Closure Property of Eigenfunctions
Closure Property of Eigenfunctions
A complete set of discrete eigenfunctions obeys the orthonormalization conditions . Complementary to these is the set of closure relations (x)()=δ(x-). For real eigenfunctions, the complex conjugate can be dropped. The finite sums (x)() for up to 100 are evaluated in this Demonstration. Four systems are considered: (1) infinite square-well potential with and eigenfunctions (x)=sin(nπx), , ; (2) linear harmonic oscillator with and (x)=n!(x), , ; (3) linear rigid rotor for fixed , , (x)=(x), , ; (4) hydrogen-like radial function with , (r), , .
∫(x)(x)dx=
*
ψ
m
ψ
n
δ
mn
∞
∑
n=0
*
ψ
n
ψ
n
x
0
x
0
N
∑
n=0
*
ψ
n
ψ
n
x
0
N
ℏ=m=a=1
ψ
n
0≤x≤1
n=1,2,3,…
ℏ=μ=ω=1
ψ
n
1
n
2
π
-2
2
x
e
H
n
-∞<x<∞
n=0,1,2,…
ℏ=ℐ=1
Θ
ℓ
(2ℓ+1)(ℓ-)!
2(ℓ+)!
P
ℓ
-1≤x=cosθ≤1
ℓ=,+1,+2,…
ℏ=μ=e=1
R
nℓ
0≤r<∞
n=ℓ+1,ℓ+2,ℓ+3,…
For the first three cases, the sum approaches an oscillatory representation of the delta function . However, the hydrogenic functions represent only the discrete bound states. They do not constitute a complete set of eigenfunctions without including the continuum. The sums usually exhibit erratic behavior but sometimes do show a peaking, particularly for larger values of .
δ(x-)
x
0
Z