When Erwin Schrödinger formulated the wave mechanics version of quantum physics in 1926 he did not specify what the wave function ψ represented. He thought its squared magnitude would represent something physical such as charge density. Max Born suggested that its squared magnitude represented spatial density of finding the particle near a particular location. Niels Bohr and his group in Copenhagen concurred and the notion that the wave function represents the intrinsic indeterminancy of the particle of the system came to be known as the Copenhagen Interpretation.

First consider a particle classically traversing periodic trajectory. Let s be path length and the velocity v be given as a function of s, say v(s). The period of time spent in an interval of length Δs is Δs/|v(s)|, where |v(s)| is the average speed of the particle in that interval. As Δs→ds the ratio ds/|v(s)| becomes something in the nature of a density. If the values of 1/|v(s)| are normalized the resulting function can be considered to be a probability density function P(s). The normalizing constant is just the time period of the trajectory cycle.

P(s) = 1/(Tv(s))
where ∫ds/|v(s)| = ∫dt = T

The kinetic energy of the particle is given by

K = ½mv²
and thus
v = (2K/m)^½
and hence
P(s) = (m/2)^½/K^½

Any constant multiplier is irrelevant in determining the probability density because it shows up in the normalizing constant as well and cancels out. The probability density for a classical particle is inversely proportional to the square root of the kinetic energy.

The Quantum Mechanical Wave Function

The time independent Schrödinger equation for a particle in a potential field V(r) is

−h²∇²ψ + V(r)ψ = Eψ

where E is energy and ψ is the wave function, the nature of which is at issue.

This equation can be multiplied on both sides by ψ and rearranged to give

h²ψ∇²ψ = −(E−V(r))ψ²
or, equivalently
h²ψ∇²ψ = −K(r)ψ²

where K(r) is the kinetic energy of the particle expressed as a function of the radial distance from the center of the potential field.

Now consider the vector calculus identity

div(ψ·grad(ψ)) = ψ·div(grad(ψ)) + grad(ψ)·grad(ψ)
which in the ∇ notation is
∇·(ψ∇ψ) = ψ∇²ψ + ∇ψ·∇ψ
or, equivalently
ψ∇²ψ = ∇·(ψ∇ψ) − ∇ψ·∇ψ

Thus the modified time independent Schrödinger equation is equivalent to

h² (∇·(ψ∇ψ) − ∇ψ·∇ψ) = −K(r)ψ²

Note that relative extremes of ψ² occur where

ψ∇ψ = 0
which is where either
ψ = 0 and ψ² is a minimum of 0
or
∇ψ = 0 and ψ² is a relative maximum

Note furthermore that

grad(ψ²) = 2ψgrad(ψ)
and hence
grad(ψ) = ½grad(ψ²)/ψ
and thus
grad(ψ)·grad(ψ) = (1/4)grad(ψ²)·grad(ψ²)/ψ²
or in ∇ notation
∇ψ·∇ψ = (1/4)∇(ψ²)·∇(ψ²)/ψ²

When the RHS of this equation is substituted for its LHS in the last modified version of the time independent Schrödinger equation the result is

h²(∇·(ψ∇ψ) − (1/4)∇(ψ²)·∇(ψ²)/ψ²) = −K(r)ψ²

There is a looped chain of maxima that are separated by minima of zero. This has to be established in general but for the case of a particle in a potential field the polar coordinates, as is shown in the Appendix, lead to an angular component of ψ² as shown below.

This is ψ² at a constant value of r and for a principal quantum number of 6.

Below is a depiction of the probability bumps in the plane of the particle's motion for a principal quantum number of 6.

The particle moves quantum mechanically relatively slowly in a probability bump, otherwise known as a state, and then relatively rapidly to the next state (bump).

The analysis now returns to the equation previously developed

h²(∇·(ψ∇ψ) − (1/4)∇(ψ²)·∇(ψ²)/ψ²) = −K(r)ψ²

If this equation is integrated from one maximum to the next or from one maximum to an adjacent minimum, the first term in the above equation evaluates to zero because at the end points of the integration either ψ is zero or ∇ψ is the zero vector. What is left is

−(1/4)h²( ∫∇(ψ²)·∇(ψ²)ds)/ψ*²) = −K(r)ψ#²Δs
or, equivalently
(1/4)h²([ ∫∇(ψ²)·∇(ψ²)ds/Δs/ψ*²) = K(r)ψ#²
or also
(1/4)h²([ ∇(ψ²)·∇(ψ²)/ψ*²) = K(r)ψ#²

where ψ*, ψ# are values of ψ within the interval of integration. The term ∇(ψ²)·∇(ψ²) stands for the average of ∇(ψ²)·∇(ψ²) in the interval of integration. Multiplying by ψ*² and taking the geometric mean of ψ*sup2; and ψ#² and denoting it as ψ² gives the equation

( ψ²)² = (1/4)h²([ ∇(ψ²)·∇(ψ²)/ψ*²)/K(r)
and hence
( ψ²) = (1/2)h([ ∇(ψ²)·∇(ψ²)/ψ*²)/K(r)^½

Here again is the probability density inversely proportional to the square root of kinetic energy.

Conclusion

The squared magnitude of the wave function is a probability density, but that probability density is the proportion of the time the particle spends in its various allowable locations.

Appendix

The Laplacian operator ∇² for polar coordinates (r, θ) is

(∂²/∂r²) + (1/r)(∂/∂r) + (1/r²)((∂²/∂θ²)

Thus the equation to be satisfied by ψ is:

−(h²/2m)[(∂²ψ/∂r²) + (1/r)(∂ψ/∂r) + (1/r²)((∂²ψ/∂θ²)] +(|E| − Q/r)ψ = 0

At this point it will be assumed that ψ(r, θ) is equal to R(r)Θ(θ). This is the separation of variables assumption. This is a mathematical convenience that is fraught with danger of precluding the physically relevant solutions. In this case it is alright because only circular orbits will be dealt with later.

When R(r)Θ(θ) is substituted into the equation it can be reduced to

−(h²/2m)[R"(r)/R + (1/r)R'(r) + (1/r²)(Θ"(θ)/Θ] + (|E| − Q/r) = 0

This equation may be put into the form

r²R"(r)/R + rR'(r)/R + (2m/(h²)(−|E|r² + rQ) = − (Θ"(θ)/Θ)

The LHS of the above is a function only of r and the RHS a function only of θ. Therefore their common value must be a constant. Let this constant be denoted as n².

Thus

(Θ"(θ)/Θ) = −n²
or, equivalently
Θ"(θ) + n²Θ(θ)

This equation has solutions of the form

Θ(θ) = A·cos(n·θ + θ₀)

where A and θ₀ are constants. Through a proper orientation of the polar coordinate system θ₀ can made equal to zero. So Θ(θ) = A·cos(n·θ). In order for Θ(θ+2π) to be equal to Θ(θ) n must be an integer. The probability density is the squared magnitude of the wave function. Therefore the probability density is proportional to cos²(nθ).

Below is the shape of this function for n=6.