Schrodinger’s Wave Mechanics

1

By 1926, Schrodinger was examining deBroglie's idea of matter waves to see whether a standard wave equation could also be used to describe matter waves. The solution he eventually derived is known as Schrodinger's wave equation and is considered to be one of the most fundamental equations of quantum theory.

[1]      1

The form of the equation above is actually described as the time-dependent wave equation, which is the general form for a system that evolves in time. There is a time-independent equation, which can be applied to a system in a stationary state, when the Hamilton [H] can also be said to be independent of time. In this case, the operator can be replaced by the Hamiltonian operator  and written as:

[2]      2

However, because [2] is essentially a special case of [1], we shall focus the discussion on Schrodinger’s development of the time-dependent wave equation, as shown in [1]. Of course, like many mathematical equations, the actual meaning of [1] is obscured by what amounts to the shorthand of mathematics and without any understanding of this shorthand, [1] is essentially meaningless. Therefore, in an attempt to try and follow the logic of Schrodinger's thinking, rather than a formalised proof, we shall start with the standard wave equation, as introduced in the preliminary discussion of ‘Wave Propagation:

[3]      3

The general form of [3] reflects a wave propagating in 1-dimensional space [x] and time [t]. However, as has also been previously been discussed under the heading ‘Wave Dispersion, matter waves as defined by deBroglie lead to a non-linear relationship between [ω] and [κ]:

[4]      4

The relationship in [4] led Schrödinger to take a different approach in the derivation of the matter wave equation, than suggested by the 2nd differential format in [3], based on a complex form first devised by Euler. In order to align with the normal notation of quantum wave mechanics, we shall also change the implied amplidude [A] to [Ψ], but defer introducing any extended meaning associated with this variable at this point.

[5]      5

Now, again, with respect to time [t]

[6]      6

So, in essence, [5] and [6] simply replicates the standard approach of a classical wave equation using complex number notation. However, based on the complex form, Schrodinger had the option to equate the 1st differential with respect to time [t], in [6], with the 2nd differential with respect to distance [x], in [5], so that the dispersive relationship in [4] could be directly explored:

[7]      7

In order to proceed from [7], we need to substitute for [ω] and [κ] based on deBroglie’s assumption for matter waves. If we start with deBroglie’s expression for the particle wavelength [λ], we can derive an expression for [κ]:

[8]      8

We can also convert Planck’s equation associating energy [E] with frequency [f] in order to create an  expression for [ω]:

[9]      9

At this point, we can now substitute the results in [8] and [9] back into [7]:

[10]    10

At this point, there is a little mathematical ‘trick’  that can be applied to [10] based on the fact than the reciprocal [1/i] equals [-i]:

[11]    11

Finally, the last step is to substitute for p2=2mE

[12]    12

In essence, [12] is the basis of Schroedinger’s time-dependent wave equation for a free particle, limited to one-dimension, although this equation does not account for the particle moving in a potential energy field. We can extend the previous derivation by expressing the total energy [E] of the particle as the sum of its kinetic [T] and potential [U] energy:

[13]    13

Rearranging in terms of [k]:

[14]    14

This equation has to be substituted back into spatial wave equation introduced in [6], which is again presented below:

[15]    15

This is considered to be the generalized form of Schrodinger’s time-dependent wave equation and applicable for a particle being acted upon by a force, although for completeness we should also present this equation in its 3-dimensional form:

[16]    16

It was stated at the outset that the steps leading to Schrodinger’s wave equation, as outlined in this section, do not constitute a proof of its physical applicability, but rather attempts to rationalise the logical steps that led to its derivation. This said, the Schrodinger equation has been applied to many systems, especially the hydrogen atom, and shown to produce excellent results that agree with experimental data. As such, Schrodinger’s equation has come to be accepted as one of the most fundamental equation in the development of quantum theory. However, while the equations have been verified by experimental data, there has always remained a nagging doubt as to what the wave function is actually telling us about the microscopic world, i.e.

What physical interpretation is now being associated with [ψ]?

We will not rush to answer this question; although it might be said that while [ψ] might still represent the amplitude of the wave function, its meanings is now only interpreted in terms of the probability of finding a particle at a given point in space-time. However, the introduction of space-time raises the issue of the role of relativity within quantum mechanics, which we will try to address in terms of the development of the Klein-Gordon and Dirac Equations.