Simple way to calculate probability amplitude for photon with known properties

Question

I’ve been doing research on quantum mechanics, but am having trouble breaking the barrier between fundamentals and mathematics. I have a good understanding of the principles and basic calculations, but don’t know how to use them.

I want to know if there is a formula, preferably an algebraic one, that allows me to plug in variables about a particle in order to calculate the amplitude that the particle will travel to a certain position.

I’ve looked at Schodinger’s equation, but don’t have a good understanding of it and don’t know what many of the inputs mean.

My main goal, which is somewhat flexible, is to simulate a photon going from place to place in position space.

At first, I would do photons as they are easier to do calculations with. I would be, so to speak, Laplace’s demon, and know everything about the system. This includes

• The position in two dimensions of the particle
• The momentum of the particle
• The mass of the particle
• The current place in time of the simulation
• Constants such as Planck’s constant, the speed of light, etc.

Although I could do this without the exact formulas, I would much rather have the accuracy that comes with using them. Also, the goal is to simulate a quantum system, so it makes the most sense to use quantum mechanical equations!

This may be an obvious question. This might not have an answer or be possible. I just want to know if there is an equation that allows me to satisfy the goals previously stated.

As I’ve said, I’m a relative beginner to the quantum world, so please bear with me. Thanks for any help.

Answer 1

First of all! Schrodinger equation is not for photons. It is a non-relativistic approximation for a single electron. Second, you can not know everything about a quantum system. Heisenberg's uncertainty principle ensures that. If you know the position of the particle with a good amount of certainty, the momentum precision of the same particle will get spoiled.

Also, if you find a way to describe the photon quantum mechanically. This description only will be valid for high energies, so, in the spectrum of ultraviolet. It is easy to see that. You know that the energy of the photon is:

$$E=\frac{h c}{\lambda}$$

So, small the wavelength of the photon $\lambda$ means big energy $E$. And small wavelength means in range of ultraviolet or smaller.

So, if you want to simulate properly electromagnetic waves, it is better to do so with Maxwell's equations, you get a better range for wavelength than by Quantum Mechanics. As a curiosity, there is an app that simulates EM waves by solving Maxwell's equations called "LightWave Studio", it is very interesting, check it out.

But, there is already a question here in StackExchange about the wavefunction of the photon, you can search for answers here:

What equation describes the wavefunction of a single photon?

There is also a book called "An Introduction to Relativistic Quantum Field Theory", by Silvan S. Schweber, in the section 5.c of the book (pag. 116). There, the author writes an equation for the photon as like as Schrodinger equation, which is:

$$i\hbar c \partial_0 \mathbf{A}(x)=\hbar\sqrt{-\nabla^2}\mathbf{A}(x)$$

with a gauge condition,

$$\nabla \cdot \mathbf{A}(x)=0$$.

Once you found a solution for the photon equation (supposing there is one), namely the state of the photon, let's say $\psi(\vec{r},t)$. You can find the probability by Born rule $p(\vec{r})=|\psi(\vec{r},t)|^2 $, which tells you the probability to find the photon a given position $\vec{r}$ and time $t$.