San José State University |
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applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
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The Satisfaction of the Uncertainty Principleby a Physical System More General than a Harmonic Oscillator |
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For a particle undergoing periodic motion the probability of finding it in an interval dx at a randomly chosen time is proportional to the time it spends in that interval in its periodic motion. That time dt is equal to dx/|v(x)|, where v(x) is the particles velocity at location x.

The probability density function for the particle's location is therefore

where T=2∫dx/|v(x)| is the time period of the motion. The factor of 2 arises from the particle traveling from a minimum to a maximum and then back down to the minimum.

There is also a probability density function for the particle's velocity ; i.e.,

where a(v) is the acceleration of the particle when its velocity is v. The time period for the period of the velocity is the same as the time period of the motion.

The particle is presumed to be moving in a potential field given by V(x). The potential is presumed to be such that V(0)=0 and V(−x)=V(x).

The energy E of the particle is given by

where m is the mass of the particle.

The force F on the particle at location x is given by

Hence the acceleration of the particle at x is

The limits of x are where all of the energy is potential and none of it is kinetic;, i.e.,

The positive solution for this condition is the maximum value of and the minimum value
for x is −x_{m}.

Likewise there is a maximum value for velocity. It occurs where all of the energy is kinetic and none of it is potential; i.e.,

and thus

v

The minimum velocity is −v_{m}.

The symmetry of V(x) is such that the mean or expected value of x is zero. Likewise the expected value of velocity is zero.

The variance of a variable z is defined as

Where E{z}=0 this reduces to

Therefore the quantities sought are

Var

Because of the symmetry these reduce to

and

Var

From the formulas for P_{X} and P_{S} these further reduce to

and

Var

Consider first Var_{S}. Since

v

and hence

p

In the formula for Var_{S} consider a change in the integration
variable from v to x; i.e.,

= 2∫

= 2∫

The crucial quantity for the Uncertainty Relation is

= (4m²/T²)[∫

Let f and g be two complex functions over the variable x. The Schwartz Inequality is then

In the Schwartz Inequality let f(x)=(x/v^{½}) and g(x)=v^{½}. Then from
the Schwartz Inequality

and thus

Var

Therefore

This is a general relationship.

Consider the special case of a harmonic oscillator, where V(x)=½kx².
For this case v(x)=(2/m)^{½}(E−½kx²)^{½}.
The term ½k may be factored out to give

But (2E/k) is equal to x_{m}² so

Thus

A change in the variable of integration to sin(θ)=x/x_{m} results in

The oscillation frequency ω for the oscillator is (k/m)^{½}. Therefore the above
relation is

Thus for this case

which reduces to

σ

which reduces to

σ

The minimum energy for the system is hω, where h is Planck's constant. Thus E/ω=h and hence

This is four times the value of h/(4π) required to satisfy the Uncertainty Principle. Thus a harmonic oscillator satisfies Uncertainty Principle.

More generally

and hence

x

and

x

Here the inverse is for only the right half of V(x); i.e., for x≥0.

Let

This is equivalent to a quadratic equation in x²; i.e.,

For λ=0 this is the case of a harmonic oscillator.

The inverses of V(x) are then

and therefore,

taking the positive root

x

or, equivalently

x

For small values of z,

Therefore for small values of E

which reduces to

x

Thus

but (k/m)=ω²

and 1/T=ω/(2π) so

(m/T)x

However the minimum energy is hω where h is Planck's constant so (E/ω)=h. Therefore

Thus, as in the case of a harmonic oscillator and regardless of the value of λ, the time-spent
probability density distributions for a particle moving in a potential field of
V(x)=½kx²+ λx^{4} with sufficiently small energies
satisfy the Uncertainty Principle. There is no problem of systems with large energies
satisfying the Uncertainty Principle. Therefore the Uncertainty Principle has no implication of the
immateriality of a particle at the quantum level.

(To be continued.)

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