# Superposition of wave functions

1. Sep 6, 2010

### Gear300

If 2 particles have wave functions w1 and w2, in which W = w1 + w2 is a superposition of the wave functions, then would the probability density of W correspond to the probability of finding both particles at the same position within some interval of space?

Last edited: Sep 6, 2010
2. Sep 7, 2010

### Mindscrape

The short answer is yes. One of the postulates of quantum mechanics is

"The state of any physical system is specified, at each time t, by a state vector $|\psi(t) \rangle[/tex] in a Hilbert space, [itex]|\psi(t) \rangle[/tex] contains all the needed information about the system. Any superposition of state vectors is also a state vector." In fact, even further, that is exactly how finding the probability of a state works for discrete spectra. For nondegenerate discrete eigenvalues the probability of obtaining one of the eigenvalues [itex]a_n$ of an operator $\hat{A}$ is given by

$$P_n(a_n)=\frac{|\langle \psi_n | \psi \rangle|^2}{\langle \psi | \psi \rangle}$$

where $\psi_n$ is the eigenstate of $\hat{A}$ with eigenvalue a_n.