What proof do we have of wave functions?

[Physics Slayer]

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##.
I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why does it work what is the reason, I am not asking where the schrödinger equation comes from rather I am asking why can we even assume that a complex function can describe a system on the smallest scale, maybe my question is nonsensical but I don't understand it.
You might refer me to the double slit wave interference experiment but that's not what I am talking about.

Thanks,

 

[PeroK]

The wave-function of the electron in the hydrogen atom, for example, provides a mathematical model of the atom's energy levels that matches that found by experimental analysis of the spectrum of hydrogen.

Physics can't be about why that works any more than it can say why Newton's laws hold. Any theory, ultimately, has its bedrock laws of nature that are taken as a starting point.

 

[vanhees71]

The problem is that you forbid us to give the answer. It's indeed simply, because the Schrödinger equation works. It has been found based on observed facts of nature together with an ingenious analogy between the relation of "geometrical/ray optics" and "wave optics" through the so-called eikonal approximation of Maxwell's equations for electromagnetic waves and the idea that the classical Newtonian equations of motion should be thought of the eikonal approximation of the to-be-found wave equation. This was triggered by a talk Schrödinger gave at the university of Zürich about de Broglie's wave-particle-duality idea (his PhD thesis), and Debye told him that if you talk about waves you should better have a wave equation that describes them. Famously Schrödinger took up this challenge and discovered "wave mechanics" during his summer vacation with one of his "muses" ;-)).

The first indication that his wave equation (now named after him) should be correct was that he could derive the hydrogen spectrum from it by solving the equation for stationary waves, i.e., where the intensity ##|\psi(t,\vec{x})|^2## is time-independent, which leads to the time-independent Schrödinger equation, i.e., the eigenvalue problem for the Hamilton operator (defining energy eigenstates).

To find a fundamental physical law or even a theory never works via logical deduction as it is presented in textbooks when one has this very theory discovered and finally formulated in a didactic way. It's rather a creative act, comparable to the intuition that leads an artist to create some ingenious piece of art!

 

[f95toli]

Would an STM image help?

https://www.nanoscience.com/techniques/scanning-tunneling-microscopy/

The STM probes the local density of states, and when done around defects you often see standing wave patterns which does imply that -as expected- you need some sort of wave equation to describe what is going on.
That said, I am not sure if that is what you are asking for.

 

[strangerep]

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##.

You've posted this as a B-level thread, but the most reliable response is that our best theoretical description of such things is by quantum fields. "What is a quantum field?", you might ask, but I don't know of a satisfactory B-level answer.

Perhaps I could say that it's an entity with infinite degrees of freedom which conforms with the principles of relativity and operates in Hilbert space. (A-level version: "Unitary Representations" of the Poincare group of special relativity and also the canonical Heisenberg commutation relations.)

[...] I am asking why can we even assume that a complex function can describe a system on the smallest scale, maybe my question is nonsensical but I don't understand it.

Even on the level of wave functions rather than quantum fields, non-trivial systems must be described by building up tensor products of simpler wave functions. But that also more of an I-level answer.

Btw, when asking questions of this kind, it often helps to tell us more of your context by saying which textbooks (or whatever) you've been studying recently.

 

[andresB]

That kind of "why" questions don't really have an answer, IMHO. For example, and using your same format, why can we even assume that a vector field E and a vector field B (or a single tensor field if you prefer relativistic notations) can describe the (classical) electromagnetic field? what other thing beyond that the mathematical framework agrees with experiments and observations allow us to such assumption?

By the way, complex wavefunctions living in Hilbert spaces are not a purely quantum thing either. Classical physics can be described using the same formalism https://arxiv.org/abs/2004.08661. So, if you want you can describe and predict the movement of Neptune using complex wavefunctions and the classical version of the "Schrödinger" equation. Do you find that notion problematic?

 

[Demystifier]

I am asking why can we even assume that a complex function can describe a system on the smallest scale

If it was real instead of complex, would it be less mysterious to you?

 

[f95toli]

This just appeared in my feed and might be relevant

https://physics.aps.org/articles/v15/7

 

[martinbn]

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##.

Is the question about how a single scalar field can be enough? A classical particle is described by just six numbers (the coordinates and the components of the momentum) at a given time. A scalar field consists of infinitely many numbers at a given time, one for each point in space. So a single scalar field is a lot of information. Of course the numbers are not indipendent and arbitrary they still have to make up a function that satisfies the equation.

 

[Physics Slayer]

Would an STM image help?

https://www.nanoscience.com/techniques/scanning-tunneling-microscopy/

The STM probes the local density of states, and when done around defects you often see standing wave patterns which does imply that -as expected- you need some sort of wave equation to describe what is going on.
That said, I am not sure if that is what you are asking for.

The father's of QM didn't have those images when they first started to formulate Quantum mechanics using the wave function.

That kind of "why" questions don't really have an answer, IMHO. For example, and using your same format, why can we even assume that a vector field E and a vector field B (or a single tensor field if you prefer relativistic notations) can describe the (classical) electromagnetic field?

There was good motivation for using the ##E## and ##B## field.(there probably was even for the wave function but i just don't see it)

If it was real instead of complex, would it be less mysterious to you?

Aren't imaginary numbers just a misnomer
So the complex nature of the wave function doesn't bother me too much(but should it?)

Is the question about how a single scalar field can be enough?

No not really, I just don't see how one can assume a quantum system can be fully described by a wave function, if one considers a classical system you need more than one function to describe a system completely.

 

[Nugatory]

No not really, I just don't see how one can assume a quantum system can be fully described by a wave function

That's not something that we assume. It's something that we accept because so far we haven't encountered any properties of any quantum system that are not described by the wave function and that's exactly what we'd expect of a full description.

Of course there is always the possibility that some new experiment/observation will expose some property of a quantum system that is not adequately described by the wave function. Unless and until that happens, however, the discussion is somewhat sterile.

 

[martinbn]

No not really, I just don't see how one can assume a quantum system can be fully described by a wave function, if one considers a classical system you need more than one function to describe a system completely.

Which classical systems do you have in mind?

 

[Physics Slayer]

Which classical systems do you have in mind?

To describe say a moving wheel we need two vectors one for it's linear velocity and one for it's angular velocity, further you would also need the moment of inertia to describe it's mass distribution, so we already have three independent functions that we need to describe the motion of the wheel completely, but that isn't the case with say a particle trapped in a potential well or even a free particle. They can all be described using a wave function alone.

If I assume de broglie's equation to be true(that particles show wave like behaviour), then I have already accepted there must be a function that describes that wave, is this correct?

 

[hilbert2]

I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why does it work what is the reason, I am not asking where the schrödinger equation comes from rather I am asking why can we even assume that a complex function can describe a system on the smallest scale, maybe my question is nonsensical but I don't understand it.
You might refer me to the double slit wave interference experiment but that's not what I am talking about.

If you start asking questions like "why does gravity and electromagnetism exist?", you end up with an infinite causal chain just like with the question "what was the cause of the Big Bang". There isn't anything that would count as a proper "root cause" when you ask the question in that way. Ultimately, we can't really even know whether anything really exists, we could just as well be held in "The Matrix", producing a simulated reality that looks real. So, the only way to test whether the wave function is a "real" object is to test whether the Schrödinger equation predicts experimental results correctly.

 

[Demystifier]

To describe say a moving wheel we need two vectors one for it's linear velocity and one for it's angular velocity, further you would also need the moment of inertia to describe it's mass distribution, so we already have three independent functions that we need to describe the motion of the wheel completely, but that isn't the case with say a particle trapped in a potential well or even a free particle. They can all be described using a wave function alone.

Are you familiar with the Hamilton-Jacobi formulation of classical mechanics? With this formalism, one describes all degrees of freedom of a classical system with one function. The quantum wave function is very similar to classical Hamilton-Jacobi function.

More generally, if you have ##N## degrees of freedom described by "coordinates" ##x_1,...,x_N##, you can describe them with one function ##F(x_1, ...,x_N)##. For example, if those coordinates have definite values ##x_1=a_1##, ..., ##x_N=a_N##, you can describe it with one ##N##-dimensional ##\delta##-function
$$ F(x_1, ...,x_N)=\delta^N(x-a)\equiv \delta(x_1-a_1)\cdots\delta(x_N-a_N)$$

 

[martinbn]

To describe say a moving wheel we need two vectors one for it's linear velocity and one for it's angular velocity, further you would also need the moment of inertia to describe it's mass distribution, so we already have three independent functions that we need to describe the motion of the wheel completely, but that isn't the case with say a particle trapped in a potential well or even a free particle. They can all be described using a wave function alone.

If I assume de broglie's equation to be true(that particles show wave like behaviour), then I have already accepted there must be a function that describes that wave, is this correct?

All these are functions of time, for example angular velosity ##\varphi = \varphi(t)##. I gave you the example of a classical particle, it is similar here. In QM the wave function is a function of position and time ##\psi = \psi (x,y,z,t)##. Which you can view as infinitely many functions of time, each labeled by the coordinates of a point. In other words you have all these ##\psi_{x,y,z} (t)## functions.

 

[vanhees71]

That's not something that we assume. It's something that we accept because so far we haven't encountered any properties of any quantum system that are not described by the wave function and that's exactly what we'd expect of a full description.

While I agree with the statement that we believe in the completeness of QT (within its realm of validity, which is huge since QT describes everything except the gravitational interaction), I don't agree with the statements that it's always "the wave function" that provides a full description. All of relativistic QFT, and that's the only successful formulation of a relativistic QT we know today, cannot be described by associating a single wave function to a quantum system as is the case in non-relativistic QT as long as you deal with systems with a fixed number of particles. In the relativistic realm the latter constraint is usually not or only approximately fulfilled since at large enough interaction energies there's always some probability to create new particles and or destroy the particles you start with. That's why you need a many-body approach for varying numbers of particles, and the most convenient concept is the description in terms of a QFT.