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Quantum Theoretic Analysis
The function is defined as a solution to the differential equation
Suppose G(x) is given by polynomial series
Then from the defining differential equation it follows that
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
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
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.)
The Hermite polynomial functions Hn(x) is relevant for determing the wavefunction for a harmonic oscillator, but always in the form
This suggests that it is this quantity that should be defined and investigated.
The Hermite polynomials are solutions to this differentia; equation
Let e−x²/2u(x)=w(x). Then
Thus the differential equation for Hermite polynomial is equivalent to
So the differential equation defining w(x) is
where μ is a real constant.
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