F = dP/dt on subatomic particles

[General Scientist]

I am trying to code a simulator for fundamental particles. The problem is that on a subatomic level, force works differently than on a macroscopic level. Then there is also the problem of Heisenberg's Uncertainty Principle. How is a force applied to a subatomic particle? I did some research and I found F = dP/dt, but I don't understand how to use it if time is discrete in the code. How can I find the next position, velocity, and acceleration based on the change in time and the force upon the particle?

 

[Vanadium 50]

You can't.

You're trying to model something quantum mechanical with classical mechanics. This won't work.

 

[General Scientist]

You can't.

You're trying to model something quantum mechanical with classical mechanics. This won't work.

Then how do I calculate the particle's motion?

 

[Vanadium 50]

You can't.

You're trying to model something quantum mechanical with classical mechanics. This won't work.

 

[PeterDonis]

I did some research and I found F = dP/dt

This is a classical formula, not a quantum formula. A quantum particle does not have a definite momentum P to begin with.

how do I calculate the particle's motion?

A quantum particle does not have a definite motion.

 

[General Scientist]

This is a classical formula, not a quantum formula. A quantum particle does not have a definite momentum P to begin with.
A quantum particle does not have a definite motion.

Then how do physicists simulate quantum mechanics?

 

[General Scientist]

Then how to do apply force to quantum particles? All the quantum fields would have an effect on it, a.k.a. a force.

 

[PeterDonis]

how do physicists simulate quantum mechanics?

Using the equations of quantum mechanics.

how to do apply force to quantum particles?

By putting them in the appropriate field. For example, you apply force to a charged particle by putting it in an electromagnetic field.

 

[Nugatory]

Then how do physicists simulate quantum mechanics?

Then how to do apply force to quantum particles? All the quantum fields would have an effect on it, a.k.a. a force.

It seems that you are thinking of quantum particles as if they are objects that move through space on paths determined by their speed and the forces acting on them, just as bullets and planets and grains of sand do.

But quantum particles don't act anything like that (and even the use of the word "particle" is a historical accident, one that has caused untold confusion over the last century). One way to see this is to look at Schrödinger's equation, which determines the evolution of a quantum system the way $F=ma$ and the rest of Newton's laws determine the evolution of a classical system. You won't see any forces or velocities or accelerations there; it's a whole different set of rules.

A cloud chamber is an example. An electron zips through it leaving a trail. If there's a magnetic field present, the trail is curved. The classical explanation is easy: the electron is moving through space like a little bullet; there's a force on it from the magnetic field; this force accelerates it sideways; the result is a curved path instead of the straight one we'd get by Newton's first law if their were no force.

The quantum mechanical analysis is completely different (and way harder, which is why we generally don't use QM when classical mechanics is a good enough approximation). Suppose the electron interacts with one droplet in the cloud chamber, making it visible. Quantum mechanics let's us calculate the probability of the electron then interacting with another nearby droplet to make that one visible as well, and another after that, and so on. When we grovel through all the computations, we will find that the droplets most likely to become visible are the ones that lie along the curved path that classical mechanics calls "the trajectory of the electron".

 

[General Scientist]

One way to see this is to look at Schrödinger's equation, which determines the evolution of a quantum system the way $F=ma$ and the rest of Newton's laws determine the evolution of a classical system.".

So I can use Schrödinger's equation? But then can you explain the V in the equations? After some quick research I see that it represents force but you said that forces aren't used?

 

[Nugatory]

So I can use Schrödinger's equation? But then can you explain the V in the equations? After some quick research I see that it represents force but you said that forces aren't used?

That $V$ is not force, it is the potential. And when you're done calculating with it you won't have a position, velocity, or acceleration, you'll have the probability of getting a particular result if you measure the position or the momentum.

As I said above, it's a whole different set of rules than Newtonian mechanics. You'll have to learn these rules, and to learn them at a level that allows any sort of quantitative simulation you'll need a real textbook, something like Griffiths that you'd encounter in the second year of college if you're working on a degree in physics.

 

[PeroK]

So I can use Schrödinger's equation? But then can you explain the V in the equations? After some quick research I see that it represents force but you said that forces aren't used?

The V is a potential. I think that to simulate anything using quantum mechanics you are going to have to study the subject.

Even the free particle, where there is no potential, is mathematically complicated.

 

[General Scientist]

That $V$ is not force, it is the potential. And when you're done calculating with it you won't have a position, velocity, or acceleration, you'll have the probability of getting a particular result if you measure the position or the momentum.

As I said above, it's a whole different set of rules than Newtonian mechanics. You'll have to learn these rules, and to learn them at a level that allows any sort of quantitative simulation you'll need a real textbook, something like Griffiths that you'd encounter in the second year of college if you're working on a degree in physics.

One last question, if Schrödinger's equation if complex valued, then it would only work in 2 dimensions right? Or do the complex values represent something else?

 

[PeroK]

One last question, if Schrödinger's equation if complex valued, then it would only work in 2 dimensions right? Or do the complex values represent something else?

The wave function is a complex-valued function of the usual coordinates: three spatial and one time. The complex values, at the simplest level, represent a pair of differential equations

 

[Nugatory]

One last question, if Schrödinger's equation if complex valued, then it would only work in 2 dimensions right? Or do the complex values represent something else?

Not right, the complex values are doing something unrelated to the number of dimensions.

Introductory treatments often present the one-dimensional version of the equation (you'll recognize it because the solution $\psi$ is a function of $x$ and $t$ - time and one space coordinate - instead of time and three space coordinates) first because it's simpler to work with and still demonstrates the most important concepts. But you'll fairly quickly be dragged into the three-dimensional version after you've understood the one-dimensional version.

Either way $\psi$ is a complex-valued function. However, any physically meaningful quantity you calculate from $\psi$ will end up using only terms like $\psi\psi^*=|\psi|^2$ so you always end up with sensible real numbers for anything you can measure.

(So you might find yourself wondering why we bother with the complex $\psi$ instead of just working with the real $|\psi|^2$ everywhere. The answer is that in solving just about every QM problem it is necessary to write the wavefunction as a sum of other wave functions, and $\psi=\psi_1+\psi_2$ does not imply that $|\psi|^2$ is equal to $|\psi_1|^2$ plus $|\psi_2|^2$. Thus, we have to do all our calculations with the complex $\psi$, and only square it when we're done.)

 

[PeterDonis]

The wave function is a complex-valued function of the usual coordinates: three spatial and one time.

This is true for a single particle (strictly speaking, for a single particle with zero spin). But for multiple particles, the wave function is no longer a function on ordinary 3-space and time; it's a function on 3N-space and time, where N is the number of particles (again with zero spin). And all this is non-relativistic QM; quantum field theory is something else again.

 

[General Scientist]

This is true for a single particle (strictly speaking, for a single particle with zero spin). But for multiple particles, the wave function is no longer a function on ordinary 3-space and time; it's a function on 3N-space and time, where N is the number of particles (again with zero spin). And all this is non-relativistic QM; quantum field theory is something else again.

How does spin affect the Schrödinger equation?

 

[PeterDonis]

How does spin affect the Schrödinger equation?

If a particle has nonzero spin, its wave function is not simply a function on ordinary 3-space and time; a complete wave function for the particle also has to include the spin degree of freedom.

Strictly speaking, the Schrödinger equation only deals with the part of the wave function that depends on the position of the particle, not spin. Sometimes, however, the behavior of the spin degree of freedom is simple enough that you can approximate it by just adding a term to the Hamiltonian; for example, see here:

https://en.wikipedia.org/wiki/Pauli_equation

 

[PeroK]

If a particle has nonzero spin, its wave function is not simply a function on ordinary 3-space and time; a complete wave function for the particle also has to include the spin degree of freedom.

Strictly speaking, the Schrödinger equation only deals with the part of the wave function that depends on the position of the particle, not spin. Sometimes, however, the behavior of the spin degree of freedom is simple enough that you can approximate it by just adding a term to the Hamiltonian; for example, see here:

https://en.wikipedia.org/wiki/Pauli_equation

That is not a "B" level reference.

 

[eudo]

Perhaps it would help if you explain what your goal is in making this simulation. As others have said, the laws of physics are very different for quantum mechanical objects.

And it's not simply a matter of substituting some different equation for f=ma to find the position. The notions of position, acceleration, and force (in the classical sense) simply don't apply. You calculate waves which tell you the probability of getting a certain outcome if you make a measurement. And the results depend on what quantity you choose to measure.

It's a large and complicated subject and the only way to do it properly is to start at the beginning and learn quantum mechanics, which can't be done in a short forum thread.

But if you tell us what features you're trying to simulate, we might be able to help a little better.

 

[General Scientist]

I am trying to calculate the dynamics of the fundamental particles as they change over time.
There would be a random assortment of particles placed randomly everywhere

 

[General Scientist]

Also, the wave function contains all the information but with ambiguity mixed in, right? Like it will tell you the the probability of a certain velocity, position, momentum, etc, right? So how would you calculate the wave function?

 

[weirdoguy]

You really need to read some proper textbook on the subject, cause it's way too much there to write it on forums...

 

[Vanadium 50]

That is not a "B" level reference.

Agreed, but it is not a B level thing he's attempting.

 

[PeterDonis]

That is not a "B" level reference.

Agreed, but it is not a B level thing he's attempting.

Both of these are valid points. But also there's this:

Also, the wave function contains all the information but with ambiguity mixed in, right? Like it will tell you the the probability of a certain velocity, position, momentum, etc, right? So how would you calculate the wave function?

Which indicates to me that the OP does not have an "A" level background (or even an "I" level background) in the subject. Which in turn indicates to me that trying to simulate the behavior of quantum particles is not something the OP should be attempting with his current background.

@General Scientist , at this point I am closing this thread since you evidently don't have the background for what you are trying to do. Anyone who is attempting to "code a simulator for quantum particles" should already know the answers to the questions in what I quoted from you just above. If you don't, you need the equivalent of several courses in quantum mechanics, which is beyond the scope of these forums.