Time-Spent Probability Distributions

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

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

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.,

P_S(v) = 1/(T|a(v)|

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 General Problem

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

E = ½mv² + V(x)

where m is the mass of the particle.

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

F(x) = −(∂V/∂x)

Hence the acceleration of the particle at x is

a(x) = F(x)/m = −(∂V/∂x)/m

Limits of the Variables

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

V(x_m) = 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.,

½mv_m² = E
and thus
v_m = (2E/m)^½

The minimum velocity is −v_m.

The Mean Values of the Variables

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 Variances of the Variables

The variance of a variable z is defined as

Var_Z = ∫ (z−E{z})²P_Z(z)dz

Where E{z}=0 this reduces to

∫z²P_Z(z)dz

Therefore the quantities sought are

Var_X = ∫_−xm^x_m(x²P_X(x))dx
Var_S = ∫_−vm^v_m(v²P_S(v))dv

Because of the symmetry these reduce to

Var_X = 2∫₀^x_m(x²P_X(x))dx
and
Var_S = 2∫₀^v_m(v²P_S(v))dv

From the formulas for P_X and P_S these further reduce to

Var_X = 2∫₀^x_m[(x²/(T|v(x))|]dx
and
Var_S = 2∫₀^v_m[(v²/(T|a(x))|]dv

Consider first Var_S. Since

½mv_m = E
v_m = (2E/m)^½
and hence
p_m =mv_m = (2Em)^½

In the formula for Var_S consider a change in the integration variable from v to x; i.e.,

Var_S = 2∫₀^v_m[v²/(T|(dv/dt)|)]dv
= 2∫₀^x_m[v²/(T|(dv/dx)(dx/dt)|)](dv/dx)dx
= 2∫₀^v_m[v²/(T|v|]dx = 2∫₀^x_m(v(x)/T)dx )

The crucial quantity for the Uncertainty Relation is

Var_X Var_P = [2∫₀^x_m[(x²/T|v(x))|]dx] m²[2∫₀^x_m[v(x)/T]dx]
= (4m²/T²)[∫₀^x_m[(x²/|v(x))|]dx]∫₀^x_m[v(x)dx]

The Schwartz Inequality

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

[∫|f|²dx]·[∫|g|²dx] ≥ |∫fgdx|²

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

Var_X Var_P ≥ (4m²/T²)[∫₀^x_mxdx]²
and thus
Var_X Var_P ≥ (4m²/T²)[½x_m²]² = (m/T)²[x_m⁴]

Therefore

σ_Xσ_P≥ (m/T)x_m²

This is a general relationship.

The Harmonic Oscillator

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

v(x) = (k/m)^½(2E/k−x²)^½

But (2E/k) is equal to x_m² so

v(x) = (k/m)^½( x_m²−x²)^½

Thus

T = 2∫_−xm^x_m(m/k)^½dx/( x_m²−x²)^½

A change in the variable of integration to sin(θ)=x/x_m results in

T = 2π(m/k)^½

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

T = 2π/ω

Thus for this case

σ_Xσ_P≥ (mω/2π)(2E/k)
which reduces to
σ_Xσ_P≥ ω(E/(k/m)/π = ω(E/(ω²π)
which reduces to
σ_Xσ_P≥ E/(ωπ)

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

σ_Xσ_P≥ h/π = 4(h/4π)

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

Generalization

More generally

V(x_m) = E
and hence
x_m = V⁻¹(E)
and
x_m² = [V⁻¹(E)]²

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

Let

V(x) = ½kx²+ λx⁴

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

λx⁴ + ½kx² − V = 0

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

The inverses of V(x) are then

x² = (−½k ± (¼k² + λV)^½)/(2λ)
and therefore,
taking the positive root
x_m² = (−½k + (¼k² + 4λE)^½)/(2λ)
or, equivalently
x_m² = (−½k + ½k(1 + 16λE/k²)^½)/(2λ)

For small values of z,

(1+z)^½ ≅ 1 + z/2

Therefore for small values of E

x_m² ≅ (½k(16λE/(2k²))/(2λ)
which reduces to
x_m² ≅ (16λE/(4k))/(2λ) = 2E/k

Thus

(m/T)x_m² ≅ (1/T)2E/(k/m))
but (k/m)=ω²
and 1/T=ω/(2π) so
(m/T)x_m² ≅ (E/ω)/π

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

σ_Xσ_P≥ (m/T)x_m² ≅ h/π = 4(h/4π)

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+ λx⁴ 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.)