quantum mechanics - Treating matter waves as light waves?

Is it valid to treat a matter wave as a light wave with wavelength equal to the de Broglie wavelength of the matter wave? Either way please can you explain why?

Answer

It's not valid to treat light and matter waves alike. Why? Apart from the obvious reason that they are not the same (we can after all see light distinctly from matter), the two have different equations of motion - the (non-relativistic) "matter wave" obeys the famous Schrodinger equation, while classical light waves go around according to Maxwell's equations (or the standard wave equation).

There are many differences because of this:

1. A 'matter wave' is (unavoidably) complex valued, unlike a light wave, where the electric and magnetic fields are always real valued.


2. The frequency of a matter wave goes as the inverse square of its wavelength, and that of a light wave goes as the inverse of its wavelength. This means that while light of all wavelengths has the same speed $c = \nu\lambda$, the speed of a matter wave depends essentially on its wavelength/frequency. (In addition to John Rennie's point - the phase velocity of a matter wave is half its group velocity, while they are the same for light).


3. There can always be a time when there is no light in the universe, as light waves can be absorbed and emitted; but the total "intensity" of a matter wave must always be a constant for all time (this follows from the probability interpretation).


4. A rather famous example is the case of refraction: when light goes into a region where it slows down, it bends towards the normal; when matter goes into a region of higher potential (where it slows down), it bends away from the normal to the surface of the region: this is because in the latter case, only the component of velocity along the normal reduces. This is once again because the wavelength increases with increasing speed for a light wave, but decreases with increasing speed for a matter wave. In fact, as the dependence of wavelength on speed is reciprocal between the two cases, matter waves obey the "inverse" of Snell's law.

Finally, there is only ever one electromagnetic field in space at any point of time, and all these waves are disturbances on that field. These disturbances all add up. When there are many ("entangled") particles, however, the wave equation itself is different (a many-particle Schrodinger equation) and it is incredibly hard to express the wave function as a simple "sum" of many matter waves, one for each particle. Instead, matter waves are not waves in real space anymore, but simply a function of various properties of a material system.