Closure Property of Eigenfunctions
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 , , , .[more]
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 .[less]
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.