[Physics] mod squared of wave function probability density

quantum mechanicswavefunction

Okay so this has been bugging me for a while now because everyone I talked to so far didn't know the answer. I'm an A level student but I got quite interested in quantum physics and so I decided to buy Griffiths introductory QM book and just try to understand the maths and how things fit together (I'm only on the first few pages, maybe 24 I think). So far the only thing I didn't understand was that why the mod squared of the wave function is the probability density. I mean I understand that it has to be real but then why don't we square root it afterwards? Is it something that just works or there is an actual reason behind the whole thing?

The main reason is that the probability density, constructed as you indicated, leads to a CONSERVED overall probability over all space.

In most QM books it is shown that the modulus square of the wave function, integrated over all space, does not depend on time. It can then be normalized to 1. The derivation of the conservation of the probability over all space being conserved is from the Schrodinger equation. I've not used Griffith but if it's a standard book it should have that derivation. The 'trick' is to find a current j in terms of the wave function and its complex conjugate such that the divergence of j plus the partial of the probability density with time is zero. By Gauss's theorem then the derivative wrt time of the integral of the density over space is zero.

You can then use the prob. density to get the average (I.e., expectation) values of dynamical valuable, or observables, like position and momentum, and get the Newton equations for those expectation values, and appropriate conserved quantities.

You can in more advanced books get the current density j and probability density from the Lagrangian for the quantum theory (Schrodinger or later relativistic quantum field theories) In a continuity of probability equation that conserves the probability integrated over the appropriate space.

There are also hysical reasons such as the similarity with the energy density in electromagnetism. It is also the the wave variables, E and B, squared, and then the energy is conserved when integrated over space.

The square root of the modulus square has been shown to not obey a conservation equation, so that was ruled out.

You can see a little of the math and essentially the same answer, stated with some differences, at Born Interpretation of Wave Function