The Schrödinger Wave Equation

schrodingerFollowing Bohr’s model for the hydrogen atom Louis de Broglie suggested that the electron could be considered as a wave rather than a particle and proposed that the wavelength of such a wave was Planck’s constant divided by the linear momentum of the orbiting electron.   In effect he was restating Bohr’s adopted assumption that angular momentum is quantised in units of Planck’s constant but de Broglie is expressing it in terms of frequency and wavelength rather than position and velocity.[1]

Erwin Schrödinger was intrigued by de Broglie’s idea of the particle as a wave and set about developing an equation to describe de Broglie’s waves.  There are many ways to derive Schrödinger’s wave equation, but by far the simplest is to substitute de Broglie’s wavelength into the canonical form of an undamped second order differential equation, in other words into the standard equation for a wave.

Schrödinger’s time independent wave equation can be written:

 Schrodinger wave equation Equation 1

Where

 Del squared Equation 2

de Broglie identifies a new type of wave defined by dividing Planck’s constant by the linear momentum.  The wavelength is then given by:-

 Lambda Equation 3

So we can identify the equivalent orbital radius associated with such a wave as

 Radius in Bohr model Equation 4

The canonical form of a second order differential equation which is associated with a continuous wave described by a body orbtiting at radius r is

 canonical form Equation 5

Substituting for r gives

 substitution Equation 6

The energy of the electron is made up of its kinetic energy E combined with its potential energy V

 total energy Equation 7

And so

 one dimensional equation Equation 8

Extending this to a three dimensional space instead of this simpler one dimensional case gives

 three dimensional Equation 9

Because the potential energy V is a function of the distance this is sometimes written:

 complete Equation 10

Hence Bohr’s original model, de Broglie’s waves and Schrödinger’s wave equation all owe their existence to Bohr’s adopted assumption that angular momentum is quantised.

If you ask almost any physicist today to prove that angular momentum is quantised, they will invariably tell you that if you apply 360 degree rotational symmetry to the solutions of Schrödinger’s equation you will obtain quantum angular momentum.  Or that the Eigenvalues of the Hamiltonian lead to a similar conclusion.  This of course is total nonsense.  Since Schrödinger’s wave equation is derived directly from the assumption that angular momentum is quantised, it cannot then be used to prove that it is quantised.  Such a proof is almost the very definition of an ontological one.  By the way the Hamiltonian referred to here is simply a way to express Schrödinger’s equations in terms of a state space using matrices, and the Eigenvalues represent the solutions to the equation when expressed in this form.

The fact of the matter is that the idea that angular momentum is quantised has never been proven. No mechanism has ever been found which could link orbital radius with orbital velocity and take account of relativity acting on the mass term to produce such quantisation. No lesser person that Louis de Broglie spent the better part of 20 years trying and never succeeded.  If it ever could be proved this would represent the missing link between classical and quantum physics. And if it ever had been proved you certainly have heard about it; the person who did so would for sure have received the Nobel Prize.

In Sampling the Hydrogen Atom a model is derived for the hydrogen atom based on the idea that it is not angular momentum that is quantised, but Gamma, the Lorentz factor and in which the orbital radius is fixed at

 fixed radius Equation 11

Since the electron is orbiting at a fixed radius, the potential energy term in the Schrödinger wave equation makes no contribution to the atomic spectra and is therefore zero.  The electron is orbiting at near light speed and so the kinetic energy of the electron is

 kinetic energy Equation 12

Substituting these into the Schrödinger wave equation gives

 substituted in wave equation Equation 13

Which simplifies to

 simplified Equation 14

Or

 simplified again Equation 15

Which is the equation of a particle in circular orbit at radius R.

Quantum theory suggests that when a particle is observed its wave front (whatever that is) collapses to reveal either the location or the velocity of the particle.  In a sense we can think of the above as the wave equation itself collapsing to reveal the particle as being objectively real.

In Sampling the Hydrogen Atom changes in energy level are accompanied by a change in orbital velocity, not by changes in orbital radius. Unlike the Bohr model, which requires the introduction of the mysterious quantum leap, or the Schrödinger model which requires that energy level is encoded in some mysterious representation of a probability, such changes as are proposed in Sampling the Hydrogen Atom are perfectly possible within the realm of conventional mechanics. They happen every time you change gear in a car.

Particles in such a model are objectively real. All of the exotic paraphernalia of quantum theory, with its wave/particle duality, inherent uncertainty, particles which are neither here nor there, representations of particles as probabilities are unnecessary and can be eliminated to yield a simple mechanical model for the atom. Particles are objectively real point particles having deterministic position and deterministic velocity exactly as Einstein would have understood them.

The debate between Bohr and Einstein, which ended prematurely and unresolved with Einstein’s death in 1955, is decided firmly in Einstein’s favour. Bohr and his cohort were simply wrong – they attempted to build on top of the Bohr model for hydrogen by carrying on with the supposition that angular momentum is quantised, a model which was fundamentally flawed, and as a consequence were forced down a path which led to such fanciful ideas as intrinsic uncertainty, wave particle duality and subjective reality. If they had recognised that it was the Bohr model itself which was at fault and sought to find a viable alternative, they would not have needed to follow this path. Einstein is not entirely blameless. While his instincts told him that Bohr and his colleagues were wrong, he was so engrossed in his work on grand unified field theory that he spent little time countering Bohr. In particular he only ever bothered to criticise Bohr’s proposals rather that to conduct his own investigation to discover the root of the problem and possibly propose an alternative theory


 

[1] It would have been far more conventional to suggest that the wavelength was angular momentum divided by linear momentum, which is what happens on any other scale.  Indeed this marks the precise point where quantum theory takes on the idea that the laws of physics are different on the scale of the atom.

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