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Time Spent Probability Distributions,their Variances and their Satisfaction of the Uncertainty Principle |
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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

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

dτ = dv/(dv/dt)

The proportions of time spent probability distributions are then

and

where T_{S} is the time period for executing the path; i.e.,

The time period for the velocity cycle T_{V} is defined as

Note that

Thus

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

hence

T

which is what intuition would have told us.

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

Thus

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

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

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

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

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

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.,

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

which reduces to

(dv/dx) = ½(dU/dx)/(TK(x)

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

which reduces to

Var

and hence to

Var

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.

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

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

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

But

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

= 2E(m/k)

which can be expressed as

σ

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

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.

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

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

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

The time spent probability function is

where T is the time period of a cycle.

The time spent probability function for velocity is given by

where α is acceleration (dv/dt). Note that

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