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21 Piecewise Interpolation Polynomials
one-dimensional quantum system. Thus the corresponding Schrödinger equation in
the coordinate representation is
−
¯
h 2
2m
d 2
dx 2 + (E − ˆ
V (x))
E|ψ = 0.
Due to the simplicity of the kinetic energy operator the coordinate representation of
the quantum system is preferred for finite elements. The shape of the potential could
be almost arbitrary complex. For simplification we set
¯
h = 1 , m =
1
2
⇒
¯
h
2
2m
= 1.
(21.9)
By multiplication of the Schrödinger equation with the wave function
from the left, we arrive at the equivalent variational equation δΠ[ψ] = 0 with
the functional
+∞
−∞
d 2
dx 2 + +ψ|x
E − ˆ
V (x)
dx = Π[ψ].
(21.10)
Because bound states = ψ(x) are normalizable we could always find a left
and right border, (x a , x b ), in coordinate space beyond which the wave functions
vanish effectively:
a |ψ = 0 = =x b |ψ ,
(21.11)
and thus we can approximate the functional by a finite integration. Integrating by
part leads to
Π[ψ] ≈
x b
x a
⎡
⎣
d 2
dx 2
++ψ|x
E − ˆ
V (x)
⎤
⎦ dx
(21.12)
= =ψ|x
d
dx
x b
x a
at the border = 0
−
x b
x a
d
dx
2
dx.
Hence,
Π[ψ] = −
x b
x a
d
dx
2
dx +
x b
x a
[E − V (x)]
σ (x)
(21.13a)