1.1 The Radiation Laws and the Birth of Quantum Mechanics

QUANTUM MECHANICS I (Dr. T. Brandes): Example/Solution Sheets 15
SOLUTION: By matrix multiplication:
( )
M11 M
M =
12
M 21 M 22
(
1 (k1 + k
=
2 )e i(k 2−k 1 )x 1
(k 1 − k 2 )e −i(k )
1+k 2 )x 1
4k 1 k 2 (k 1 − k 2 )e i(k 2+k 1 )x 1
(k 1 + k 2 )e −i(k 2−k 1 )x 1
.
( (k2 + k
×
1 )e i(k 1−k 2 )x 2
(k 2 − k 1 )e −i(k )
2+k 1 )x 2
(k 2 − k 1 )e i(k 1+k 2 )x 2
(k 2 + k 1 )e −i(k 1−k 2 )x 2
1
[
M 12 = (k 1 + k 2 )(k 2 − k 1 )e i(k 1−k 2 )a−i(k 2 +k 1 )a
4k 1 k 2 ]
+ (k 1 − k 2 )(k 2 + k 1 )e i(k 1+k 2 )a−i(k 1 −k 2 )a
= k2 2 − k2 [
]
1
e −2ik2a − e 2ik 2a
= k2 1 − k2 2
2i sin(2k 2 a)
4k 1 k 2 4k 1 k 2
1
[
M 11 = (k 1 + k 2 )(k 2 + k 1 )e i(k 1−k 2 )a−i(k 2 −k 1 )a
4k 1 k 2 ]
+ (k 1 − k 2 )(k 2 − k 1 )e i(k 1+k 2 )a+i(k 2 +k 1 )a
Correspondingly for M 21 and M 22 .
2.10.2 * Transmission coefficient (15 min)
= e2ik 1a [
]
(k 1 + k 2 ) 2 e −2ik2a − (k 1 − k 2 ) 2 e 2ik 2a
4k 1 k 2
[
= e 2ik 1a k
2
1 + k2
2 ]
i sin(−2k 2 a) + cos(2k 2 a) . (2.80)
2k 1 k 2
Verify the expressions for the transmission coefficients of the tunnel barrier, given in the lecture
notes.
2.10.3 Transmission coefficient (10 min)
a) Draw the transmission coefficient of a tunnel barrier (roughly) as a function of energy E.
What are transmission resonances
b) Draw the transmission coefficient of a potential step (roughly) as a function of energy E.
2.10.4 ** Determinant of M (10 min)
Consider the case k 1 = k N+1 in (2.68). Use the definitions for T 1 (T n correspondingly) and M
T 1 = 1 ( )
(k1 + k 2 )e i(k 2−k 1 )x 1
(k 1 − k 2 )e −i(k 1+k 2 )x 1
2k 1 (k 1 − k 2 )e i(k 2+k 1 )x 1
(k 1 + k 2 )e −i(k 2−k 1 )x 1
, M = T 1 T 2 ...T N , (2.81)
to show that the determinant of the transfer matrix det(M) = 1.
SOLUTION: The determinant of a matrix product is the product of the determinants,
Furthermore,
det(M) = det T 1 ... det T n . (2.82)
det T 1 = 1 [
4k1
2 (k1 + k 2 ) 2 − (k 1 − k 2 ) 2] = k 2
k 1
det T 1 ... det T n = k 2
k 1
k 3
k 2
... k N+1
k N
= k N+1
k 1
= 1. (2.83)