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EE 400B Homework #3 Solution
1. (10%) Quantum well is a critical structure that allows
semiconductor lasers to operate in room temperature
(see figure on the right for an example of multiplequantum-well
semiconductor
device).
Let’s
approximate the conduction band of a GaAs quantum
well of width 10 nm as a 1-D infinite square well
potential. The effective mass of electrons in the
conduction band is mc  0.07m0 . Find the 1st three
energy levels for electrons in the conduction band.
En 
n2 2 2
2ma 2
E1 (eV ) 
 6.62 10
34
/ 2   /108 
2
2
2  0.07  (9.11031 )  (1.6 1019 )
E2  4E1  216 meV
E3  9E1  486 meV
=54 meV
2. (30%) Griffiths, Problem 2.5.
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3. (10%) Show that the wave function of a particle in the infinite square well returns to its
original form after a quantum revival time T  4ma 2 /  . That is, ( x, t  T )  ( x, t ) for
any state (not just a stationary state).

( x, t )   cn n ( x)ei ( n 
2 2
/ 2 ma 2 ) t
n 1
n
n2 2 4ma 2
T


 2 n2
2ma 2
2ma 2 
2 2
2
2 2
2
2
2 2
 ei ( n  / 2ma )(t T )  ei ( n  / 2ma )t ei 2 n  ei ( n 
( x, t  T )  ( x, t )
2
2
/ 2 ma 2 )t
4. (30%) Griffiths, Problem 2.21.
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5. (20%) Griffiths, Problem 2.27, but change (b) to: How many bound states does it possess if
(i)   2 / 2ma ; (ii)   2 / 2ma ? Sketch the eigenfunctions.
Hint:
(1) Since the potential is symmetrical, the eigenfunctions are either even or odd functions.
 Ae x
( x  a)
  x  x
(2) For even function,  ( x)   B(e  e ) (a  x  a) . You will arrive at a transcendental
 Ae x
( x  a )

equation e2 a 

 1 after applying boundary conditions and discontinuous derivative
m
2
at x  a (please derive this instead of just using the result).
 Ae x
( x  a)
  x  x
(3) Likewise, for odd function,  ( x)   B(e  e ) (a  x  a) . The transcendental
  Ae x
( x  a )

equation becomes e
2 a
 1

.
m
2
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