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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) = Σ_{m=0}^{∞}c_{m}x^{m}

Then from the defining differential equation it follows that

(m+2)(m+1)c_{m+2} + (m+λ)c_{m} = 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 G_{n}(x) is then

(d² GC/dx²) + x(d G_{n}/dx) −n G_{n} = 0

This differential equation is homogeneous so nothing is lost by taking the first nonzero coefficient
to be unity. Two sets of G_{n} functions may be defined; one in which c_{m} is zero
if m is even and another in which c_{m} 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 G_{n}(x) functions obey the same ones.
One of particular significance says that as n increases without bound G_{n}(x)² asymptotically approaches

2(2n/e)^{n}cos²(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 H_{n}(x) is relevant for determing the wavefunction
for a harmonic oscillator, but always in the form

exp(−x²/2)H_{n}(x)

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

The Hermite polynomials are solutions to this differentia; equation