Physics 4D Take Home Exam
Due: Friday, 12/14/12 by 1:00 pm
Chapter 7: The Schrödinger Equation in One Dimension
Reading: Taylor, Zafiratos, Dubson, Chapter 7; Tipler and Llewellyn, Chapter 6
Question A
When we introduced the Schrödinger equation, which is complex, our oscillatory
solutions went from sines and cosines to solutions of the form exp(±ikx). Show that this
is still an oscillatory solution by expressing the general wave function  = Aeikx + Beikx
into the form  = acos(kx) + bsin(kx). What is the relation between a, b and A, B?
Answer: a = A + B; b = i(A – B)
Question B
It is stated that a finite potential well always has at least one bound level, no matter how
shallow the well. Does this mean that as U0→0, E1→0? Does this violate the Heisenberg
uncertainty principle? Explain
Question C
For a particle in a box, the probability function is zero at the points x = 0 and x = L, the
“walls” of the box. Does this mean that the particle never strikes the walls? Explain.
Problem 1
The wavelength of light emitted by a ruby laser is 694.3 nm. Assuming that the emission
of a photon of this wavelength accompanies the transition of an electron from the n = 2
level to the n = 1 level of an infinite square well, compute L for the well.
Problem 2
An electron moving in a one-dimensional infinite square well is trapped in the n = 5 state.
(a) Compute the probability of finding the electron between x = 0.2L and x = 0.4L. (b)
Compute the probability of finding the electron within the “volume” x = 0.01L at x = L/2.
Problem 3
A proton is in an infinite square well potential with L = 1 fm. (a) Find the ground-state
energy in MeV. (b) Make an energy diagram for this system. Calculate the wavelength of
the photon emitted for the transition (c) n = 2 to n = 1, (d) n = 3 to n = 2, and (e) n = 3 to
n = 1.
Problem 4
(a) Write down and sketch the probability distribution |(x)|2 for the second exited state
of a particle in a rigid box of length L. Interpret your sketch. (b) Calculate the most
probable positions, <x>; does it agree with the results of (a)? (c) Calculate the
probabilities of finding the particle in the intervals [0.75L] and [0.76L]. (d) Repeat (a) –
(c) for the third excited state?
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Problem 5
Consider the potential well shown in figure (5a). (a) Sketch the wave function for the
eighth exited level of this well, assuming that it has energy E > U0. [Hint: the wave
function has eight nodes – not counting the nodes at the walls of the well.] (b) Sketch the
wave function for the fourth excited level of the well, assuming that it has energy in the
interval 0 < E < U0.(c) Do the same task as in (a) and (b), but for figure (5b).
Problem 6
A beam of particles, each with kinetic energy E, approaches a step potential of height V0.
(a) Derive the coefficient of reflection R and transmission T for the step potential:
k  2mc 2E / c
(k  k )2
4k1k 2
 1
R  1 2 2 and T 
where

2
2
(k1  k 2 )
(k1  k 2 )

k 2  2mc (E  V0 ) / c
(b) A beam of protons, each with kinetic energy 40 MeV, approaches a step potential of
height 30 MeV. What fraction of the beam is reflected and transmitted? (c) How does
your answer change if the particles are electrons? (d) Interpret the meaning of your
results for part (a) and (b).
Problem 7
Consider the finite potential barrier, having height U0 and width L. Assume E
> U0. The probability amplitudes in region I (x < 0) are A1 and A′1, in region II
(x < 0) are A2 and A′2, and in region III (x < 0) is A3.
a. Write down the wave functions for the three regions I (x < 0), II (x < 0),
and III (x < 0), apply the boundary conditions and derive the transmission
coefficient above the barrier for E > U0, where
E
2 2
1
k
;
2m
E  V0 
2 2
2
k
;
2m
A3

A1
eik1L
k12  k 22
cosk 2L  i
sink 2L
2k1k 2
b. From the transmission coefficient (make sure it is a function of sin2(k2L)), under what
conditions is the transmission coefficient unity as a particle crosses above the
potential barrier? What does this physically mean?
1

  k 2  k 2 2

2
2

T  1  sin k 2L    1

1


  2k1k 2 




c. Now carefully plot the wave function accurately for regions I, II and III. Explain what
you plot in terms of part (1b).
Problem 8
Consider the same finite potential barrier as in problem 7, however, now Assume U0 >
E. A beam of electrons, each with kinetic energy E, is incident on a potential barrier of
height U0 and width L.
a. Derive the coefficient of transmission T2:
2
1

sinh2 k 2L 
E
T  1 
where k 2  2m(U0  E)/ and X 

U0
 4X 1  X  
b. A beam of electrons, each with kinetic energy 2.0 eV, is incident on a potential
barrier of height U0 = 6.5 eV and width 5.0 × 1010 m. What fraction of the electrons
in the beam will be transmitted through the barrier? Hint: Show that for k2L ≫ 1, a
somewhat simpler form can be shown to be
T 2  16X 1  X  e 2k 2L
c. An Alpha decay from heavy nuclei was first explained by regarding it as a tunneling
process through the nuclear barrier. The Coulomb barrier is
roughly 16 MeV high and its width is of the order of 10-14 m. A
typical value of the energy of an emitted alpha particle is 6
MeV, which is considerably below the summit of the barrier.
(i) Explain physically the shape of this potential function for
the heavy nuclei.
(ii) Explain how this model is able to account for the
experimental fact that the mean lifetime of alpha emitters vary
from 1 μs to 1010 years. Explain your reasoning using short
concise sentences.
 transmission   frequency of 
mean decay lifetime  


 coefficient   hitting the walls 
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