Massachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 4

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1 Massachusetts Institute of Technology Quantu Mechanics I (8.04) Spring 2005 Solutions to Proble Set 4 By Kit Matan 1. X-ray production. (5 points) Calculate the short-wavelength liit for X-rays produced by an electron acceleration voltage of 30 kv. Is the work function of the etal relevant? Wavelength of the ost energetic photon is given by λ = hc E = = (1) We can ignore the work function because it is uch saller than the electron energy. The work function of ost etals is about a few ev while the electron energy is of the order of kev, 10 3 ties larger. 2. Suerfeld-Wilson quantization for linear potential in one diension. (25 points) Consider a particle of ass linear potential V (x) = C x, C > 0. We want to deterine the quantized energy levels in such a potential. (a) Assue x(t = 0) = A, A > 0, and p(t = 0) = 0. Calculate x(t) and p(t) for one period T. How large is T? The force on the particle is, Using the Newton s equation, we obtain, F = dv dx { C for x > 0 F = C for x < 0. (2) and, p(t) = F = dp(t) dt Ct for 0 < t < T/4 C(t T/2) for T/4 < t < T/2 C(T/2 t) for T/2 < t < 3T/4 C(T t) for 3T/4 < t < T, dx(t) dt = p(t), (3) (4) 1

2 A C 2 t2 for 0 < t < T/4 C x(t) = (t 2 T/2)2 for T/4 < t < T/2 C (T/2 2 t)2 A for T/2 < t < 3T/4 A C (T 2 t)2 for 3T/4 < t < T. (b) Calculate p(x)dx, i.e. the integral over one period, as a function of C, particle ass, and aplitude of otion A. Note that dx < 0 (dx > 0) if the particle oves towards negative (positive) x values. Calculate p(x) fro the total energy of the syste, E = CA = p2 + Cx for 0 < t < T/4 and 0 < x < A (5) 2 = p(x) = 2C (A x). (6) So we obtain, p(x)dx = 4 A 0 2C (A x)dx (7) = 8 3 2CA 3 (8) Note that we integrate p(x) fro x = 0 to x = A, which is a quarter of one period. (c) Now use the Soerfeld-Wilson quantization condition p(x)dx = nh to deterine a quantu echanical condition on the aplitude A n. What is the value of the aplitude for the ground state A 1, i.e. the aplitude for n = 1? Use the Soerfeld-Wilson quantization consition, nh = p(x)dx (9) = 8 3 2CA 3 n (10) = A n = ( 9 h 2 n 2 ) 1/3. (11) C For n=1, ( 9 h 2 ) 1/3 A 1 =. (12) C (d) Calculate the quantized energy levels E n. Sketch the potential V (x) and the quantized energy levels E n. Copare the dependence of the spacing between energy levels on quantu nuber n to the Bohr ato. Calculate the quantized energy levels, E n = CA n = ( 9 2 C 2 h 2 ) 1/3 n 2/3 αn 2/3, (13)

3 Figure 1: The potential V(x). Figure 2: Energy levels of V(x) and the Bohr ato. 3

4 ) C 2 h 2 1/3 Figure 3: The otion in phase space. where α = ( 9 The energy spacing for this potential is, E = E n2 E n1 = ( 9 C 2 h 2 ) 1/3 ( 2/3 n 2 n 2/3 ) 1. (14) Fig. 2 shows energy levels of the potential V (x) in coparison with those of the Bohr ato. (e) Plot the otion in phase space, i.e. in a oentu-versus-position diagra. What is the geoetrical eaning of the Soefeld-Wilson quantization condition? In one sentence, how would you describe the stationary states in phase space? Fig. 3 shows the otion of the particle in the pase space. The Soerfeld-Wilson quantization condition requires that the integral enclosed by each curve ust be an integer ultiple of h. The stationary states are states in which the area enclosed by the curve satisfies the quantization condition. The transition energy can also be calculated fro an area enclosed by any two curves representing the stationary states. 3. Fictitious Bohr ato. (20 points) What would the Baler forula look like for a fictitious Bohr ato where the electron is bound to the nucleus by a potential V (r) = C 6? Use the Bohr quantization condition L = n h for the r 6 angular oentu for circular orbits to calculate the energy levels corresponding to different 4

5 quantu nuber n, and reeber which transitions the Baler forula corresponds to. Find the quantity that would correspond to the Rydberg constant, and express it in ters of C 6, the electron ass, and h. Calculate a radial force fro the given potential, For a circular orbit, a centripetal force is the force given above, F = dv dr = 6C 6 r 7. (15) v 2 Using the Bohr quantization condition, we obtain, r = 6C 6 r 7 (16) = 2 v 2 r 2 = 6C 6 r 4. (17) n 2 h 2 = 6C 6 (18) r 4 ( ) 1/2 6C6 = r n = n 2 h 2. (19) Substitute r n to the Bohr quantization condition and solve for v n, Fro r n and v n, we can calculate the total energy levels, v n = n h = n h ( n2 h 2 ) 1/4 r n 6C 6 (20) = ( n6 h 6 ) 1/ C 6 (21) E n = 1 2 v2 n C 6 (22) r 6 = 1 ( n6 h 6 ) 1/2 ( n2 h 2 ) 3/2 2 C (23) C 6 6C 6 = 1 h 3 n 3 3 3/2. (24) 6C 6 The Baler forula corresponds to the transitions fro n > 2 to n = 2. where R 0 corresponds to the Rydberg constant, and is equal to, 1 λ = R ( 0 n 3 2 3), (25) R 0 = E n=1 hc = 1 h 2 3 2πc 3/2. (26) 6C 6 5

6 4. Childish precision experient. (10 points) A child on top of a ladder of height H is dropping arbles of ass to the floor and trying to hit a crack in the floor. To ai, the child is using equipent of the highest possible precision. Assue that the effects of air resistance and breeze are entirely negligible. Show that the arbles will iss the crack by a typical distance of order ( h/) 1/2 (2H/g) 1/4, where g is the acceleration due to gravity. How large is this distance for H = 3. = 10 2 kg? An experientalist decides to perfor the sae experient with 87 Rb atos and a drop height H = 0.1. Will she be able to observe the effect? Calculate the tie a falling object takes fro height H, H = 1 2 gt2 (27) = t = 2H g. (28) If a starting uncertainty in the oentu p x, after tie t the uncertainty in position is, Substitute p x = h/ x and solve for x, For H = 3 and = 10 2 kg, x = x = p x t = p x 2H g. (29) x = h For 87 Rb atos, H = 0.1 and = kg, 2H g. (30) = (31) x = = (32) This spread of 10 u is easily observable with an optical icrsoscope 6