Physical interpretation of the wave function

We have assumed that the wave associated with a particle in motion is represented by a complex variable quantity called the wave function $\psi(x,t)$. Therefore, it can not have a direct physical meaning. Since it is a complex quantity, it may be expressed as

$\psi(x,y,z,t)=a+ib \qquad(1)$

Where $a$ and $b$ are real functions of the variable $(x,y,z,t)$. The complex conjugate of wave function $\psi(x,y,z,t)$

$\psi^{*}(x,y,z,t)=a-ib \qquad(2)$

Multiply equation $(1)$ and equation $(2)$

$\psi(x,y,z,t).\psi^{*}(x,y,z,t)=a^{2}+b^{2} \qquad(3)$

$ \left| \psi(x,y,z,t) \right|^{2}=a^{2}+b^{2} \qquad(4)$

If $\psi \neq 0$ Then the product of $\psi$ and $\psi^{*}$ is real and positive. Its positive square root is denoted by $\left|\psi(x,y,z,t) \right|$, and it is called the modulus of $\psi$.

The quantity $ \left| \psi(x,y,z,t) \right|^{2}$ is called the probability density $(P)$. So for the motion of a particle, the probability of finding the particle in the region $d\tau$ will be:

$\int {P d\tau}= \int {\psi(x,y,z,t).\psi^{*}(x,y,z,t).d\tau}=\int {\left| \psi(x,y,z,t) \right|^{2}d\tau}$

Here $P$ are the probability that tells us that the particle will be found in a volume element $d\tau(=dx.dy.dz)$ surrounding the point at position $(x,y,z)$ at time $t$.

For the motion of a particle in one dimension, the probability of finding the particle in the region $dx$ will be:

$\int{P dx}= \int {\psi(x,t).\psi^{*}(x,t).dx}=\int {\left| \psi(x,t) \right|^{2}dx}$