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I was told that a quantum mechanical system is completely determined by its wave function. But superposition principle says that given two wave functions of some system, a linear combination of them is also a wave function for the same system.

I'm confused, how can it be that the same system have more than one wave function. How to clarify this confusion?

I think that this means that every wave-function corresponds to a particular state of the system, but I don't understand what does that mean? What is the exact meaning of a state of a system in this context? Can you give some examples of states of a system?

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    – SuperCiocia
    Commented Apr 2, 2023 at 7:56
  • $\begingroup$ trivially, and probably not what you asking about, the wave functions $\psi (x)$ and $ e^{i \phi} \psi (x)$ have the same probability density. $\endgroup$
    – jim
    Commented Sep 5, 2023 at 13:15
  • $\begingroup$ I was told that there are 8 planets. But 8=3+5. How can there be two different numbers of planets? $\endgroup$
    – WillO
    Commented May 13 at 6:09

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The superposition principle is not unique to quantum mechanics. Let's look at the wave equation: $$\frac{1}{c^2}\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0.$$ It is also a linear differential equation, so it obeys the superposition principle. This means that if $u_1$ solves the wave equation, and $u_2$ is another function that also solves the wave equation, then $\xi=u_1+u_2$ also solves the wave equation. You can try this for yourself. If we use $\xi$ as our solution, does this mean that $\xi$ has "two states"? No: it is a single state which behaves as both $u_1$ and $u_2$ at the same time. See also this picture below. The bottom wave is a superposition of right- and left travelling waves. But there is no quantum mechanics here: it's just a normal wave!

enter image description here

To reiterate this point more: just because a state is a solution to the system, does not mean the system is in that state. A system has many solutions, each corresponding to a state the system can be in. But at each point in time, the system can only be in one state.

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One example for a quantum mechanical system is "one free electron in 3-dimensional space". The state "the electron is at the point $\vec x_1$" is described by one wave function $\psi_1(\vec x)$, the state "the electron is at the point $\vec x_2$" is described by another wave function $\psi_2(\vec x)$, and a third wave function $\psi_3(\vec x) \sim \psi_1(\vec x) + \psi_2(\vec x)$ describes the superposition of those two states.

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Just considers a one dimensional system, where the Hamiltonian is given by $$H = \frac{-\hbar^2 \partial_x^2}{2m}.$$ You want to solve the time-independent Schrödinger equation $$H \Psi = E \Psi.\tag{1}\label{eq:1}$$ The solution are simply given by plane-wave, i.e. $e^{\pm ikx}$ where $k = \sqrt{2mE}$. So you see that there is two wave function solutions to ($\ref{eq:1}$). But because ($\ref{eq:1}$) is a linear equation the superposition of the two will be also be a solution. So you can write the general solution

$$\Psi = A e^{ ikx} + B e^{ -ikx},$$ where $A$, $B$ will be given by some boundary condition.

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