Wave Functions

Wave functions can be functions of many different physical parameters of the system of interest. For example, one can define a wave function in coordinate space, in momentum space, in spin space, and so on as long as the unit vectors of the space are orthogonal. For a single particle, the wave function ψ(x₁,y₁,z₁) is the QM amplitude for finding the particle at the three dimensional configuration space point (x₁,y₁,z₁), which directly corresponds one-to-one to position space coordinates x₁, y₁, and z₁ for this one-particle system. For the two-particle system, the wave function ψ(x₁,y₁,z₁; x₂,y₂,z₂) defines a six-dimensional configuration space. Is there a direct correspondence to three-dimensional position space coordinates for this two-particle wave function as well? What about the multiparticle wave function?

Answer

No. Beyond three dimensions there is no direct one-to-one correspondence between many-dimensional configuration space coordinates and the three dimensional coordinates of position space.

The misconception referred to here shows up in discussing the wave function for two-particle systems, especially when the discussion refers to the two-particle wave function reducing to the classical result. One often encounters questions about how the wave function can reduce instantaneously to the result, as if there has been some faster-than-light information transfer. Fortunately, the two-particle wave function reduces in configuration space, not in position space!