Probability density from Wave Function

[Mandelbroth]

A friend of mine recently tried to tell me that the square of the wave function for a particle (that is, $\Psi^2$) gives the probability density of finding a particle in space.

I disagree. I always thought that the wave function multiplied by its complex conjugate (that is, $\Psi \Psi^*$) yielded the probability density for the particle. They are definitely not the same, because $\forall a,b \neq 0, \ (a+bi)^2 = a^2 + 2abi + b^2 \neq a^2 + b^2$.

So, is the probability density given by $\Psi^2$ or $\Psi \Psi^* = |\Psi|^2$?

 

[cattlecattle]

A friend of mine recently tried to tell me that the square of the wave function for a particle (that is, $\Psi^2$) gives the probability density of finding a particle in space.

I disagree. I always thought that the wave function multiplied by its complex conjugate (that is, $\Psi \Psi^*$) yielded the probability density for the particle. They are definitely not the same, because $\forall a,b \neq 0, \ (a+bi)^2 = a^2 + 2abi + b^2 \neq a^2 + b^2$.

So, is the probability density given by $\Psi^2$ or $\Psi \Psi^* = |\Psi|^2$?

It's the $|\Psi|^2$, a real number

 

[Jano L.]

Perhaps your friend referred to wave function that is eigenstate of some atomic or molecular Hamiltonian. These can be chosen to be real, so then $\Psi^2$ gives the correct density as well as $|\Psi|^2$. But the latter is the general expression.