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Probability Distributions, Quantum and Classical |
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This material is to introduce a topic in simplified form that will later be analyzed rigorously. Consider the probability density distribution for a harmonic oscillator as derived from time independent Schrödinger equation.
The heavy line is for the time-spent probability density function derived from the classical analysis of a harmonic oscillator. As can be seen the spatial average of the quantum mechanical probability density function is the classical distribution function.
Let P(x) be the time-spent distribution function ranging from L up to H. Let E{x} and E{x²} denote
Note that the variance of x is given by
The quantum probability density function is approximately
Let the cumulative distribution be defined as
Note that R(L)=0 and R(H)=1.
Use will be made of the formula for integration by parts for a definite integral; i.e.,
Now consider
In the integration by parts formula let U=(2cos²(λx)) and dV=P(x)dx. Thus V=R(x). Then
Then
If ∫_{L}^{H}P_{S}=1 then
This is eminently reasonable and established rigorously in Asymptotic limit of a sine integral. The quantity cos(λx)sin(λx) is equivalent to sin(2λx) and the rapid fluctuation of this quantity between positive and negative values eliminates any accumulation of nonzero values.
Now consider
Let
Note that S(L)=0 and S(H)=E(x).
Then
Since ∫_{L}^{H} R(x)cos(λx)sin(λx)dx can be expected to be zero
By a similar argument
Thus the variances are equal
The same analysis applied to the momentum of the system yields
Thus the classical distributions either satisfy the uncerainty principle together or fail it together.
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