# Can a quantum mechanical system have more than one wave-function?

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?

• A wave function is like a vector. Vectors can be made from other vectors by linear superposition and they can be decomposed into other vectors. That doesn't mean that any given vector isn't unique, it only means that it can be decomposed into infinitely many different combinations of other vectors and no combination is any more special than any other. You are really not doing anything else here than in linear algebra, except, maybe, that the basis can be infinite dimensional, but that's mostly a problem for the mathematicians. – CuriousOne Jun 16 '16 at 8:20

## 1 Answer

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.