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Probability Distributions for a Harmonic Oscillator |
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A harmonic oscillator is a device for which the restoring force on a particle mass is proportional to its displacement from equilibrium; i.e.,
where m is the mass of the particle and k is a constant, usually called the stiffness coefficient.
The potential energy function is then V(x)=½kx².
The Hamiltonian H for the harmonic oscillator is then
where p is the momentum of the particle.
This means that the Hamiltonian operator for a harmonic oscillator is
where φ is the wave function and h is Planck's constant.
The energy E is an eigenvalue of the equation and is equal to (n+½)h.
The wave function is a complex-valued function such that its squared value is the probability density.
The solutions give the probability density functions in terms of the dimensionless variable ζ=x/σ
where H_{n}(ζ) is the Hermite polynomial of order n.
The probability density function in terms of the displacement x is then given by
where
It can be shown that in the limit as n→∞ the squared values of the Hermite polynomials H_{n}² approach
where e=2.7218....
This means that the probability density functions for the harmonic oscillators asymptotically approach
The average of cos²(z) is 1/2. So the spatial average of (φ(x))² is essentially
The classical time-spent probability density function for a hamrmonic oscillator is
where x_{max} is the maximum deviation. The energy of the oscillator is equal to ½kx_{max}². Therefore x_{max}=(2E/k)^{½}. By the appropriate choice of units P(x) can be made proportional to (Φ(x))². For probability distributions constant factors do not matter because they cancel out in the normalization process.
The conclusion is then therefore that the spatial average of the quantum theoretic probability distribution for a harmonic oscillator is asymptotically equal to the classical time-spent probability distribution for such an oscillator. ` `
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