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The Uncertainty Principle and a Harmonic Oscillator |
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This is an application of the concepts 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 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
The parameter k is called the stiffness coefficient. The solution to the above equation is
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 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
where ½mv² is the kinetic energy and ½kx² is the potential energy. The velocity is then
When the term (k/m)^{½} is factored out of the above expression the result is
The extreme displacements, ±x_{m}, are where the entire energy is potential and none kinetic; i.e.,
Thus the expression for velocity takes the form
Now it can be noted that the probability density is inversely proportional to [(x_{m}²−x²]^{½}. The displacement ranges from −x_{m} to +x_{m}. The normalizing factor T for the probability density function is then
The variable of integration can be changed to z=x/x_{m} to obtain
The change z=sin(θ) gives dz=cos(θ) and (1−z²)^{½}=cos(θ) so
The probability density function P(x) for displacement is then
Here is its shape.
From this distribution the variance of displacement σ_{x}² may be computed; i.e.,
By the same changes of variable as was used in determining the normalizing factor the formula for variance reduces to
The time the system spends in a velocity interval dv is given by
where a(v) is acceleration. For a harmonic oscillator the acceleration is given by
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
The extreme velocities occur where all of the energy is kinetic and none is potential; i.e.,
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
where S is the normalizing factor. It is given by
This is identical to the evaluation of the normalizing factor T for displacements. The value of S is π. Thus
This is the shape of the probability distribution for particle velocity.
Since the momentum p of the particle is equal to mv
The Uncertainty Principle requires that
When σ_{x} and σ_{p} are replaced by their formulas in a harmonic oscillator the result is
The minimum energy of the oscillator equal to hω and therefore the expression (E/ω) is equal to Planck's constant h and hence
Thus the Uncertainty Principle is satisfied by the time-spent probability distributions for displacement and velocity of a harmonic oscillator. The satisfaction of the Uncertainty Principle thus does not imply any intrinsic indeterminism of the system.
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