Unformatted text preview: Physics 143 Fall Term 2007 HOMEWORK 6
Due: October 29 Problem 1: (BFG Ch.7 Problem 1, 3)
1. Consider the very simple wave packet x A exp i p p x A exp i p p x Show that x takes the form of a single plane wave of momentum p multiplied by a function. What is that function? x A exp A exp 2A cos i p p x A exp exp / i p p x exp exp The term The factor 2 is a single plane wave. is the multiplier function which is the amplitude of the wave. 3. Consider the wave packet of Problem 1. The time-dependent form of the packet is i p x E t i p x E t x, t A exp A exp where p p p , p p p , E E E , p E E . (a) Show that takes the forms of a plane wave times a time-dependent modulating factor. (b) Show that the modulating factor has a time dependence that can be interpreted as the propagation of an "envelope" mobbing with speed v E/p . The group velocity of a packet of waves is generally dE/dp. given as v (a) In the same with as Problem 1: p p x x A exp i A exp i 2A cos The term 2 (b) From t to t
E E E E t
E A exp i exp / i p
E p x E E t exp i exp is the timedependent modulating factor. t, the factor has the phase shift: t t t t moves forward which has the phase shift: x x x t x t
P a g e | 13 In this time interval, the "envelope" Two shifts are the same. Therefore, Thus the speed of the "envelope" is, x x Problem 2: (BFG Ch.7 Problem 14)
The binding energy of an electron in a crystal lattice is 1.2 10 eV . Use the uncertainty relation to estimate the size of the spread of the electron wave function. How many lattice sites will that include, assuming that the spread is spherically symmetric and the spacing between the ions is 0.12nm? Assume it as non-relativistic case. E So the corresponding momentum is, P Based on the uncertainty relation: 2mE 5.9 P 2m 10 /2 8.9nm kg m/s x p x Therefore, x /2 /2 ~ p p 8.9nm 74 0.12nm which indicates that hhe diameter of the sphere of the spread is 74a. It includes N lattice sites, 4 74 2 N 3 ~2.1 10 (Notice that it is a 3Dimensional Problem) Problem 3: (BFG Ch.7 Problem 16)
Monochromatic light of wavelength 683nm passes through a fast shutter that opens for 10 longer be monochromatic. What will the spread in wavelength? s. The light that emerges will no Based on the uncertainty relation: E t We know that E E Therefore, the spread in wavelength is, || /2 E~ hc hc t 1.2 10 m hc hc /2 Problem 4: (BFG Ch.8 Problem 1)
A beam of electrons is send along the x-axis from with kinetic energy E=4.2eV. The beam encounters a potential barrier if height V 3.2eV and width 2a=1.2nm. What fraction of the incident beam is reflected? Refer to (BFG 8-12) |R| In the formula above, k 2mE 1.05 10 m q 2kq cos 2qa k sin 2qa q k sin 2qa P a g e | 23 q plug in the formula |R| . We obtain, The probability of reflection is 1.2%. 2m E V 0.51 10 m |R| 0.012 1.2% Problem 5: (BFG Ch.8 Problem 3)
Show that the reflection coefficient, given by Eq.(8-12), can never exceed unity. Is this reasonable result? Refer to (BFG 8-12) R And therefore, |R| q 2kq Cot 2qa Cot 2qa |R| We know that q k q q Therefore, |R| Which means, q q k k 1 q k k k 1 4q k 0 q q k k k q 0 k i q 2kq cos 2qa k sin 2qa k sin 2qa i q |R| 1 It is reasonable because the reflected part should be less than incident due to the conservation of energy. Problem 6: (BFG Ch.8 Problem 7) Be sure to apply boundary conditions and solve for coefficients.
Consider the step potential discussed in Example 8-1. Solve the Schrodinger equation for the case where v the meaning of your answer. , and discuss We introduce k and q defined by k q 2mE 2m V E . Notice that V Region I: x 0, we hypothesize an incident and reflected wave: e Re I x Region II: x 0, solve the Schrodinger equation: d II V II EII 2m dx
P a g e | 33 We obtain, d II dx The general solution is, When x 0, II 0, therefore B 0, II Te II I | dI | dx We obtain the equations about R, T. 1 ik The solution is: T R 2k k iq k k iq iq R ik R T qT Te II | dII | dx Be q II 0 Plug in the boundary conditions at x = 0 In conclusion, e k k iq e iq x 0 0 2k e k iq x We see that the wave function decays exponentially in the classical "forbidden" region II(x>0) where E<V . (or you can just let and plug it in the answer in Example 8-1) October 29, 2007 P a g e | 43 ...
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