8.5 The Galerkin Method
485
Find the minimum of the energy E and the ground state wave function ϕ(x), where,
h, m and ω are all positive constants. The boundary condition is
lim
x→±∞
ϕ = 0 or bounded
The constraint condition is
+∞
−∞
ϕ
2
(x)dx = 1
Given the definite integral formula
+∞
−∞
e
−ax
2 dx =
π
a
,
+∞
−∞
x
2 e
−ax
2 dx =
1
2a
π
a
Solution Choosing the coordinate function ϕ(x) = ce
−ax
2 , substituting it into the
constraint condition, we obtain
c
2
+∞
−∞
e
−2ax
2 dx = c
2
π
2a
= 1
( 1 )
Solving for c =
2a
π
1
4 , substituting it into the expression of the coordinate
function, there is ϕ(x) =
2a
π
1
4 e
−ax
2 .
The Galerkin equation of the Schrödinger equation is
+∞
−∞
−
h
2
2m
d
2
ϕ(x)
dx 2 +
1
2
mω
2 x
2
ϕ(x) − Eϕ(x)
ϕ(x)dx = 0
( 2 )
Solving for the energy E from Eq. (2), we have
E =
+∞
−∞
−
h 2
2m
d 2 ϕ(x)
dx 2 + 1
2 mω 2 x 2 ϕ(x)
ϕ(x)dx
+∞
−∞ ϕ 2 (x)dx
=
2a
π
1
2
+∞
−∞
−
h 2
2m
d 2 e −ax 2
dx 2 +
1
2
mω 2 x 2 e −ax 2
e −ax 2
dx
=
2a
π
1
2
−
h 2
2m
+∞
−∞
(−2a + 4a 2 x 2 )e −2ax 2
dx +
2a
π
1
2 1
2
mω 2
+∞
−∞
x 2 e −2ax 2
dx
=
h 2 a
m
−
h 2 a
2m
+
mω 2
8a
=
h 2 a
2m
+
mω 2
8a
(3)