Consider a particle traversing a periodic trajectory. Let s(t) and v(t) denote its path position and velocity, respectively, as functions of time t with

v(t) = (ds/dt)

The times dt and dτ spent in path and velocity intervals ds and dv are given by

dt = dx/v
dτ = dv/(dv/dt)

The proportions of time spent probability distributions are then

P_S(s) = 1/(T_S|v(s)|)
and
P_V(v) = 1/(T_V|dv(s)/dt|)

where T_S is the time period for executing the path; i.e.,

T_S = ∫ ds/|v(s)|

The time period for the velocity cycle T_V is defined as

T_V = ∫dv/|dv(s)/dt|

Note that

|dv(s)/dt| = |(dv(s)/ds)(ds/dt)| = |(dv(s)/ds)v|

Thus

T_V = ∫dv/|(dv(s)/ds)v|

Consider a change in the variable of integration from v to s based on the relation dv=(dv/ds)ds

From this it follows that

T_V = ∫dv/|(dv(s)/ds)v| = ∫(dv/ds)ds/|(dv(s)/ds)v| = ∫ ds/|v|
hence
T_V = T_S

which is what intuition would have told us.

Formulation of a Model

Consider a particle of mass m moving in one dimensional space whose position is denoted as x. The potential field given by U(x) where U(0)=0 and U(−x)=U(x). Let v be the velocity of the particle, p its momentum E its total energy. Then

E = ½mv² + U(x)

Thus

v = (2/m)^½(E−U(x))^½

For a particle executing a periodic trajectory the time spent in an interval dx of the trajectory is dx/|v|, where |v| is the absolute value of the particle's velocity. Thus the probability density of finding the particle in that interval at a random time is

P(x) = 1/(Tv(x))

where T is the total time spent in executing a cycle of the trajectory; i.e., T=∫dx/|v|. It can be called the normalization constant, the constant required to make the probability densities to sum to unity. Thus

P(x) = [(m/2)^½/T]/(E−U(x))^½

It is convenient to represent (E−U(x)) as K(x), the kinetic energy of the particle expressed as a function of location. Therefore

P(x) = [(m/2)^½/T]/(K(x))^½

The constant factor (m/2)^½ is irrelevant in determining P(x) because it is also a factor of T and thus cancels out.

The variance of P(x) is defined as

Var_P(x) = ∫(x−x)²P(x)dx

where x is the mean value of x; i.e., x=∫xP(x)dx. For the case being considered x is zero, so

Var_P(x) = ∫(x²/(T(K(x)^½)dx

The Uncertainty Principle also involves the variance of momentum and that is determined by the variance of velocity. As noted previously, the time the particle spends in a velocity interval dv is the interval length divided by the acceleration of the particle; i.e.,

dτt = dv/|(dv/dτ)| = dv/[|(dv/dx)(dx/dt)|] = dv/[|(dv/dx)vV|]

Since K(x) is equal to E−U(x) and v(x) = (2/m)^½(K(x))^½) it follows that

(dv/dx) = −½(−dU/dx)/[T(K(x))^½]
which reduces to
(dv/dx) = ½(dU/dx)/(TK(x)^½)

The mean value of the velocity is zero. The variance of velocity is therefore

Var_P(v) = ∫ [v²/(|T((dv/dx)v)|)|dv
which reduces to
Var_P(v) = ∫ |v/(T(dv/dx))|dv
and hence to
Var_P(v) = ∫ [K(x)^½/(T(dU/dx))]dv

So the variance of velocity is directly dependent on (K(x))^½ whereas the variance of location is inversely dependent on (K(x))^½. Anything that increaes K(x) decreases the variance of x but increases the variance of v. Thus the product of the variances is subject to the contrary influences due to any changes in K(x). Likewise the product of the variance of x and the variance of momentum is subject to those contrary influences.

Harmonic Oscillators

The levels of K(x) depends upon the total energy and the potential energy function U(x). The potential energy function can be considered by its Taylor's series

U(x) = U(0) + (dU/dx)₀x + ½(d²U/dx²)₀x² + …

It is assumed that U(0)=0. As the level of energy E goes to zero the system under consideration converges to a harmonic oscillator with a stiffness coefficient k equal to (dU/dx) at x equal to zero.

From a previous analysis it was found that σ_x=x_max/(2π)^½ where x_max is the maximum deviation of the oscillator from equilibrium. That maximum deviation is where all of the energy is potential and none of it is kinetic; i.e., x_max=(2E/k)^½. Likewise the standard deviation of velocity is

σ_v=v_max/(2π)^½

The maximum velocity v_max occurs where all of the particle's energy is kinetic and none of it is potential; i.e., at x=0 so v_max=(2E/m)^½. Thus

σ_x = (2E/k)^½/(2π)^½
σ_v = (2E/m)^½/(2π)^½

But

σ_p = mσ_v = (2mE)^½/(2π)^½

Thus the product of the standard deviations of displacement and momentum for a harmonic oscillator is equal to

σ_pσ_x = [ (2mE)^½/(2π)^½][ (2E/k)^½/(2π)^½
= 2E(m/k)^½/(2π)
which can be expressed as
σ_pσ_x = (E/(k/m)^½)/π

It has already been established that the harmonic oscillator has a minimum quantum of energy equal to hω, where ω is the frequency of the oscillator and h is Planck's constant. The frequency of a harmonic oscillator is equal to (k/m)^½ and hence the minimum E divided by the frequency (k/m)^½ is equal to h, Planck's constant.

Thus, from the above

σ_pσ_x = h/π

In order to sastisfy the uncertainty relation that product has to be greater than or equal to h/(4π). Thus the product of the standard deviations of displacement and momentum for a harmonic oscillator exceeds the required amount for a factor of 4.

Generaliztion

It wasn't necessary to limit the analysis to that of a simple harmonic oscillator but by properly choosing the coordinate system the analysis can be made almost as simple.

Consider a particle of mass m whose potential energy is a function of its location on a one dimensional coordinate. Let the zero point of that cordinate system correspond to the unique point of minimum potential energy and x be the diviation from that minimum. Let the potential energy U br the potenial energy in excess of that minimum. So the minimum U(x) is zero.

The relevant total energy E is given by

E = &fract12;mv² + U(x)

where v is the velocity of the particle (dx/dt). The velocity of the particle is given by

v = (2/m)^&fract12;(E−U(x))^&fract12;

It is convenient to let (E−U(x)) be denoted as K(x), particle kinetic energy as a function of location. Thus

v = (2/m)^&fract12;K(x)^&fract12;

The time spent probability function is

P_X(x) = 1/(T|v|)