h-bar O^ IS the operator H^. Now that we know this, we can write i h-bar partial_t psi(x,t) = H^ psi(x,t) for general states psi, because we knew that by definition i h-bar partial_t psi(x,t) = i h-bar O^ psi(x,t), and we just learned that i h-bar O^ = H^. |> *On #1, there are several similar processes and integrals to grunge. |> If after doing a couple, we are enlightened, and see the |> secrets, do we really have to grunge through them all? Yes, you have to. You should use this as an opportunity to work on your style and reduce the mathematical operations on the page to the bare minimum. It is good for you to get this experience so that when you do more complex manipulations in your future career, you will be able to do the steps more easily and with confidence. |> *Also on #1, I believe parts 2 and 3, you have given psi of p |> instead of the usual psi of k. I know this only changes |> things by a factor of h bar, but does it also change the |> normalization of psi for use in the Fourier transform |> as given in problem set 4? |> Yes, very acute of you to notice! The normalization for the Fourier Transform is changed. The sqrt(2 pi)'s will now read sqrt(2 pi h-bar). The normalization constants do not enter into the expressions for the operators, however. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Meaning of infinite wavelengths In-reply-to: Your message of "Wed, 20 Mar 1996 21:56:02 EST." <9603210256.AA00617@m11-116-12.MIT.EDU> -------- |> Hello Prof Arias, |> |> I am confused with a result obtained on the last P.S. In question 2.4 you ha |> ... d us |> plot the real part of Psi(k) vs. k, where the wavelength of the cos term in |> ... k |> space is 2pi/x(0). If X(0) is arbitrary -simply the center of the probabilit |> ... y of x |> - why is there a singularity of the wavelength at X(0)=0, and what does that |> ... |> mean? |> |> thank you |> Julio Hi Julio! No problem to answer your question. 2pi/x0 --> infinity as x0 -->0, yes. The meaning of a wavelength that --> infinity simply means a phase that isn't changing at all as we vary our variable (in this case k). This means that that only phase involved in the integral giving psi(x) is the exp(ikx) term. Taking the derivative of that, then, tells us that psi(x) is "centered" near x=0, which makes sense because the center of the packet is given by x0, which in this case -->0. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Re: PS #5 In-reply-to: Your message of "Tue, 19 Mar 1996 01:44:39 EST." <9603190644.AA03700@MIT.MIT.EDU> -------- |> THANK YOU!! |> THANK YOU!! |> THANK YOU!! |> THANK YOU!! |> THANK YOU!! |> YOU'RE WELCOME!! YOU'RE WELCOME!! YOU'RE WELCOME!! YOU'RE WELCOME!! YOU'RE WELCOME!! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: How to hand in PS5 if you are away for Spring Break In-reply-to: Your message of "Tue, 19 Mar 1996 01:46:51 EST." <9603190646.AA09592@m1-142-17.MIT.EDU> -------- |> Pr. Arias, |> I am going to be away (at home) on Friday March 29. Does that mean that just |> for that reason my deadline is this Friday? (I am planning to leave this Fri |> ... day) |> Is it possible to submit it on Sunday March 31? (that's when I come back) Dear xxxxxx, Normally, yes, your due date would be different because your travel plans are your own doing. Sunday, March 31 will be too late to insert your problem set into the cycle for the graders. However, because I feel a personal commitment to fairness in such cases, I will do everthing that I can to accomodate you. You can either 2-day express mail your solutions for $2.90 US-post (if home is in the US) or FAX them, so long as they get to my offices before 11:59p on Friday, and I will take the time to personally deliver your solutions to the graders. =============================================================================== Mail them to ------------------------------------------------------------------------------- Prof. Arias Rm. 12-110 MIT 77 Mass Ave Cambridge MA 02139 =============================================================================== Or FAX to ------------------------------------------------------------------------------- Prof. Arias (617) 253-2562 =============================================================================== , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Clarifications of assuptions in PS#5 In-reply-to: Your message of "Mon, 18 Mar 1996 21:24:27." <9603190224.AA26629@MIT.MIT.EDU> -------- |> Dear Dr. Arias, |> Sorry to bother you with questions so early in the week, but here goes: No problem. It is good to get an early start. |> In 1.3, can we assume that f(p) is of the same form as f(x)? Yes. |> In 1.4, must we really "derive" this relation or just extrapolate from one |> dimension in to three? Yes, please "derive" them going through the formal steps. If you need help with that, be sure to ask in recitation. Also, you should expect to get the forms that you would "extrapolate". |> In 2.3, do we assume that x and p commute, i.e. [x,p]=0? Actually, [x,p] = i hbar. You can prove that by applying p and then x to a function psi(x) and then from that subtracting the application of x and then p on the same wavefunction. That is, compute x[(p psi)]-p[(x psi)] You are welcome, , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Re: 8.04 In-reply-to: Your message of "Sun, 17 Mar 1996 20:54:40 EST." <9603180154.AA03366@w20-575-86.MIT.EDU> -------- |> -------- |> Profesor Arias, |> I am very confused about the use of the * function on the last problem |> set. I could not find an explanation of it in the class notes. Could you p |> ... lease |> give an explanation of *? Thank you. |> Sure, no problem. I wish you had asked about this earlier. The "*" function just means take the complex conjugate. In many high-school texts this is indicated by placing a horizontal bar on top of the complex number, but that soon becomes unwieldy in complicated expressions, so we just use the "*" notation instead. So, for instance (a + b i)* = (a - b i) Let me know if this still isn't clear. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Problem Set # 5 Changes -------- Dear Class, It looks as though we will not have time to cover probability currents next week. Please cancel problems 3.5, 3.6 and 4.4 from Problem Set 5 (the one due NEXT Friday, March 22), which deal with this topic. I will assign them on a future problem set. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: The "monster" convolution in problem 3 In-reply-to: Your message of "Thu, 14 Mar 1996 22:08:25." <9603150308.AA00528@MIT.MIT.EDU> -------- |> Dear Dr. Arias, |> I have come to the end of my rope with problem 3.4. I get an expression |> for psi(K) that has a very complicated exponent, and it does not seem like |> there is any way to take the fourier transform of it. I believe my |> expression to be correct, but I can not see how you would want us to |> integrate something so ugly! Can I just use a hand waving argument to show |> that this expression and psi(x)*phi(x) are equivalent, or must I grind that |> monster out? |> MSG |> Dear Michael, The "monster" shouldn't be so bad, all the integrals on the problem set end up being Gaussians time plane waves of various sizes and various centers. Just keep completing the squares of those exponents and it will all come out quite pretty. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Form of result for Problem 3.2 In-reply-to: Your message of "Thu, 14 Mar 1996 19:40:37 EST." <9603150040.AA00820@alfredo.MIT.EDU> -------- |> |> I deleted my messages on the corrections to the problem set so this |> might be my problem. For 3.2, I find that the result is |> exp[iko(X-Xo]f(x) but the ps has a slightly different result. I can't |> find my mistake so I was wondering what you think the problem might be. |> Sorry I can't be more specific. |> |> Thanks, |> Bob Trenczer Dear Bob, In case you delete them again, I put COPIES of ALL 8.04 CORRECTION MESSAGES and HINTS on the web page at http://web.mit.edu/8.04/8.04s/www/questions/FAQ You can read them with Mosaic or netscape: athena% Mosaic http://web.mit.edu/8.04/8.04s/www/questions/FAQ & will do it. In way of a specific response, your result for 3.2 should look something more like exp(iko(X-Xo)) f(X-Xo), where f(x) is a Gaussian. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Convolution Problem In-reply-to: Your message of "Thu, 14 Mar 1996 18:29:49." <9603142329.AA24125@MIT.MIT.EDU> -------- |> Dr. Arias, |> 3.4 and I'll be done! I already found phi(x) using a fourier transform, |> and found it to be 1 |> --- * exp{(-k^2(x+x1)^2)/4W)} |> (2W)^.5 |> |> I then multiplied these psi(x) and phi(x) to get another expression. My |> question, first is, to find the convolution do I integrate the expression: |> |> |> constant(exp{-ikxo -D(k-ko)^2 - i(K-k)x1 - W(K-k)^2}dk |> |> when i integrate the above expression I get some constant with an |> exponential whose exponent does not depend on k at all? Can this be right? |> Please help me! |> MSG |> Dear Michael, You are definitely on the right track! Note that the result of the convolution, Psi(K) (capital Psi) is a function of capital "K" not lower case "k", so all is well. Once you have Psi(K), you can get Psi(x)=Int{ dK exp(iKx)/sqrt(2 pi) Psi(K)}, integrating over capital K in the usual fashion. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Hints for PS#4: 2.4, 3.3, 3.4 In-reply-to: Your message of "Thu, 14 Mar 1996 02:27:27." <9603140727.AA28970@MIT.MIT.EDU> -------- |> Dr. Arias, |> I am nearly done with the problem set except for a few things upon which |> I need clarification. That sounds great Michael! > First, in 2.4 how do you compute the center of the |> psi(k) graph, it does not seem like there is any straightforward way to |> find the maximum? I actually mean for you to find the center of the envelop, not necessarily the precise position of the maximum of the function itself. |> Second: In 3.3, I am not sure how to explain the effect in terms of |> stationary phase, can you give me a hint please? Sure, use stationary phase to show that the center of the function f(x) gets shifted by x_0 if you multiply f^(k) by exp(-i k0 x). You should know by now how to find the position of the center of f(x) given as an integral of complex numbers by using stationary phase. |> Third: I am not sure which operation you are referring to as the |> convolution, are you saying that the convolution of psi(k) and phi(k) is |> integral(dk*psi(k)*phi(k-K'))? or something else The convolution, Psi(k), of psi(k) and phi(k) is defined to be the integral Psi(k)=int{ dk psi(k) phi(K-k) } , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Problem 4.2 -------- Dear Class: Problem 4.2 on Problem Set # 4 states that the solution from last year's problem set will not be accepted. This remains the policy. The solution which will not be accepted is decribed in the hint for problem 4b) on last year's Problem Set # 5, which you can down load from the web. There is little chance that you would come up with last year's solution unless you were using a bible from last year. For those of you not yet used to using the web, I highly recommend that you learn to use it. The hint from last year read something like-- (Hint: Define I(alpha)=(f+alpha g,f+alpha g), minimize I with respect to alpha and use the fact that (h,h)>=0 for any h.) --but is much more easily read from the web. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Response to voice mail question: ``What is a pure state?'' -------- Dear class, A few students have been asking me about the precise definition of a pure state, especially wondering what the difference is between a "pure" state and a "quantum" state. This is a very good question. It is a very important issue at the heart of what we are doing, and I am happy to explain it as many times as necessary. GENERAL QUANTUM STATE: Basically, once you know the QUANTUM STATE of a system, you know everything there is to know about that system, namely the probability distributions for all possible observables (x, p, E=p^2/2m+V(x) are the most interesting ones in one dimension). Note that these probability distributions could be almost anything, as long as they don't violate certain basic principles such as the HUP. We could have (and normally do) some spread in the distributions for x, p, and E. We learned in class on Tuesday that you can specify the QUANTUM STATE of a system by giving the wavefunction psi(x). We will discuss this more in depth on tomorrow. PURE QUANTUM STATE: PURE QUANTUM STATES are very special examples of QUANTUM STATES. In a PURE STATE, one of the different probability distributions has ZERO SPREAD, you always get the same result when you measure that observable. An example of this would be a case where Delta X = 0. That would be a pure state with respect to X. Of course, by the HUP, we know that a pure state with respect to X implies that Delta P = Infinity, so that you cannot have a pure state with respect to X and with respect to P at the same time. Genereally, it is not possible to be a pure state with respect to two different variables at the same time, except for very special and important cases which we will learn about later. Because being a pure state with respect to one variable does not mean that the state is pure with respect to others, when talking about a PURE STATE, you should always state the observable with repect to which the state is pure. If Delta X = 0, for instance, we say that we have a pure state WITH RESPECT TO X. If there are any more questions about this, please feel free to ask. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: mgolinko@MIT.EDU (Michael S. Golinko ) Fcc: 8.04s96 Subject: Re: Extra Hint on Problem 4.2 (Cauchy-Schwartz Inequality) In-reply-to: Your message of "Wed, 13 Mar 1996 00:11:24." <9603130511.AA16252@MIT.MIT.EDU> -------- |> Dr.Arias, |> in question 4.1, what does the star mean next to f(q) and (f,g)*? |> MSG |> "z*" means the complex conjugate of the complex number z. f*(x) means the complex conjugate of the function f(x). Note: f*(x) = [f(x)]* , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Extra Hint on Problem 4.2 (Cauchy-Schwartz Inequality) -------- Dear Class: For problem 4.2 you will find the following inequality useful (be sure to include this part of the proof in your solutions) 0 <= (|f(x)||g(y)|-|f(y)||g(x)|)^2 = |f(x)|^2 |g(y)|^2 + |f(y)|^2 |g(x)|^2 - 2 |f(x)||g(y)||f(y)||g(x)| Thus, |f(x)|^2 |g(y)|^2 + |f(y)|^2 |g(x)|^2 >= 2 |f(x)||g(y)||f(y)||g(x)| (Note that in the above, by "<=" and ">=" I mean "less than or equal to" and "greater than or equal to", respectively.) , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Justin Cave Fcc: 8.04s96 Subject: Re: Quiz 1 HUP Question 2 In-reply-to: Your message of "Tue, 12 Mar 1996 21:35:00 PST." <31465E84.2380@mit.edu> -------- Dear Justin, |> Continuing with the question I asked after lecture today about the |> slit HUP question on the quiz... |> |> Why is it that when we used HUP in questions on problem sets (ie |> confining a ball in a potential well) we determined that localizing a |> single object to a small area gave us a large uncertainty in its |> momentum if HUP only applies to a distribution? On the problem sets what we were localizing was the probability distribution of where the particle could be found. What we then did was to compute THE AVERAGE ENERGY we would expect to find under those circumstances and then minimized that quantity. |> |> My interpretation of these problems is that individual particles have |> some "fuzziness" in their position and momentum such that it is |> impossible to precisely measure either quantity. That is, in the |> photographic plate example, an electron will make a smear on the plate |> in recording its position |> |> photon photographic |> wave plate |> _ _ | |> / \ / \ | ] |> / \ / \ | ] |> \ / \ | ] /_\ x |> \_/ \_ | ] |> | What the fuzziness is is not actually the fuzziness in the momentum or position that we get when we measure those observables (we will get definite answers on our momentum/position meters), the "fuzziness" as you call it is in the range of possible outcomes (i.e., the width of the prob dists) we can expect. |> PS This problem has been eating away at my concentration since yesterday. |> I can't concentrate on my other problem sets, my mind keeps wandering |> back to this question. PLEASE HELP!!! It's a good question to keep thinking about, but don't let your other courses suffer either!!! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: mgolinko@MIT.EDU (Michael S. Golinko ) Fcc: 8.04s96 Subject: Re: Dr. Arias In-reply-to: Your message of "Tue, 12 Mar 1996 01:47:21." <9603120647.AA07591@MIT.MIT.EDU> -------- |> Dr. Arias, |> I tried question two on the problem set and was wondering for 2.1, if N |> should be in terms ofm some constant times an exponential function, or just |> the constant itself, I would think it would be the entire expression one |> gets from imposing the normalization condition. |> Michael S. Golinko |> Dear Michael, I'm glad to see you are getting an early start. N should be some constant with a few square roots, and pi and D and some constats like 2 in it. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject:Quiz I Solutions -------- Dear Class: Written solutions to Quiz I are now out at the UGPO. I suggest you pick one up before attending Monday's recitation, when you will go over the quiz. , -Tomas =============================================================================== Subject:Typo on Problem Set 4. -------- Dear Class: I have just noticed a typo in the fourth problem set. In Equation (4), "x0" should read "x1" and "(k-k0)^2" should read "k^2". , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Any one interested in helping...? In-reply-to: Your message of "Fri, 08 Mar 1996 06:39:10 EST." -------- PS: You can cut and paste all of the class grades from the web page using the mouse. If any one is interested in making some suitable *.gif files of more detailed plots, I will be happy to post them... , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Re: Grades for Quiz I are Posted In-reply-to: Your message of "Fri, 08 Mar 1996 06:39:10 EST." -------- |> I just have to tell you that the web page and the system you are using to |> post grades on it is amazing |> Also I would not mind seeing a better histogram for exam grades even if it |> is not by grades a,b,c,d,and a+ |> if you could also make histograms for problemsets (not just one) |> see you should not have even made the first histogram for a ps now we are |> asking you for all |> and also a cumulative histogram would help |> |> |> Thank you in advance and my best wishes in making it all work |> Thank you. I am glad you appreciate the effort. Thank you also for the advice, I will take it under consideration.... , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Grades for Quiz I are Posted -------- Dear Class: Your official grades for Quiz I are now posted on the web page. Within the next 30 mins, I will also post over-all statistics. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Re: Formula Sheet for Quiz I In-reply-to: Your message of "Wed, 06 Mar 1996 10:54:04 EST." <199603061554.KAA00573@aaront.mit.edu> -------- |> |> >>The new formula sheet does not contain things |> >>which you should know by heart such as h=p/lambda. |> |> |> Profesor Arias, |> Don't you mean to say "p=h/lambda"? I realize this was |> simply a typo, but I found it humorous since this is |> something everyone "should know by heart." :) |> |> Aaron Ha-ha! Yes, actually, it is quite a humorous typo I must admit!!! :^) Too many all-nighters up at 4:30a with the flu... I suggest that everyone not follow my example and the class does make sure to get enough sleep before the quiz. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc:8.04s96 Subject:Formula Sheet for Quiz I -------- Dear Class: Note that I have decided to use a different formula for Quiz I from the one used last year. The new formula sheet does not contain things which you should know by heart such as h=p/lambda. The new sheet is now available at the UGPO with the other course hand-outs for 8.04. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Subject: Re: Studt Break for 8.04 Students after Quiz I/Small dp, large dx. In-reply-to: Your message of "Wed, 06 Mar 1996 02:08:31 EST." -------- |> |> |> |> Hello Prof. Arias, |> |> |> |> Two things, |> |> First, I was wondering if you could announce that SPS will be having the |> |> study break after the 8.04 exam and it will be held in the physics |> |> commons room. |> |> I will let the students know at the start of the exam! |> |> > (p.s. your welcome to stop by and saying hello :-). |> |> also, I was talking to a friend about the uncertainty principle, and |> |> I gave him an example of a position measurement making the |> |> momentum uncertain, but to my dismay, I couldn't think of any |> |> example of the reverse (measuring momentum and so losing info about the |> |> position). I was wondering if you could think of an example (I couldn't |> |> find any example of this in any of my text books). |> |> Sure, if you want to know very well the momentum of a particle, you |> have to measure its position at two different times (a long time |> apart) so that you can take the difference and divide by the time to |> get the velocity and then momentum. Now, because you wait a long |> time so that the positions are far separated, you don't really have to |> know the positions at the end points very well. This turns out to be |> a good thing, because if you measure the poitions with photons, for |> instance, you must use very long-wavelength photons so that you don't |> disturb the momentum of the particle during the measurement. |> |> But, because you must use long wavelength photons so as not to disturb |> your momentum measurement, you cannot know very well where the |> electron is. So you have p defined very well but x not very well at |> all!!! |> |> |> |> , |> Tomas Arias, PhD |> Assistant Professor |> Department of Physics |> Massachusetts Institute of Technology |> Cambridge, Massachusetts , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Printing BIG 8.04 files/Solutions to Last Year's quiz In-reply-to: Your message of "Tue, 05 Mar 1996 12:10:33 EST." <9603051710.AA00729@m66-080-6.MIT.EDU> -------- |> |> Prof. Arias, |> |> I have two comments/questions. |> |> 1) I am unable to print the formula sheet for last spring's exam. |> I get an error message saying the file is too big...I only get |> the cover sheet. How can I fix that or if it isn't fixable can |> you hand out a copy in class? The following commands should print the documents you desire if they are too big: add 8.04 lpr -Pyour_favorite_athena_printer /mit/8.04/8.04s/www/quiz1_1995/quiz.ps lpr -Pyour_favorite_athena_printer /8.04/8.04s/www/quiz1_prac/quiz.ps |> |> 2) I cannot find solutions to any of last year's quizzes. If they |> are not online could you make them available in some other fashion? |> There are no solutions available for last years quiz. The average on it was almost 100%, so we never bothered to make up solutions. Any questions you have on it I will be happy to review at the review session. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Fcc: 8.04s96 Subject: Fixing Grades on Official List/Nature of Quiz I In-reply-to: Your message of "Tue, 05 Mar 1996 22:05:18 EST." -------- |> Professor Arias |> |> I have two quick questions for you regarding 8.04. The first is about |> problem set 1. Originally I didn't have a grade on the ""pseudonym" page, |> but now I have a zero in that column. I received a xx/100 on the |> assignment.... > ... I can bring it in to the test to |> verify the grade if you want. Anyway, I just wanted to get that taken care |> of now, so when the end of the term rolls around, I'm not scrambling. |> |> My second question is regarding the quiz on Thursday. I was looking over |> last years test and it appears to be more "touchy-feely" than the average |> MIT test. By this I mean that there are more questions on why something is |> so and fewer on actually calculating something. In preparing for the test, |> I'm going to try to be able to do both types of problems well. But, I was |> wondering if it will be advantageous to prepare myself for a particular |> type of question. Any guidance would be appreciated. Thanks for your |> time. |> DO NOT USE LAST YEAR'S TEST OF ANY KIND OF GUIDE TO THIS YEAR'S TEST!!! I only provide you with those materials to give you some practice problems beyond the problem sets and the texts. THIS YEAR'S QUIZ I WILL BE MORE DIFFICULT THAN LAST YEAR'S! To prepare there are three general areas to concentrate on... 1) Questions relating to understanding of the experiments we discussed. 2) Questions related to physical concepts we learned, (HUP, etc.) 3) Questions involving computations (such as BS Quantization.) , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Anti Matter Fcc: 8.04s96 In-reply-to: Your message of "Mon, 04 Mar 1996 09:37:57 EST." <9603041437.AA27856@m66-080-5.MIT.EDU> -------- |> |> if energy of photon is hv |> then hv=mc^2 |> |> therefore mass of photon is hv/c^2 |> |> but then photon couldn't possibly have mass |> since it would take infinite amount of energy |> to move it at speed of light(gamma*rest mass*c^2). |> |> question: why does hv=mc^2 fail? |> |> beng Hi Beng, The problem is your first equation. It should read hv=sqrt((mc^2)^2+(cp)^2) Then, in the limit m -> 0, we get the familiar hv=cp => p=h/lambda. What you are missing is that the kinetic energy of the photon far outweighs any contribution from its mass. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: lucylim@MIT.EDU Fcc: 8.04s96 Subject: Re: One final question! In-reply-to: Your message of "Fri, 01 Mar 1996 00:42:22." <9603010542.AA08910@MIT.MIT.EDU> -------- |> I apologize for the multiple messages. I'll try to learn to |> consolidate them! |> In Question 3, what is the meaning of "infinite square well |> potential"? At first it seems to be referring to the Particle |> in a Box, but in the notes it says that there is no potential |> in that model. |> |> Thanks for your patience, |> Lucy Lim |> No problem Lucy, "Infinite Square Well" is the technical term for a particle in a box. The idea is that if the potential looks like: ^ V(x) ^ | ^ . | . . | . . | . . | . . | . . | . . | . . | . . | . . | . . | . .............................................. ---------------------------------------------------------------------------> -L/2 L/2 x where the potential is zero V(x)=0 from -L/2 to L/2 and then is infinite outside of that range, then the particle is effectively "trapped" in the region -L/2 to L/2 and can not get out. That is what it means to be in a "box", trapped a region where there is "no potential," V(x)=0. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: lucylim@MIT.EDU Fcc: 8.04s96 Subject: Re: PS-3 again In-reply-to: Your message of "Thu, 29 Feb 1996 23:49:52." <9603010449.AA08007@MIT.MIT.EDU> -------- |> Sorry to bother you again! |> In problem 2.1, are we supposed to assume uniform |> velocity for the ball, or to go through the messy algebra |> needed to come up with the actual probability density of z, |> or am I completely on the wrong track with this problem? |> Also, what simplifying assumptions can we make on |> problem 1? |> |> Sincerely, |> Lucy Lim |> No problem that you wrote again. In problem 2.1, there is no uniform velocity for the ball. Each time you measure its momentum, you get different values with a width specified by the HUP. You do not know enough yet to compute the correct quantum mechanical shape of the distribution -- you will start to learn that next week -- for now all you can do is draw approximate general conclusions from the width of the distribution. For problem 1, you may assume R< Professor Arias, |> The 'minimum energies' in Problem 2 aren't really the |> minimum possible energies, are they? The question asks for |> average energies (at least for the potential energy) and |> then asks for allowed total energies. But why couldn't |> the energies be lower than the 'allowed total energies' |> if they just weren't average? |> |> Thanks, |> Lucy |> Good question, Lucy. By "Allowed Total Energies" in the later parts of the question, I mean "average expected energies not ruled out by the HUP." In this type of problem, people generally confuse the two. It is a very good sign that you appreciate the difference! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Anti Matter Fcc: 8.04s96 Subject: Re: 804 extension date possible? In-reply-to: Your message of "Fri, 01 Mar 1996 07:51:45 EST." <199603011251.HAA05911@ringworld.MIT.EDU> -------- |> |> hi prof. i only got around to doing the ps yesterday |> due to 3 midterms this week and since the solutions |> will not be availabled until monday, may i request an |> extension? i will try to see if i can finish it by 5pm |> today but if you are kind enought to giv4e me an extension |> it may result in better work. thanks. |> |> bengteck I am sorry bengteck, but I cannot approve an extension for you, it would not be fair to the other students who also had exams and lots of work due this week too. Please hand in the best work which you can by 5:00p today. Remember that I drop you single lowest problem set from your average, and schedule your time better in the future. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Dave Matthews Fcc: 8.04s96 Subject: Re: Problem 2 =) In-reply-to: Your message of "Wed, 28 Feb 1996 22:48:55 EST." <199602290348.WAA05200@ashland.MIT.EDU> -------- |> I'm a little unsure what you're looking for in part 2...especially in |> light of part 3, which is looking for the delta Z that gives the lowest aver |> ... age |> energy. Without using the HUP relation that it looks like I am to verify in |> part 4, I get a KE graph that is just the opposite of the graph of V, creati |> ... ng |> a square graph for the total E. If I were to use the uncertainty relationsh |> ... ip |> between delta Z and delta P I would get a curve for the KE vs delta Z graph, |> but should I use this relationship if I am to verify it in part 4? And if n |> ... ot, |> what am I doing wrong in finding the KE vs delta Z to be a triangle with |> the inverse slope of that of the potential graph? |> I'll be looking for your response...I'm up until 1:30 |> Sincerely, |> Aaron Sanders To do part 2 you are expected to use the HUP to indicate the region of allowed kinetic energies as a function of Delta Z. You should have for the plots for 2 and 3 indicated a region of allowed energies lying all above two curves, one for 2 and one for 3. Then, in 3 you are supposed to show the point indicating the minimim allowed total energy. In part 4, you are just supposed to verify that the kinetic energy you compute for the minimum energy point in 3 is just given by the "Energy of Confinement" formula given in class. The point of problem 4 is physical, THE COST OF CONFINEMENT IS INDEPENDENT OF THE AGENT CAUSING THE CONFINEMENT. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Michael S Golinko Fcc: 8.04s96 Subject: Re: 8.04, yet again In-reply-to: Your message of "Wed, 28 Feb 1996 21:03:59 EST." <9602290204.AA14434@sci-read-room.MIT.EDU> -------- No problem, Michael, I am happy to answer your questions! |> First, on Problem 3, I understand that the current jumps will correspond to |> ... peaks of the energy, I think that for the range given there should be t |> ... hree peaks.Do we need to calculate, can we, the actual current, or just |> ... show a sketch indicating the quantization of energy? Just show a sketch with the current dropping when the electrons have enough energy to excite the "atomic" systems. |> Second, also on problem three, I assume the intensity is also proportional t |> ... o tthese quantized states, except that the intensity keeps rising at pea |> ... ks, while the current should fall back down again, is this correct, and |> ... again do we need a quantitative relationship for Intensity? Again, just a sketch will do. The intensisty should be proportional to the number of atoms first excited by the electrons and then lowering their energy by emitting photons. Thus, for instance, there should be no light intensity until the electrons are able to excite the "atoms" into their fist excited states. |> Third, on question 4, 4.3, the condition I used was 2(pi)(R)=nhc/p |> is this correct to assume, or must we use the integral Bohr qunatization equ |> ... ation, i.e. nh=int(pdq). You should not have that factor of "c" there. (Maybe it was a typo.) With out that factor of "c", then the two expressions are equivalent. Namely, nh = int(p dq) = p * 2*pi*r => 2 pi R = nh /p Once I did that I got a expression something li |> ... ke, |> (sin theta)^4 (nhc)^2 |> ------------- = ------------------------ |> (cos theta)^2 4pi^2(m^2L^3g + kL^4m) |> |> is there any way to easily solve for theta, other than quadratic, must we ma |> ... ke some sort of small angle approximation? There is no need to solve this relation (without commenting on its correctness -- the correct answer is very messy and cannot be solved for theta in a simple way) for theta, just state the relationship. |> |> Sorry my inquiry was so long, please e-mail me back when you get the chance, |> ... hopefully sometime tonight!!!! |> Sincerely, Michael S. Golinko I am sorry I didn't get back to you sooner. I've been out with the flu. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: Dave Matthews Fcc: 8.04s96 Subject: Re: Problem #3 In-reply-to: Your message of "Tue, 27 Feb 1996 21:01:58 EST." <199602280205.VAA15713@ashland.MIT.EDU> -------- |> Professor Arias, |> I am working on problem 3 of problem set 3 and I must be doing |> something wrong, although I don't know what that might be. We just covered |> ... the |> experiment in class, so I decided to do it fresh. I found the equation for |> E(n) of a particle in an infinite mass well (tell me if this is the wrong on |> ... e) |> and I am using E(n)=(h^2)*(pi^2)*(n^2)/(2*m*a^2) |> and with the voltage given in problem 3, I have that the maximum KE for the |> electrons is KE(max)=(5*h^2)/(4*m*a^2) |> How can the graph have any of the drop-offs in class due to the acceptance o |> ... f |> energy from the electrons by the particles in the wells if the first "excite |> ... d |> state" of the particles is greater than the maximum KE of the electrons? |> |> Please respond as soon as you can; I'll probably still be on athena if what |> I'm asking isn't clear. |> Thank you, |> Aaron Sanders |> |> - --VAA15683.825472921/ashland.MIT.EDU-- |> |> |> ------- End of Forwarded Message |> Dear Aaron, please keep in mind that the expression on the problem set involves Planck's constant h, whereas the formula you gave is in terms of h-bar. The lowest energy transition in the wells should thus be: E2-E1=pi^2 * h-bar^2/(2ma^2)*(2^2-1^2) = h^2/(ma^2)*(3/8) < (5*h^2)/(4*m*a^2) , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== To: mgolinko@MIT.EDU (Michael S. Golinko ) Fcc: 8.04s95 Subject: Re: 8.04 In-reply-to: Your message of "Tue, 27 Feb 1996 16:40:48." <9602272140.AA07429@MIT.MIT.EDU> -------- |> Dr. Arias, |> I tried to get the answer on the problem set to question number 1.1, |> the one I asked you briefly about in class today. I still don't know if I |> am approaching the problem right. First, the real value of psi is big. |> something like, |> |> exp( -c1(x-xo)^2 + c2(x-xo) + c3), first is this right? I got your message about advising and I understand. Now about this problem, the trick is to complete the square in the real part of the exponent. You should be able to get things in a form like psi(x,t)=A * exp ( i phase(x,y) ) * exp (-c1*(x-x0-hkt/m)^2 ) by completing the square in your exponent. After doing this, things should get easy... Let me know if this helps! , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts =============================================================================== Review of calculation of deflection angle in JJ Thomson's c.r.t. experiment and what might a student do by mistake to get (1/2) times the correct result? |> Hi Prof. Arias, |> |> Earlier during the lecture you mentioned that the angle of an |> electron deflected as it goes in between two plates of a capacitor |> as |> |> tan theta = (Ft)/(mv) |> |> I have two questions, |> |> first of all, how does this work? Is Power/Momentum some special |> quanitity always? |> |> Second question: I used mechanics, x = .5at^2, and calculated |> that the tan of theta is half that value. Why? |> |> Answers would be greatly appreciated. No problem to answer your question. 1) First let me analyse the problem in a more familiar way to you. You will see that the answer is the same. Say the electrons were originally moving along the y-diection and the z-direction is pointing "up" (along the direction of the electric field E). The angle at which the electron leaves the plates and then travels down the c.r.t. is tan theta = v_z' / v_y' where v_y' and v_z' are the components of the velocity of the particle in the correcsponding directions after leaving the plates. BEFORE entering the plates we have v_y=v_y and v_z=0. Now, as the electric field is in the z-direction ONLY, there is no acceleration in the y-diection and we have v_y'=v_y. On the other hand, the electric field pointing in the z-direction causes an acceleration (qE/m) which lasts for a time t. Therefore, v_z'=v_z+a*t=0+(qE/m)*t and tan theta = (qE/m)*t/v_y=(qEt)/(mv_y)=(Ft)/mv. 2) Now, let me answer your specific questions: > Is Power/Momentum some special |> quanitity always? Actually, (Ft)/(mv) is not power/momentum. "Power" would be P=Fv. "Ft" is impulse, a concept closely related to momentum. |> Second question: I used mechanics, x = .5at^2, and calculated |> that the tan of theta is half that value. Why? What you must have done was calculate the angle between the points where the electron just enters and leaves the parallel plates. You forgot that what is observed is the angle of deflection of the beam as it travels all the way down the c.r.t. I hope this helps. I will post this correspondence to the web (with out your name, of course) to help others with the same question. , Tomas Arias, PhD Assistant Professor Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts