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Electron Motion in Atoms, Quantum Mechanics and the Uncertainty Principle 

The Copenhagen Interpretation of Quantum Mechanics holds that electrons in atoms do not have motion. The argument based upon the Uncertainty Principle that electrons in atoms do not have definite trajectories or even motion of any kind goes as follows. The outer limits of atoms are less than 10^{8} cm apart. The uncertainty of the location of an electron, σ_{x}, therefore cannot more than 10^{8} cm. By the Uncertainty Principle the product of the uncertainties of location and momentum must be greater than or equal to Planck's constant divided by 4π. Since Planck's constant divided by 4π is 5.275×10^{28} ergsec the uncertainty of momentum must be at least 5.275×10^{19} gcm/s.
An electron's mass is 9.11×10^{28} g. Therefore the uncertainty in electron velocity, σ_{v}, must be at least 5.79×10^{8} cm/s. This is translated naively into an uncertainty in the kinetic energy as
The correct procedure for computing the uncertainty in the kinetic energy would give an even higher value. The above value however is asserted to be so large compared with the magnitudes of 15 eV for potential energy and kinetic energy of electrons that no location and velocity for an electron in an atom is meaningful.
The problem with this argument is that the mean value and standard deviation for a a probability distribution are generally independent. This applies to the probability distribution for velocity. Velocity is not constrained in the way that location is, except by the speed of light. There are in fact perfectly legitimate probability distributions that have infinite variances and infinite standard deviations.
If P(x) is the probability density function for a variable x then the expected value of x, μ_{x} is defined as
The variance of x, Var(x), is then defined as
The standard deviation σ_{x} of x is the square root of its variance. Thus all that is required for a distribution to have infinite variance is that its probability density function go to zero slower than 1/x² as its argument x increases without bound either in a positive direction or a negative direction.
Here is an illustration from the special set of probability distributions called the stable distributions. A type of distribution is called stable if when x and y have that type of distribution then x+y also has that type of distribution. The Gaussian (normal) distribution is one type of stable distribution.
The Gaussian (normal) distribution is the only type of stable distribution that has a finite variance.
The Uncertainty Principle also implies that energy cannot be precisely conserved. The product of the uncertainties of energy and the time range over which value of energy is known must also be greater than or equal to Planck's constant divided by 4π. Thus if energy is known precisely to be a constant value the uncertainty concerning time would have to be infinite.
The analysis starts with formulating the energy in terms of kinetic and potential energy. The kinetic energy is expressed as a function of the momenta of the system. This total energy is called the Hamiltonian function of the system. The Hamiltonian function is converted into the Hamiltonian operator by replacing momenta conjugate to a state variable q by −i(∂ /∂q) and an exponent m by the mth order derivative. Then the timeindependent Schrödinger equation. The Schrödinger equation is then solved for the wave function. The squared magnitude of the wave function is the probability density function.
Thesolutions depend upon several parameters, one of which is a positive integer n called the principal quantum number. Another parameter is l, the orbital quantum. A third quantum number is the magnetic quantum number m. Below is a depiction of the probability density function for n=4, l=3 and m=±1.
The probability density function is in the nature of a set of shells rather than of blobs. This can be seen by viewing radial factor of functions.
Each peak of the radial factor corresponds to a separate shell.
According to the standard interpretation of quantum mechanics a probability density function represents a static (motionlesss) description of the electron. Consider the situation. The solution was derived from a Hamiltonian function involving motion for the electron. By the normal rules of logic this would invalidate the solution. The alternate interpretation is that the solution represents the time average of the motion of the electron as it passes successively through the allowed states.
The depictions of the solutions for other parameters are shown below.
Quantum movement involves systems dwelling in allowed locations and then rapidly moving to the next allowed location. A harmonic oscillator according to classical analysis follows a sinusoidal function of time. The quantum mechanical analysis of a harmonic oscillator with principle quantum number of 4 would have a pattern of motion as illustrated below.
For the quantum probability density function depicted above the pattern of electron motion might be as illustrated below.
The quantum probability density function depicted above is not the standard representation. That representation is as follows.
There is no obvious pattern of electron movement consistent with such a probability density function.
However since the Schrödinger equation is based upon the electron and proton of a hydrogen atom having motion if such patterns of motion cannot be found then the solutions are invalidated.
(To be continued.)
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