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The Uncertainty Principle and Harmonic Oscillators |
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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 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)dθ 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. Since the average value of x is zero this takes the form of
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. Thus 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; i.e.,
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. Therefore the satisfaction of the Uncertainty Principle does not imply any intrinsic indeterminism for the particle of the system.
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