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A Rotating Harmonic Oscillator
and the Uncertainty Principle

A particle of mass m is attached by a spring of stiffness coefficient k to the origin of polar coordinate system. The restoring force on the particle is −kr. The polar angle is θ and its rate of change is denoted as ω. The potential energy of the system is ½kr² and its kinetic energy is ½m[(dr/dt)² + (rω)²). The angular momentum L of the system is equal to mr²ω. Conservation of angular momentum means that

ω = L/(mr²)
and hence
E = ½m[(dr/dt)² + (rω)²) + ½kr²
E = ½m(dr/dt)² + ½L²/ (mr²) + ½kr²

The term ½L²/ (mr²) + ½kr² can be construed as a potential function V(r). The minimum of V(r) occurs where V'(r)=0; i.e.,

−L²/ (mr³) + kr = 0
or, equivalently
r4 = L²/(mk)

Let R=L½/(mk)1/4.

The second derivative of V(r) at R is crucial.

V"(r) = 3L²/ (mr4) + k
and thus
V"(R) = 3L²/ (mL²/(mk)) + k = 3k + k = 4k

Thus the frequency of oscillation of the particle about R is twice the natural frequency of the harmonic oscillator.

From what is already known for a particle in a potential field the probability density functions derived from the proportion of time spent in the allowable states for the location and momentum satisfy the Uncertainty Principle. There is no intrinsic indeterminacy of the particles for the system considered above. The result only requires the quantization of energy.

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