Misinterpretation of Louis de Broglie hypothesis

Question

De Broglie hypothesis says that a particle also has wave like properties and gives a relation for calculating its wavelength. I have following doubts regarding it:-

1. wave is defined as the propagation of a disturbance through space. What is the disturbance here?
2. what about the other characteristics of the wave like frequency, wave speed etc. Is the wave even sinusoidal or not? What about its amplitude (is it related to amplitude or not?)
3. is the wave transverse or longitudinal?
4. If two elementary collide (come very close) with each other, will there wave counterparts show interference or something like that or not?

Answer 1

1. In "quantum field theory," the vacuum has a number of properties which are defined at every point in space. You might be familiar with the "electric field," whose values have three degrees of freedom and transform like a vector under rotations. We often say "the electric field is a vector" as a shorthand for these relationships.

   You can also imagine a "scalar field," where there is just one number at each point in space, and that number doesn't change under rotations. In a gas, you can choose a location and define a temperature and a pressure, so those quantities act like scalar fields.

   The field associated with the electron, or other spin-half particle, transforms under rotations like a "spinor." A spinor field has four components, which correspond to the two spin orientations of the matter and antimatter particles corresponding to the field.

2. The frequency of a matter wave is governed by the Schrödinger equation, or whichever relativistic replacement you are using. In the interpretation where the square of the wavefunction is a probability density, the amplitudes are "normalized" so that there is a 100% chance of detecting the particle somewhere.

   The solutions to the Schrödinger equation for a free particle are sinusoids,

   $$ \psi(x,t) = e^{i(\vec k\cdot\vec x - \omega t)}, $$

   but the solutions for bound particles are more complicated.

3. "Transverse" and "longitudinal" only make sense for vector fields. Note that for the pressure waves which correspond to sound, the relevant vector is the average velocity of some little blob of particles, so we can talk about sound waves being longitudinal even though the pressure is not a vector.

   For the wavefunction above, what evolves over time is the phase of the wavefunction, like $e^{-i\omega t}$. The value of that particular wavefunction at any point in space follows a circle in the complex plane, but the complex plane doesn't correspond to any physical direction.

4. If two electrons collide, because they are excitations of the same property of the vacuum, the electron field does show interference. Famously, the total wavefunction for two electrons must be "antisymmetric" if the two particles swap places.

Answer 2

All of those questions could asked identically for electromagnetic waves. Also, you seem to be assuming a lot of properties that waves don't always have.

1. Wave is a disturbance in space, but that space doesn't have to be the old three-dimensional space of classical mechanics. A wavefunction lives in a Hilbert space, which is not ordinary space.
2. Those characteristics are computed for probability waves exactly in the same way as they're computed for other waves: solve the propagation equation, find the dispersion relation, compute the celerity of a wave front, and so on. Just like electromagnetic waves, they can be sinusoidal, or not, it depends on the situation. The amplitude is mostly controlled by the normalization of the probability density defining the particle.
3. Just like an electromagnetic wave or a mechanical wave, it depends. You won't know until you solve the equation for your specific problem. The answer may be more complicated for a probability wave, since they propagage in an abstract space.
4. The answer requires some knowledge of optics, since quantum interference, while physically different from optical interference, share a similar mechanism (waves have to be coherent, the physically observable quantity is the square of the modulus of the wave). So quantum interference can happen. With a single particle it's rather easy (self-interference), but with two particles it's harder because their wavefunctions must be coherent.