## What is the wave function?

For reasons I can’t recall I was rereading my quantum text last night (Ballentine). After presenting the coordinate representation of Schrodinger’s wave equation (eq. 4.4) (presuming the absence of a magnetic field):

$\left [\frac{-\hbar^2}{2M}\nabla^2+W(\vec{x})\right ]\Psi(\vec{x},t)=i\hbar\frac{\partial}{\partial t}\Psi(\vec{x},t)$

Ballentine writes the following:

“Because (4.4) has the mathematical form of a wave equation, it is very tempting to interpret the wave equation $\Psi(\vec{x},t)$ as a physical field or “wave”…Moreover, it may seem plausible to assume that a wave field is associated with a particle, and even that a particle may be identified with a wave packet solution of (4.4). To forestall such misinterpretations…”

Whoops. I’ve been finding out that I have a lot of such misconceptions about quantum mechanics (and much else) and perhaps this is another one. Ballentine’s basic argument is that if you look at a system of N particles you obtain a single wave function describing the entire system, not N separate wave functions. The confusion, he claims, arises because for a single particle we can identify the three spatial dimensions the particle moves in with the three dimensional configuration space its ‘trajectory’ lies in. However for a system of N particles, those particles still ‘move’ in three spatial dimensions but the system’s ‘trajectory’ lies in a 3N dimensional configuration space.

This suggests to me that when we are learning or teaching quantum we should be careful. Its common to ‘sketch’ wave functions for messy potentials in 1D to build some intuition. We can relate aspects of the wave function (its wavelength) to the energy and momentum of the particle for example. But would the wavelength of the wave function of the N particle system mean? I’m not sure at the moment.

Something to ponder (and hopefully resolve else this blog becomes little but a list of my own shortcomings)

1. A further note: Upon discussing this with some friends I have some new bits to add. Clearly $\langle x\mid\psi\rangle$ and $\langle p\mid\psi\rangle$ have different units (justified via the normalization argument). Where do those units come from? They must come from the momentum and position kets/bras. There is a comment on a previous post which has made me wonder if the units of the bra differ from that of the ket. I’m not sure, but I don’t think so. I can’t see that the adjoint of the Schrodinger equation gives us any different information. That’s not justification for having the same units however.