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Superposition of wave functions

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  • #1
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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?
 
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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 [itex]a_n[/itex] of an operator [itex]\hat{A}[/itex] is given by

[tex]P_n(a_n)=\frac{|\langle \psi_n | \psi \rangle|^2}{\langle \psi | \psi \rangle}[/tex]

where [itex]\psi_n[/itex] is the eigenstate of [itex]\hat{A}[/itex] with eigenvalue a_n.
 

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