How to programmatically simulate the propagation of a wave packet?

Question

Programming

Please help me in the next problem.
I need to mathematically describe the propagation of a wave packet, i.e. the very "something" that unites the view of matter as a wave, and as a particle. It is clear that even on Wikipedia there are equations of quantum mechanics - I need the simplest possible model so that it can be programmed without any special tricks. That is, we need a minimally sufficient formula that at least somewhat adequately describes the process of motion of a stable wave in space. Perhaps this will be the equation of an ordinary wave (through cos), I'm not sure ..
So, the input is:
- wave frequency
- its spatial distribution (density function of position in space)
- wave energy
- wave velocity vector
At the output:
we see how the wave moves in space (along the x-axis).
Ideally, we can change the conditions of the medium (density, for example) and see how the wave speed changes.

Answer 1

So, we are dealing with electromagnetic waves in curved space-time. OK. The solution is indeed represented by the sum of cosines, but as an argument there will be an integral of the phase increment of the wave along the direction of its propagation, which is a geodesic curve. In short, the solution to this problem is known, but I can’t explain it “on the fingers” so that you can immediately model it without diving into covariant integration.
However, if you want to play with this task "for yourself", set the harmonic in the same form as I wrote above, only take the phase velocity as a constant, and multiply the wave number by
1 - (1/x_0 - 1/x) r_g/2 ,
where x_0 is the coordinate where the wave originates; r_g - grav. the radius of a massive body, and x_0 > r_g. The wave propagates in the direction of increasing x. It can be seen that the wave number decreases as the wave propagates. This means that the wavelength increases and the frequency decreases. This is gravitational redshift.
The sum of cosines, because: (a) the wave equation is linear, (b) photons do not interact with each other. The latter is true in vacuum for low-energy photons, when we do not take into account the creation of particles that are knocked out of nowhere by a flying photon.

Answer 2

As always, first you need to choose an acceptable mat model.

Answer 3

Correctly noted Nc_soft, we need a model. What do you call a wave? Is it a solution to the wave equation or is it a propagating perturbation in some non-linear system? If you're gentle enough to make something that looks like a wave without any particular physics involved, then maybe you can think of a wave as a sum of cosines with different initial phases and phase velocities:
f(t,x) = sum_n A_n cos[(x - c_n t) k_n + phi_n]
Here is the sum of harmonics numbered with index n;
A_n - harmonic amplitude;
c_n - phase speed, i.e. the speed at which this harmonic moves as a whole;
k_n - wave number of the given harmonic, it defines the wavelength as lambda_n = 2*pi/k_n;
phi_n - initial phase, offset of the graph of this harmonic along the x-axis at the moment t=0.
If the phase velocity is different for different harmonics, then they say that the waves in such a system have dispersion, then the wave packet will spread out over time. Otherwise, the representation of the wave above can be reduced to the form
f(t,x) = g(x - ct) ,
where g(x) is the wave profile at the initial moment of time.
Such a model can describe a fairly large layer of physics.

Answer 4

There is no such thing as a "green particle". If by this term you mean some macroscopic bodies (for example, of small mass) that fly out like a cannon, then: (a) their motion will be described by the GR equations for a free particle; (b) everything emitted by such particles will experience both Doppler and gravitational shifts.
If you mean that quantum properties are essential for particles and particles do not interact with each other, then we represent each particle as a wave (hello, de Broglie), which experiences gravity. bias.
In the general case, when particles interact, we write the Schrödinger equation in a curved metric. The radiation that is generated in such a system will be subject to the same energy-momentum transformations, known as the Doppler effect and gravity. bias.