San José State University |
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applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
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The Irrelevance ofConstant Multiliers in an Equation which Determines Probability Densities |
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Suppose P_{X}(x) is the probability density function for the variable X. This means that
the probability of the variable X being found to be between a and b is

The function _{X}(x) has to be such that

If the variable X is changed to the variable Z by the transformation z=γx then the change in the variable of integration in

gives

This means that P_{Z}(z), the probability density function for the variable Z
is given by

If F(x) is a candidate for P_{X}(x) then P_{x}(x) is obtained by first computing

Then

This is the process of *normalization*.

If G(x)=γF(x), where γ is a constant, then G(x) leads to the same probability density function as does F(x).

The wave function for a physical system is a solution to its time-independent Schrödinger equation. If ψ(x) is the wave function then the probability density function is the normalization of |ψ(x)|². ,

Let φ(x) be the wave function for a system. which is given by the solution to

and let ψ(x) be the wave function given by

Now consider the function β(x)=φ(α^{½}x).
Note that

and

(d²β/dx²) = α(d²φ/dx²) .

But (d²φ/dx²) is equal to f(α^{½}x) which is β(x).
Thus

The probability density associated with φ(x) is

But (d²φ(x)/dx²) equals φ(x) so

Thus the equation

leads to the same probability density function as the solution to the equation

In other words, the constant multiplier of α is irrelevant except for scale.
That is to say, if the nonzero values of ψ(x) run from x_{min} to x_{max} then those of
φ(x) run from α^{½}x_{min} to α^{½}x_{max}.

The usual presentation of the time-indepednet Schrödinger equation is that it arises from the substitution of
i~~h~~(∂/∂x) for momentum p in the Hamiltonian for the system, where i is the imaginary unit and ~~h~~
is the reduced Planck's constant. For a particle of mass m moving in a potential field of V(x) that gives the equation

The inclusion of ~~h~~ is thought to garantee that the analysis is quantum mechanical, but the analysis above indicates
that the coefficient ~~h~~²/(2m) is irrelevant except for scale. The same shaped wave function and hence probability density function would
arise if i(∂/∂x) were substituted for p instead of i~~h~~(∂/∂x). It is also notable that the probability
density function is independent of the mass m of the particle.

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