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The Irrelevance of
Constant Multiliers in an
Equation which Determines
Probability Densities

Probability Density Functions

Suppose PX(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

abPX(x)dx

The function X(x) has to be such that

−∞PX(x)dx = 1

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

abPX(x)dx

gives

γaγb[PX(z/γ)/γ]dz

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

PZ(z) = PX(z)/γ

Normalization

If F(x) is a candidate for PX(x) then Px(x) is obtained by first computing

T = ∫−∞+∞F(x)dx

Then

PX(x) = F(x)/T

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).

Wave Functions

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

(d²φ/dx²) = f(x)φ(x)

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

(d²ψ/dx²) = αf(x)ψ(x)

Now consider the function β(x)=φ(α½x). Note that

(dβ/dx) = α½(dφ/dx)
and
(d²β/dx²) = α(d²φ/dx²) .

But (d²φ/dx²) is equal to f(α½x) which is β(x). Thus

(d²β/dx²) = αβ(x)

The probability density associated with φ(x) is

P(x) = (φ(x))²/∫(φ(z))²dz

But (d²φ(x)/dx²) equals φ(x) so

(d²β/dx²) = α²φ(x) = β(x)

Thus the equation

(d²φ/dx²) = αf(x)φ(x)

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

(d²ψ/dx²) = f(x)ψ(x)

In other words, the constant multiplier of α is irrelevant.

Quantum Mechanics

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

h²/(2m)(∂²ψ/dx²) + V(x)ψ(x) = Eψ(x)

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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