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The Solutions to the Time-Independent Schrödinger
Equation and their Relationship to the Classical
Time-Spent Probability Distributions

Background

Schrödinger equations, time dependent and independent, are the standard basis for quantum theory but they do not have a derivation from first principles; they are merely constructed according to definite rules. The purpose of this material is to derive a time-averaged version of the time-independent Schrödinger equation for a particle in a potential field from the classical analysis.

The Correspondence Principle

According to the Correspondence Principle as articulated by Niels Bohr, for a quantum analysis to be valid it t must equal or asymptotically approach the results of classical analysis when it is scaled up by energy to a macroscopic level. What Bohr left out is that the macro level observation necessarily involves spatial and time averages. The correspondence is between the spatial average of the scaled up version of the quantum analysis and the classical analysis.

Since the result from Schrödinger's Equation is a probability density distribution, scaled up it is still a probability density distribution. What probability density distribution is there for classical analysis that the quantum analysis could correspond to? The answer is the time spent probability density distribution. This is the proportion of the time a system spends in its allowable states while traversing a periodic path. For a particle traveling at a velocity v in a spatial interval the time spent in that interval is its length divided by the magnitude of the velo+city. If the length of the interval is ds then the time spent dt is ds/|v|. If the total time spent traversing the periodic path is T then the proportion of the time spent in the interval is ds/(T|v|). This is the probability of finding the particle in the specified interval at a randomly chosen time.

The Quantum Analysis

Consider a particle of mass m in a one dimensional space subject to a potential V(x). The total energy E of the system is:

E = ½mv² + V(x)

where v is the velocity (dx/dt).

The Hamiltonian function H for this system is the:

H = ½p²/m + V(x)

where p is momentum mv.

This is converted into the Hamiltonian operator H^ by replacing the squared momentum p² with −h²(d²/dx²), where h is Planck's constant divided by 2π. Thus the time independent Schrödinger equation for the particle is

H^φ = −h²(d²φ/dx²) + V(x)φ = Eφ

where φ is such that its squared magnitude is equal to the probability density function for the particle.

This equation can be put into the form

(d²φ/dx²) = − (K(x)/h²)φ

where K(x)=E−V(x), the kinetic energy of the particle as a function of location.

The solution to this type of differential equation is characterized by rapid oscillations as shown below for a harmonic oscillator.

For typographic convenience let k²(x)=K(x)/h².

Elsewhere it is found that the average value of φ² about an interval around a point xi at which φ² has a local maximum is given by

av(φ²(xi) = (1/Li)∫xi−½Lixi+½Liφ²(z)dz = ½φ²(x0i/k(xi+½Li)

where Li=2π/k(xi).

Γi = Πj=1i [k(yj+½Lj)/k(yj−½Lj)]

For some cases φ² is at a local maximum at x=0; for other cases it is at a local minimum there. It is a local maximum when the principal quantum is odd as is assumed above.

As E → ∞ so does k(x)→∞ for all x. Thus Lj → 0 and hence Γj → 1 for all j. Likewise (xi+½Li)→xi. Therefore asymptotically the spatially averaged probability density distribution is inversely proportional to k(x)=K1/2(x).

The Classical Analysis

The time-spent probability distribution is the proportion of the time a system spends in its allowable states while traversing a periodic path. For a particle traveling at a velocity v in a spatial interval the time spent in that interval is its length divided by the magnitude of the velocity. If the length of the interval is ds then the time spent dt is ds/|v|. If the total time spent traversing the periodic path is T then the proportion of the time spent in the interval is ds/(T|v|). This is the probability of finding the particle in the specified interval at a randomly chosen time.

The total energy E is given by

E = ½mv² + V(x)

Therefore

|v| = [(2/m)(E − V(x)]½

The quantity (E−V(x)] can be represented as K(x), the kinetic energy of the particle as a function of location.

Thus

|v| = (2/m)½K(x)½

The factor of (2/m)½ is irrelevant for determining the probability density function since it multiplies everything.

The probability density function PC(x) for the particle is then

PC(x) = K(x)−½/T

where T = ∫K(x)−½dx.

In the display shown below for a harmonic oscillator the heavy line is the time-spent probability distribution for a harmonic oscillator from classical analysis. As can be seen, the spatial average of the quantum analytical probability distribution is equal to the classical probability distribution.

Conclusion

The spatially averaged quantum analytic probability distribution for a particle in a potential field is asymptotically proportional to K−½(x), where K(x) is the kinetic energy of the particle expressed as function of location. The time-spent probability distribution for the particle from classical analysis is also proportional to K−½(x).


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