This is an application of the concept of Heisenberg's Uncertainty Principle to a classical system. A classical system is deterministic and does not inherently involve probabilities. However for a system that goes through a cycle the time spent in the allowable states is in the nature of a probability distribution. It represents the probability of finding the system in its various states at a randomly chosen time. The system utilized for this application is a harmonic oscillator.

A Harmonic Oscillator

A harmonic oscillator is a system with an equilibrium and such that the restoring force for a displacement from the equilibrium is proportional to the displacement. Let x be the displacement. Then

F = m(d²x/dt²) = −kx
and hence
(d²x/dt²) = −(k/m)x

The parameter k is called the stiffness coefficient. The solution to the above equation is

x(t) = A·cos(2πωt) + B·sin(2πωt)

where A and B are constants and ω, the frequency, is equal to (k/m)^½. For now the important point is the formula for the frequency; i.e., ω=(k/m)^½.

The Time-Spent
Probability Distribution

The time spent in an interval dx is dx/|v(x)| where v(x) is the velocity at point x. Thus the probability density is inversely proportional to the velocity v(x). The velocity is found from the total energy function

E = ½mv² + ½kx²

where ½mv² is the kinetic energy and ½kx² is the potential energy. The velocity is then

v(x) = [(2E−kx²)/m]^½

When the term (k/m)^½ is factored out of the above expression the result is

v(x) = (k/m)^½[(2E/k)−x²]^½

The extreme displacements, ±x_m, are where the entire energy is potential and none kinetic; i.e.,

E = ½kx_m²
and hence
x_m = (2E/k)^½

Thus the expression for velocity takes the form

v(x) = (k/m)^½[(x_m²−x²]^½

Now it can be noted that the probability density is inversely proportional to [(x_m²−x²]^½. The displacement ranges from −x_m to +x_m. The normalizing factor T for the probability density function is then

T = ∫_−xm^x_mdx/[(x_m²−x²]^½

The variable of integration can be changed to z=x/x_m to obtain

T = ∫₋₁⁺¹dz/(1−z²)^½

The change z=sin(θ) gives dz=cos(θ)dθ and (1−z²)^½=cos(θ) so

T = ∫_−π/2^π/2cos(θ)dθ/cos(θ) = ∫_−π/2^π/2dθ
T = (π/2) − (−π/2) = π

The probability density function P(x) for displacement is then

P(x) = (1/π)[1/(x_m²−x²]^½

Here is its shape.

From this distribution the variance of displacement σ_x² may be computed. Since the average value of x is zero this takes the form of

σ_x² = ∫_−xm^x_mx²dx/[(x_m²−x²]^½

By the same changes of variable as was used in determining the normalizing factor the formula for variance reduces to

σ_x² = (x_m²/π)∫_−π/2^π/2sin²(θ)dθ
and since
∫_−π/2^π/2sin²(θ)dθ = 1/2
σ_x² = x_m²/(2π)
and
σ_x = x_m/(2π)^½

The Probability Density
Function for Velocity

The time the system spends in a velocity interval dv is given by

dt = dv/|(dv/dt)| = dv/|a(v)|

where a(v) is acceleration. For a harmonic oscillator the acceleration is given by

a = F/m = −kx/m = −(k/m)x

Thus the probability density for velocity is inversely proportional to the magnitude of displacement. There is a perfect symmetry between displacement and velocity for a harmonic oscillator. The displacement as a function of v is given by

x(v) = [(2E−mv²)/k]^½

The extreme velocities occur where all of the energy is kinetic and none is potential; i.e.,

E = ½mv_m²
and hence
v_m = (2E/m)^½

Velocity goes through a cycle from 0 to v_m and then back down again to 0 and decreasing to −v_m before increasing back to 0. All of the values of velocity between −v_m and +v_m are covered. The probability density function Q(v) for velocity is then

Q(v) = (1/S)[1/(v_m²−v²]^½

where S is the normalizing factor. It is given by

S = ∫_−vm^v_mdv/[(v_m²−v²]^½

This is identical to the evaluation of the normalizing factor T for displacements. Thus the value of S is π. Thus

Q(s) = (1/π)(1/[(v_m²−v²]^½)
and hence
σ_v² = v_m²/(2π)
and
σ_v = v_m/(2π)^½

This is the shape of the probability distribution for particle velocity.

Since the momentum p of the particle is equal to mv

σ_p = mv_m/(2π)^½

The Uncertainty Relation

The Uncertainty Principle requires that

σ_xσ_p ≥ h/(4π)

When σ_x and σ_p are replaced by their formulas in a harmonic oscillator; i.e.,

σ_x = x_m/(2π)^½ and x_m = (2E/k)^½
σ_p = mv_m/(2π)^½ and v_m = (2E/m)^½

the result is

σ_xσ_p = (2E/(k/m)^½)/(2π)
and since (k/m)^½=ω
this reduces to
σ_xσ_p = (1/π)(E/ω)

The minimum energy of the oscillator equal to hω and therefore the expression (E/ω) is equal to Planck's constant h and hence

σ_xσ_p = h/π = 4(h/(4π))

Thus the Uncertainty Principle is satisfied by the time-spent probability distributions for displacement and velocity of a harmonic oscillator. Therefore the satisfaction of the Uncertainty Principle does not imply any intrinsic indeterminism for the particle of the system.