How to correctly imagine a quantum particle, what does it look like?

Question

I am a student, besides, only in the first year, so I apologize if the question seems stupid. How can you imagine a quantum particle, like an electron? This is a microscopic "ball" with a certain size, which at every moment of time has a certain coordinate, but instantly teleports or changes direction and speed, so we have an undefined impulse? Or is the electron what our devices register, and "what does it look like" is generally a meaningless question? How to understand all this correctly?

Answer 1

It is tempting to think of an electron as some ball-like¹ particle with a definite position and momentum. Unfortunately, such an analogy does not work in quantum mechanics.

In QM, an electron is described by a wave function which is usually denoted by the Greek letter Psi: $\Psi$ or $\psi$. What this means is that the electron is not a particle, but a wave which can be very confusing first. The wave function can be used² to obtain the probability distribution for this particle, i.e. where in space it is more likely to be found when measuring its position.

So how does the electron "look" like? There is not an actual answer to this, I think. One could however visualize the wavefunction (or rather, its probability distribution) in a way that one traces out a shape such that the probability of finding the electron is for example 90%. This is often done for atomic orbitals³:

(Source)

One might now notice that I talked about the probaility of finding the electron somewhere. This seems to indicate that the electron, contrary to the above explanation, does have some specific position and momentum. But this is not the case, except for one measures the position. Such a measurement causes what is called a wave function collapse. One could say that interacting with the electron "forces" it to take on some position. But before that, the electron does not have such a position; it is in a superposition of all possible states. In some sense, one could say that the electron is at every possible position at the same time.

So to conclude, in QM an electron is described by a wave function and does not have a well-defined position until one measures it. One can use the wave function to obtain the probability distribution for the electron. It should also be noted that the above image is only a representation and should not be taken literally, since the question of how an electron actually looks like cannot be answered.

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¹ However, to be precise, even in non-QM an electron is described to have no spatial dimension, i.e. to be point-like
² One can obtain the probability distribution by the squared modulus of the wavefunction, $|\psi|^2$
³ If you are interested, I wrote a bit more in-depth about atomic orbitals in my answer to What do atomic orbitals represent in quantum mechanics?

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Just a quick side note: As a highschool student, all I know about QM is self-taught. If someone catches a mistake, I would be glad if they pointed it out so I can correct it.

Answer 2

Edit: I know there are problems with this answer, I'm hoping it's helpful still but I may delete

Besides a purely mathematical description, any further description will be an interpretation of quantum mechanics and/or of what the wave function represents.

If you go with the standard textbook, Copenhagen-family of descriptions, the wave function is epistemic. This means any quantum object's state can't be thought of as 'real' (as in real in the world, i.e. realism) prior to measurement. Only measurement outcomes are real. So no, the electron does not have a specific coordinate at every moment. It is in a state at every time, but a state in quantum mechanics "must be specified by fewer or more indefinite data than a complete set of numerical values for all the coordinates and velocities of some instant of time." And in QM, these states are such that "whenever a system is definitely in one state we can consider it as being partly in each of two or more other states". And "conversely any two or more states may be superposed to give a new state". The Principles of Quantum Mechanics 4th Ed. pp 11-12

Such states do not exist classically. The wavefunction gives you the state(s) the object is in.

One heuristic you hear is "quantum systems evolve as waves and are observed as particles". But there is no perfect picture. And even these waves are not entirely classical. Any picture assumes some interpretation. When you do look at a photon detector, you see photons (or evidence from them). They register as points. What you can say prior to measurement is much less "pictureable".

Ultimately the math does capture the observations. So you can always math it out. But what that math represents is not classical and is unclear. You can't form a classical picture. The particles never teleport instantaneously too.