Photons were originally assumed to be waves and even after they were found to be particles, photons still exhibited the same qualities that allowed them to be considered as waves before. de Broglie considered that, if a photon could be considered both a particle and a wave, then perhaps other forms of matter could be treated as waves.
de Broglie’s finding later became a pillar of quantum mechanics. The Schroedinger equation becomes an equation for such ‘matter waves’ as proposed by de Broglie. The plane wave as mentioned above later becomes the ‘wavefunction’ ψ(x,t) as is fundamental to Schroedinger’s equaiton. Is the wavefunction ψ(x,t) measurable? What is the meaning of the wavefunction ψ(x,t)? To come to this understanding, we should unpack some of de Broglie’s matter wave formulas.
Consider that the wavelength is inversely proportional to the momentum in de Broglie’s formula. Further, the momentum is found to be equal to h_bar multiplied by the wavenumber.
de Broglie’s wave formula presented further complications however. It was found that certain aspects of the wave would in fact be correct, but other parts of the wave would not be accurate. For instance, the phase of de Broglie’s matter wave would be accurate. The phase of the wavelength is understood to adhere to Galilean, non-relativistic physics, meaning that the result is not altered according to perspective. An example of a system would be relitivistic is the case for instance where the speed of a vehicle is determined relative to another moving vehicle. The phase of de Broglie’s calculation is therefore is a type of ‘objective’ calculation.
Further calculations below show that the waves proposed by de Broglie are not directly measurable. The wavefunction ψ(x,t) is also known to be multiplied by an imaginary number and this makes it difficult to measure. Further, the system is not Galilean invariant. This means that matter waves may differ according to reference. Finally it is concluded that the wavefunction ψ is not like sound waves, water waves, mechanical or electromagnetic waves. A difference according to reference however does not mean that all hope is lost in the calculation of the wavefunction, however. By taking account for the difference in reference, two points can be compared to allow for a wavefunction that may vary according to reference.
What does a matter wave look like?
Consider it should look like any other wave. It should have some kind of sinusoidal representation. Considering one aspect of matter, it should be that matter is not allowed to exist in any one place. This is the case for instance, when a particular piece of matter is not present in a location. However, to restrict a matter wave by time could be problematic. This is to say that, at a certain moment, no matter is allowed to exist anywhere or for all positions.
There are four cases listed below that, being sinusoidal, may seen to be possible respresentations for matter waves. In the case however for a sin(kx-wt) or cos(kx-wt), it is implied then that for time wt eual to pi/2, 3*pi/2, etc matter is not allowed to exist anywhere. Therefore these functions are not acceptable representations.
For the case however of e^(ikx-iwt) + e^(-kx-iwt) and it’s counterpart e(-ikx+iwt)+e^(ikx+iwt), this is not the case. Could both be used together in superposition? The answer is that if they were added together, matter would be restricted to one direction, which is undesireable as well. Therefore, either representations would be acceptable, but not both. The boxed answer below is the normative convention taken by physicists for the matter wave wavefunction for a particle. The other representation would work as well.
Principle of Stationary Phase
Consider the case of a narrow peak, modulated by a sinusoidal function centered about zero. If the sinusoid frequency is too high with relation to the narrowness of the peak or if the phase is rapidly changing, the averaging of the system will cause the narrow peak to disappear. If however the frequency is low – or if the phase is stationary at the narrow peak, any averaging that is done will still allow the narrow peak to exist.
As related to quantum mechanics, while determining characteristics of a wave, the principle of stationary phase becomes important. The wave Φ(k) as a function of wavenumber will be detected as a narrow peak. In order to properly detect this narrow peak, the modulating sinusoid must have a phase that varies much less with respect to (in this case, k) the x-axis. Otherwise, the narrow signal will be lost.
Towards developing a generalized Wavefunction
Performing a specific operation to the wavefunction interestingly produces the momentum multiplied by the wavefunction. The operator used is termed the “momentum operator.”
As we know from linear algebra, if a matrix multiplied by a vector is equal to a number multiplied by the vector, then the vector is termed an eigenvector of the matrix.
Given the above momentum operator relationship, it may be concluded that ψ(x,t) is an eignevector, or more specifically an eigenstate of the momentum operator p_hat. p is then also the eigenvalue.
And finally, using the eigenstate condition, Schroedinger’s Equation in general form for a free particle is derived: