Superposition of wave functions

[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?

 

[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 [itex]|\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 $a_n$ 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.