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The Classical and Quantum Mechanical Probability Density Function for the Velocity of the Particle of a Harmonic Oscillator |
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A probability density function is an intrinsic aspect of the quantum mechanical analysis of a physical system. Classical analysis is deterministic and probability is not involved, but a a probability density function can be introduced in terms of the probability that a particle is in an infinitesimal interval at a randomly chosen time. This probability density function represents the proportion of the time that a particle is in the various states of the system.
The case analyzed is that of a harmonic oscillator. Let x be the dispacement of a particle from equilibrium. The restoring force is proportional to the displacement; i.e.,
where k is a positive constant, m is the mass of the particle and t is time.
The kinetic energy of the particle is ½mv² and the potential energy is ½kx², where v is the velocity of the particle. The total energy E is constant and hence
gives the dynamics of the oscillator. In particular this relationship can be solved for velocity v as a function of displacement x.
The time dt that the particle spends in an interval dx is dx/|v|. Thus the probability density is proportional to 1/|v|. The proportionality constant has to be chosen so that the total probability is equal to unity.
The amount of time that the system operates with a velocity in the interval dv is proportional to:
but for a harmonic oscillator the acceleration of the particle (dv/dt) is equal to (k/m)x. Just as v may be expressed as a function of x, so x may be expressed as a function of v; i.e.,
The velocity v ranges for zero at the limit of the oscillation to
The particle's velocity increases from 0 to v_{max} at x=0 then decreases back down to zero. With normalization the probability density function P_{vel}(v) for the velocity of a particle undergoing harmonic oscillation is then
The Hamiltonian function for a harmonic oscillator is
Therefore the time independent Schrödinger equation for a harmonic oscillator is
where h is Planck's constant divided by 2π and φ(x) is the
wave function for the system. The probability density is the squared magnitude of
φ(x). The energy E is quantized and is given by
where n is a positive integer and ω is the frequency of the oscillator (k/m)^{½}. Shown below is the probability density function for the case of n=4.
There are some displacements for which the probability density function is zero. In a sense, the probability density function is in the nature of a quantization of the allowable locations for the particle of the harmonic oscillator.
The full function extends beyond the range shown, including values of x corresponding to negative kinetic energy.
The difficult problem is establishing some basis for relating particle velocity to the probability density. In the classical analysis the probability density is proportional to the reciprocal of particle velocity. Elsewhere it is established that a spatial average of the quantum mechanical probability density is asymptotically equal to the classical probability density. Shown below is a spatially averaged version of the probability density function shown above.
There are no longer any points at which the probability density is zero.
According to the classical analysis the velocity of the particle would be inversely proportional to the probability density. The relationship for the quantum mechanical analysis would then be as shown below.
The particle more or less dwells at certain locations and then skips abruptly over other locations which could be characterized as forbidden locations.
The standard deviations of the location and momentum for a particle are significant quantities because the Uncertainty Principle requires that their product be greater than Planck's constant divided by 4π. The standard deviation of particle velocity and hence of momentum has to be derived from the probability density of velocity as a function of velocity. An analytical derivation of such a probability distribution is not available but the data from the above example of a harmonic oscillator with principal quantum number of 4 can be used to show the general shape of such a distribution. What is plotted below is the reciprocal of the spatially averaged quantum mechanical probability density sorted by magnitude. It gives the same sort of information as a histogram without arbitrarily specifying the quantitative intervals. The display is like a histogram turned on its side.
There are a relatively large number of locations with low but not zero velocity, a few with high velocity and a significant number with velocities ranging from low to high. The frequent instances in which several estimates have same or nearly the same value is evidence of quantization.
Werner Heisenberg and his associates probably thought of the probability distributions for position and momentum as being something like normal distributions centered on their expected values. The range of the distribution for position might be limited by the physical limits of the system. This is not true for the distribution of velocity and hence of momentum. There are perfectly legitimate probability distributions that even look similar to normal distributions which have infinite variance. A distribution has infinite variance if the density function goes to zero for increasing values slower than the reciprocal of those values squared. Thus the functional form of the distribution of velocity is critical in determining whether the Uncertainty Principle is trivially satisfied or not satisfied at all for particles in orbits.
(To be continued.)
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