Example 5.1 Suppose that the charged particle is confined within a one-dimensional potential well. This problem has already appeared in Example 1.2. Now let us
consider how an energy of a particle carrying a charge e is shifted by the applied
electric field. This situation could experimentally be achieved using a “nano capacitor” where, e.g., an electron is confined within the capacitor, while applying a
voltage between the two electrodes.
We adopt the coordinate system the same as that of Example 1.2. That is, suppose
that we apply an electric field F between the electrodes that are positioned at x ¼ Æ L
and that the electron is confined between the two electrodes. Then, the Hamiltonian
of the system is described as
H ¼ À
h
2
2m
d
2
dx 2 À eFx,
ð5:22Þ
where m is a mass of the particle. Note that we may choose ÀeF for the parameter λ
of Sect. 5.1.1. Then, the coordinate representation of the Schrödinger equation is
given by
À
h
2
2m
d
2
ψ i x
ð Þ
dx 2 À eFxψ i x
ð Þ ¼ E i ψ i x
ð Þ,
ð5:23Þ
where E i is the energy of the i-th state from the bottom (i.e., the ground state); ψ i is its
corresponding eigenstate. We wish to seek the perturbation energy up to the second
order and the quantum state up to the first order. We are particularly interested in the
change in the ground state j0i. Notice that the ground state is obtained in (1.101) by
putting l ¼ 0. According to (5.21) we have
E 0 % E
0
ð Þ
0 þ λ 0jVj0
h
iþ λ
2
X
k6 ¼0
1
E
0
ð Þ
0 À E
0
ð Þ
k
h
i kjVj0
h
i
j
j
2 ,
ð5:24Þ
where V ¼ À eFx. Since the field F is thought to be an adjustable parameter, in
(5.24) we may replace λ with ÀeF (vide supra). Then, in (5.24) V is replaced by x in
turn. In this way, (5.24) can be rewritten as
E 0 % E
0
ð Þ
0 À eF 0jxj0
h
iþ ÀeF
ð
Þ
2
X
k6 ¼0
1
E
0
ð Þ
0 À E
0
ð Þ
k
h
i kjxj0
h
i
j
j
2 :
ð5:25Þ
Considering that j0i is an even function [i.e., a cosine function in (1.101)] with
respect to x and that x is an odd function, we find that the second term vanishes.
Notice that the explicit coordinate representation of j0i is
5.1 Perturbation Method
157
consider how an energy of a particle carrying a charge e is shifted by the applied
electric field. This situation could experimentally be achieved using a “nano capacitor” where, e.g., an electron is confined within the capacitor, while applying a
voltage between the two electrodes.
We adopt the coordinate system the same as that of Example 1.2. That is, suppose
that we apply an electric field F between the electrodes that are positioned at x ¼ Æ L
and that the electron is confined between the two electrodes. Then, the Hamiltonian
of the system is described as
H ¼ À
h
2
2m
d
2
dx 2 À eFx,
ð5:22Þ
where m is a mass of the particle. Note that we may choose ÀeF for the parameter λ
of Sect. 5.1.1. Then, the coordinate representation of the Schrödinger equation is
given by
À
h
2
2m
d
2
ψ i x
ð Þ
dx 2 À eFxψ i x
ð Þ ¼ E i ψ i x
ð Þ,
ð5:23Þ
where E i is the energy of the i-th state from the bottom (i.e., the ground state); ψ i is its
corresponding eigenstate. We wish to seek the perturbation energy up to the second
order and the quantum state up to the first order. We are particularly interested in the
change in the ground state j0i. Notice that the ground state is obtained in (1.101) by
putting l ¼ 0. According to (5.21) we have
E 0 % E
0
ð Þ
0 þ λ 0jVj0
h
iþ λ
2
X
k6 ¼0
1
E
0
ð Þ
0 À E
0
ð Þ
k
h
i kjVj0
h
i
j
j
2 ,
ð5:24Þ
where V ¼ À eFx. Since the field F is thought to be an adjustable parameter, in
(5.24) we may replace λ with ÀeF (vide supra). Then, in (5.24) V is replaced by x in
turn. In this way, (5.24) can be rewritten as
E 0 % E
0
ð Þ
0 À eF 0jxj0
h
iþ ÀeF
ð
Þ
2
X
k6 ¼0
1
E
0
ð Þ
0 À E
0
ð Þ
k
h
i kjxj0
h
i
j
j
2 :
ð5:25Þ
Considering that j0i is an even function [i.e., a cosine function in (1.101)] with
respect to x and that x is an odd function, we find that the second term vanishes.
Notice that the explicit coordinate representation of j0i is
5.1 Perturbation Method
157