Closure Property of Eigenfunctions
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A complete set of discrete eigenfunctions obeys the orthonormalization conditions . Complementary to these is the set of closure relations
. For real eigenfunctions, the complex conjugate can be dropped. The finite sums
for
up to 100 are evaluated in this Demonstration. Four systems are considered: (1) infinite square-well potential with
and eigenfunctions
,
,
; (2) linear harmonic oscillator with
and
,
,
; (3) linear rigid rotor for fixed
,
,
,
,
; (4) hydrogen-like radial function with
,
,
,
.
Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA
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Details
The closure relation can be derived by considering the expansion of an arbitrary function obeying the same analytic and boundary conditions as the eigenfunctions
. If the set of eigenfunctions is complete, one can write
, with expansion coefficients determined from
. Substituting the last relation into the expansion, we find
, with the summation equivalent to the delta function
.
Reference: Any graduate-level text on quantum mechanics.
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