﻿ A Function Relevant for Quantum Theoretic Analysis
San José State University

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Thayer Watkins
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A Function Relevant for
Quantum Theoretic Analysis

The function is defined as a solution to the differential equation

#### (d²G/dx²) + x(dG/dx) + λG = 0

Suppose G(x) is given by polynomial series

#### G(x) = Σm=0∞cmxm

Then from the defining differential equation it follows that

#### (m+2)(m+1)cm+2 + (m+λ)cm = 0

The coefficient series will terminate (become all zeroes after a particular index) only if λ is equal to a negative integer, say n. The defining differential equation for Gn(x) is then

#### (d² GC/dx²) + x(d Gn/dx) −n Gn = 0

This differential equation is homogeneous so nothing is lost by taking the first nonzero coefficient to be unity. Two sets of Gn functions may be defined; one in which cm is zero if m is even and another in which cm is zero if m is odd. These correspond to the odd-evenness of the parameter n.

The Hermite polynomials obey a number of relationships. The Gn(x) functions obey the same ones. One of particular significance says that as n increases without bound Gn(x)² asymptotically approaches

#### 2(2n/e)ncos²(x(2n)½ − nπ/2)(1 − ½x²/n))½

This has the implication that the spatial average of the quantum theoretic probability distribution for a harmonic oscillator asymptotically approaches the classical time-spent probability distribution for a harmonic oscillator.

(To be continued.)

## Appendix

The Hermite polynomial functions Hn(x) is relevant for determing the wavefunction for a harmonic oscillator, but always in the form

#### exp(−x²/2)Hn(x)

This suggests that it is this quantity that should be defined and investigated.

The Hermite polynomials are solutions to this differentia; equation

#### (e−x²/2u')' + λe−x²/2u = 0

Let e−x²/2u(x)=w(x). Then

#### u(x) = ex²/2w(x) and u'(x) = ex²/2w'(x) + x·ex²/2w(x) e−x²/2u'(x) = w'(x) + xw(x)

Thus the differential equation for Hermite polynomial is equivalent to

#### (w'(x) + xw(x))' + λw(x) = 0 and w"(x) + xw'(x) + w(x) + λw(x) = 0 w"(x) + xw'(x) + (1 + λ)w(x) = 0

So the differential equation defining w(x) is

#### w"(x) + xw'(x) + μw(x) = 0

where μ is a real constant.