Ph2b Quiz - 1. Instructions
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1 Ph2b Winter Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm Ph2b Quiz - 1 Instructions 1. Your solutions re due by Mondy, Jnury 29th, 218 t 4pm in the quiz box outside 21 E. Bridge. 2. Lte quizzes will not be ccepted, except in very specil circumstnces. 3. The quiz is open textbook (Griffiths, open lecture notes, open section notes, open homework sets nd open homework solutions. Plese check the collbortion policy on the course website phys2 for more detils. 4. Clcultors my be used. Computers nd smrt devices, other thn your personl brin re not llowed. 5. There re totl of 3 questions nd you hve 2 hours (in single sitting to work on the quiz. Not ll questions re of equl length, so plese go through them ll before you strt nd llocte your time ppropritely. 6. Plese justify your nswers nd show ll work. Time Limit: 2 hours Plese write your nme, section number, nd your TA s nme on the front sheet of your submission. Good Luck! 1
2 Ph2b Winter Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm 1 Tbles Turn on Schrödinger (5 points After surviving brush with deth, Schrödinger s ct (Albert is plying in Schrödinger s living room nd decides to perform little experiment on Schrödinger. Albert plces Schrödinger in one-dimensionl box. At some time t, Schrödinger s wvefunction (Ψ(x, t, not Ψ 2 is given in Figure 1. You my ssume the imginry prt of the wvefunction Ψ(x, t is, nd the figure is drwn to scle. For x to the left of F nd for x to the right of E, the wvefunction is. Albert performs mesurement of Schrödinger s position. For ech of the following, stte Figure 1: Schrödinger s Wvefunction Ψ(x, t for Problem 1. whether the quntity on the left is smller thn, equl to, or greter thn the quntity on the right. <, > or = (. [2 points] will be found left of A will be found right of A. (b. [1 point] will be found between F nd E 1 (c. [1 point] will be found between F nd B 2
3 (d. [1 point] will be found t A will be found between C nd E. 3
4 Ph2b Winter Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm 2 Fun with the Infinite Squre Well (1 points A prticle of mss m in n infinite squre well of width with potentil energy function, Figure 2: Potentil Energy Function V (x for the Infinite Squre Well of Problem 2. V (x = { : x + : elsewhere (1 s shown in Figure 2 hs s its initil wvefunction, Ψ(x, = αψ n (x + βψ m (x, (2 where n, m re fixed positive integers (n m nd α nd β re rel, non-zero numbers with α β. Remember tht the normlized sttionry sttes (s solutions of the time-independent Schrödinger eqution for n infinite squre well of width re, 2 ( nπx ψ n (x = sin, n = 1, 2, 3,, (3 nd the corresponding energy levels re, where ω = π2 2m 2. E n = n 2 ω, (4 (. [1 point] Find reltionship between α nd β such tht Ψ(x, is normlized. (b. [1 point] Write down the wvefunction Ψ(x, t for the prticle t some lter time t. (c. [3 points] Compute the expecttion vlue of the position of the prticle, x t t =. 4
5 (d. [1 point] Now we mke position mesurement t t = on this stte Ψ(x, nd find the prticle in the left hlf of the well. If we mesure the position of the prticle gin, immeditely fterwrds, wht is the probbility tht the prticle will be found in the right hlf of the box? (e. [1 point] Suppose we tke the prticle in its originl stte Ψ(x, nd mesure its energy right fter preprtion (i.e. very close to t =. Wht possible energies cn we get nd wht is the probbility of ech possibility? (f. [1 point] Wht is the expecttion vlue of the energy E of the prticle when it is in the stte Ψ(x, t. Express your nswer in terms of α, β nd the energies E n nd E m of the sttes ψ n nd ψ m, respectively; nd time t, if necessry. (g. [1 point] As we just computed in prt (f, the verge energy of the prticle in the well is simply E. A clssicl prticle in this well would bounce bck nd forth between the wlls t constnt velocity. Wht is the frequency of oscilltion for the clssicl motion? (h. [1 point] Assume tht mesurement is mde of the energy of this prticle in nd we find energy E n. We then wit until time t = t 1 > nd mesure the energy gin. Wht possible energies cn we get nd wht is the probbility of ech possibility? You my express your nswer in terms of t 1, if necessry. Some reltions you might find useful: And some useful integrls: cos θ = eiθ + e iθ 2 sin θ = eiθ e iθ 2i, (5. (6 ( sin 2 nπx dx = 2, (7 ( x sin 2 nπx dx = 2 4, (8 x sin ( nπx sin ( mπx [ ] dx = 2 1 π 2 (n + m 2 1 (n m 2 where n m nd n + m = odd number. (9 x sin ( nπx sin ( mπx where n m nd n + m = even number. dx = (1 5
6 Ph2b Winter Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm 3 Uncertinty! (1 points Consider the following normlized wvefunction, Ψ(x, t = { k e kx e iet : x k e kx e iet : x (11 (. [3 points] Clculte x nd x 2. (b. [3 points] Clculte p nd p 2. (c. [2 points] Determine σ x nd σ p, nd evlute the uncertinty reltion. (d. [2 points] If E = 2 k 2 2m, then for wht function V (x does the wvefunction stisfy the Schrödinger Eqution in the region x >? 6
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