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Constant 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
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.
The usual presentation of the time-indepednet Schrödinger equation is that it arises from the substitution of
ih(∂/∂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 h²/(2m) is irrelevant. The same wave function and hence probability density function would
arise if i(∂/∂x) were substituted for p instead of ih(∂/∂x). It is also notable that the probability
density function is independent of the mass m of the particle.
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