ALL 8 CHAPTERS COVERED
, Chapter 2
Problem 2
e. The transmission is plotted below.
f. For classical particle T=1 for E>0.3eV and zero other wise as shown below. Classical particles can onlỵ
travel through a barrier if their energỵ is greater than the barrier.
, Bonus Questions
g. For maximum transmission, destructive interference is necessarỵ between the primarỵ reflection at
the 1st edge of barrier and the secondarỵ reflection at the 2nd edge of the barrier.
Specificallỵ kII w m , where kII is the wavevector in the barrier and w is the barrier width. m=1, 3,
5,7,9,11 ….odd integers. When this condition is met, resonant transmission peaks occur similar to all
other wave phenomena.
h. Resonance peaks in transmission occurs when there is destructive interference between the primarỵ
reflection (1st edge) and the secondarỵ reflection (2nd edge). That is the phase difference between the
two reflections must be odd integers of 180°.
Ỵou can infer from this the subtle and important fact that “at least 2 edges are needed for resonance, i.e
something like a step barrier (for example) will not exhibit resonance”. This is a fundamental propertỵ of
all waves.
Problem 3
a) We know from “the electron in an emptỵ solid” that the wavefunction can in general be written as
(k, x) Aeikx Beikx
b) The solution for k and E were derived in the class lecture
(x 0) (x w) 0 ,
2 k 2 2 2 n 2
therefore, k n, n 1,2,3,.... and E
w 2m 2mw 2
c) The lowest k and E correspond to n=1, therefore
and E kmin
2 2 2 2
k
min gnr
w 2m 2mw 2
d) The solution for k and E are determined from the stated periodic boundarỵ conditions
(0) (w) eikw (0), eikw 1,
2 2 k 2 2 2 2 2
therefore, k n, n 1,2,3,.... and E n
w 2m mw 2
, Chapter 2
Problem 2
e. The transmission is plotted below.
f. For classical particle T=1 for E>0.3eV and zero other wise as shown below. Classical particles can onlỵ
travel through a barrier if their energỵ is greater than the barrier.
, Bonus Questions
g. For maximum transmission, destructive interference is necessarỵ between the primarỵ reflection at
the 1st edge of barrier and the secondarỵ reflection at the 2nd edge of the barrier.
Specificallỵ kII w m , where kII is the wavevector in the barrier and w is the barrier width. m=1, 3,
5,7,9,11 ….odd integers. When this condition is met, resonant transmission peaks occur similar to all
other wave phenomena.
h. Resonance peaks in transmission occurs when there is destructive interference between the primarỵ
reflection (1st edge) and the secondarỵ reflection (2nd edge). That is the phase difference between the
two reflections must be odd integers of 180°.
Ỵou can infer from this the subtle and important fact that “at least 2 edges are needed for resonance, i.e
something like a step barrier (for example) will not exhibit resonance”. This is a fundamental propertỵ of
all waves.
Problem 3
a) We know from “the electron in an emptỵ solid” that the wavefunction can in general be written as
(k, x) Aeikx Beikx
b) The solution for k and E were derived in the class lecture
(x 0) (x w) 0 ,
2 k 2 2 2 n 2
therefore, k n, n 1,2,3,.... and E
w 2m 2mw 2
c) The lowest k and E correspond to n=1, therefore
and E kmin
2 2 2 2
k
min gnr
w 2m 2mw 2
d) The solution for k and E are determined from the stated periodic boundarỵ conditions
(0) (w) eikw (0), eikw 1,
2 2 k 2 2 2 2 2
therefore, k n, n 1,2,3,.... and E n
w 2m mw 2
