What are wave-functions in QM. corresponding to?

I was learning about a particle trapped in a double well potential

$$V(0) = \infty, V(x1)=\infty$$

which can be described by $$\psi_n$$ for n=0,1,...,$$\infty$$ with corresponding Energies $$E_n$$.

Just conceptually I understand that in qunatum mechanics a particle can be described by a wave function $$\psi$$. What I am not so sure about is how $$\psi_1$$ and $$\psi_2$$ are different from each other.

I know that they'll have different energies and a different shape. But I can't really think of an intuitive classical analog to help me understand this better.

Any ideas?

The wave function ψ is related to probability since P (x ∈ (0, x1)): = ∫ |ψ|² dx; x ∈ (0, x1), where P (x ∈ I) is interpreted as the probability of finding the particle in the interval I. note that the variation of this function |ψ|² with the x-axis, indicating the most likely places.

note that in trying to solve the Schrödinger's EDP we get a family of solutions that depend on an n; n ∈ N.

This phenomenon is interpreted as a quantization of energy, in the sense of having discrete amounts of energy.

Therefore, the differences in ψ_n are related to different distributions of probability (I am not entering into the merit of who implies who).

ψ for physicists is related to something that they call "probability amplitude", this ψ function, does not assume only real values, but also assumes complex values.

what it means? is a reason for discussion in many books... but the main point is its relation to probability (described above). for probability we can speak of many analogues, but for the amplitude of probability, I can't!

you can think of ψ as a mathematical device to generate the "right" probability distribution function.

You can think of the different wavefunctions as different "patterns of motion". It is true that the wavefunctions (the eigenfunctions) you are referring to are stationary in time. However, it is possible to calculate the velocity at specific points in space by taking derivatives of the wavefunction. This velocity very roughly correponds to something like "if the particle were in this location in space it would be moving with this velocity". Of course, we must think of the particle position as being smeared out across all space so we must also think of there being a velocity spread for the particle.

Think of the patterns of motion as similar to simple harmonic motion of a harmonic oscillator. The oscillator will exhibit periodic motion with a certain amplitude $$A$$. however, if you give the system more energy (pull on the mass and then release) it can now oscillate at a new amplitude $$2A$$. In both cases the oscillator exhibits a "pattern of motion".

The quantum mechanical case is similar. For the square well you mention you will that $$\psi_1$$ does not achieve the same maximum velocities as $$\psi_2$$. This can be seen mathematically by noticing that $$\psi_2$$ has a higher frequency so it has larger derivatives or it can be understood physically be realizing that since $$\psi_2$$ is higher energy we expect higher velocity components.

The square is nice because the energy is directly related to the spatial derivative of the wavefunction (the momentum of the particle) and there is no dependence on the position of the particle. When you introduce a potential you will see that the "patterns of motion" now have to take into account both position and momentum. For example, you'll find that the momentum (spatial derivative of the wavefunction) is low where the potential energy is high and vice versa. This of course what you also find classically for a harmonic oscillator, for example.

You cannot. The product of the wave function and its complex conjugate gives a real value which is the probability. There is no "physical analogue" of a wave function and no way to understand this intuitively