Probability Density Functions

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

∫_a^bP_X(x)dx

The function _X(x) has to be such that

∫_−∞^∞P_X(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

∫_a^bP_X(x)dx

gives

∫_γa^γb[P_X(z/γ)/γ]dz

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

P_Z(z) = P_X(z)/γ

Normalization

If F(x) is a candidate for P_X(x) then P_x(x) is obtained by first computing

T = ∫_−∞^+∞F(x)dx

Then

P_X(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(α^½x). Note that

(dβ/dx) = α^½(dφ/dx)
and
(d²β/dx²) = α(d²φ/dx²) .

But (d²φ/dx²) is equal to f(α^½x) 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 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^½x_min to α^½x_max.

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 except for scale. The same shaped 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.