j 0i ¼
ffiffiffi
1
L
r
cos
π
2L
x,
ð5:26Þ
which is obtained by putting l ¼ 0 in j l cos i
ffiffi
1
L
q
cos
π
2L þ
lπ
L
À
Á
x l ¼ 0, 1, 2, Á Á Á
ð
Þin
(1.101).
By the same token, hkj xj 0i in the third term of RHS of (5.25) vanishes if jki
denotes a cosine function in (1.101). If, on the other hand, jki denotes a sine
function, hkj xj 0i does not vanish. To distinguish the sine functions from cosine
functions, we denote
j n sin i
ffiffiffi
1
L
r
sin
nπ
L
x n ¼ 1, 2, 3, Á Á Á
ð
Þ
ð 5:27Þ
that is identical with (1.102); see Fig. 5.1 with the notation of the quantum states.
Now, putting E
0
ð Þ
0 ¼
h
2
2m Á
π
2
4L
2 and E
0
ð Þ
2nÀ1 ¼
h
2
2m Á
2n
ð Þ
2 π
2
4L
2
n ¼ 1, 2, 3, Á Á Á
ð
Þin (5.25),
we rewrite it as
E 0 % E
0
ð Þ
0 À eF 0jxj0
h
iþ ÀeF
ð
Þ
2
X 1
k¼1
1
E
0
ð Þ
0 À E
0
ð Þ
k
h
i kjxj0
h
i
j
j
2
⋯
( ) =
ℏ
( ) =
ℏ
( ) =
ℏ
( ) =
ℏ
|
⟩
0
≡ | ⟩
0
|
⟩
1
≡ | ⟩
1
|
⟩
2
≡ | ⟩
3
|
⟩
1
≡ | ⟩
2
|
⟩
2
≡ | ⟩
4
( ) =
ℏ
Fig. 5.1 Notation of the
quantum states that
distinguish cosine and sine
functions.
E
0
ð Þ
k
k ¼ 0, 1, 2, Á Á Á
ð
Þ
represents energy
eigenvalues of the
unperturbed states
158
5 Approximation Methods of Quantum Mechanics