San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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Quantum Theoretic Analysis |
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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 G_{n}(x) is then
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
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 H_{n}(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^{−x²/2}u(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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