Closure Property of Eigenfunctions

​
system
square well
harmonic oscillator
rigid rotor
hydrogen-like atom
x
0
(or
r
0
)
0.75
m (rotor)
0
ℓ (hydrogen)
0
Z (hydrogen)
1
number of terms N
20
A complete set of discrete eigenfunctions obeys the orthonormalization conditions
∫
*
ψ
m
(x)
ψ
n
(x)dx=
δ
mn
. Complementary to these is the set of closure relations
∞
∑
n=0
*
ψ
n
(x)
ψ
n
(
x
0
)=δ(x-
x
0
)
. For real eigenfunctions, the complex conjugate can be dropped. The finite sums
N
∑
n=0
*
ψ
n
(x)
ψ
n
(
x
0
)
for
N
up to 100 are evaluated in this Demonstration. Four systems are considered: (1) infinite square-well potential with
ℏ=m=a=1
and eigenfunctions
ψ
n
(x)=sin(nπx)
,
0≤x≤1
,
n=1,2,3,…
; (2) linear harmonic oscillator with
ℏ=μ=ω=1
and
ψ
n
(x)=
1
n
2
n!
π
-
2
x
/2
e
H
n
(x)
,
-∞<x<∞
,
n=0,1,2,…
; (3) linear rigid rotor for fixed

,
ℏ=ℐ=1
,
Θ
ℓ
(x)=
(2ℓ+1)(ℓ-)!
2(ℓ+)!

P
ℓ
(x)
,
-1≤x=cosθ≤1
,
ℓ=,+1,+2,…
; (4) hydrogen-like radial function with
ℏ=μ=e=1
,
R
nℓ
(r)
,
0≤r<∞
,
n=ℓ+1,ℓ+2,ℓ+3,…
.
For the first three cases, the sum approaches an oscillatory representation of the delta function
δ(x-
x
0
)
. 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
Z
.

Details

The closure relation can be derived by considering the expansion of an arbitrary function
f(x)
obeying the same analytic and boundary conditions as the eigenfunctions
ψ
n
(x)
. If the set of eigenfunctions is complete, one can write
f(x)=
∑
n
c
n
ψ
n
(x)
, with expansion coefficients determined from
c
n
=∫
*
ψ
n
(
x
0
)f(
x
0
)d
x
0
. Substituting the last relation into the expansion, we find
f(x)=
∑
n
∫
*
ψ
n
(
x
0
)f(
x
0
)
ψ
n
(
x
0
)d
x
0
=∫f(
x
0
)
∑
n
*
ψ
n
(
x
0
)
ψ
n
(x)d
x
0
, with the summation equivalent to the delta function
δ(x-
x
0
)
.
Reference: Any graduate-level text on quantum mechanics.

External Links

Orthonormal Functions (Wolfram MathWorld)

Permanent Citation

S. M. Blinder
​
​"Closure Property of Eigenfunctions"​
​http://demonstrations.wolfram.com/ClosurePropertyOfEigenfunctions/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011