- For a system consisting of particles moving in a field of a conservative force, there is an associated complex wave function $\psi(x, y, z, t)$ where $x$,$y$,$z$ space coordinates, and $t$ is the time. This wave function enables us to obtain a description of the behavior of the system, consistent with the principle of uncertainty.
- There is an operator with every observable dynamical quantity. The operator corresponding to the pertinent dynamical quantities is:-

Dynamical Variable Symbol Quantum Mechanical Operator Position $x$

$y$

$z$$x$

$y$

$z$Momentum $P_{x}$

$P_{y}$

$P_{z}$

Generalised Form $\overrightarrow{P}$$\frac{\hbar}{i}\frac{\partial}{\partial x}$

$\frac{\hbar}{i}\frac{\partial}{\partial y}$

$\frac{\hbar}{i}\frac{\partial}{\partial z}$

Generalised Form $\frac{\hbar}{i}\overrightarrow{\nabla}$Total Energy $E$ $i\hbar \frac{\partial}{\partial t}$ Total Energy $E$ $-\frac{\hbar^{2}}{2m} \nabla^{2}+V(x)$

This is also known as Hamiltonian Operator $H$.Kinetic Energy $K$ $-\frac{\hbar^{2}}{2m} \nabla^{2}$ Potential Energy $V(x,y,z)$ $V(x,y,z)$ All the operators have eigen functions and eigen values. - The wave function $\psi(x,y,z,t)$ and its partial derivatives $\frac{\partial \psi}{\partial x}$, $\frac{\partial \psi}{\partial y}$, $\frac{\partial \psi}{\partial z}$ must be finite, continuous and single-valued for all values of $x$, $y$, $z$ and $t$
- The product of $\psi(x,y,z,t)$ and $\psi^{*}(x,y,z,t)$ is always a real quantity. The product is called the probability density and $\psi\psi^{*} d\tau $ is interpreted as a probability that the particle will be found in volume element $d\tau$ at $x$,$y$,$z$ and time $t$. Since the total probability of finding particles somewhere in the entire space must be equal to 1.
$\int_{-\infty}^{+\infty} \psi \: \psi^{*}\: d\tau = 1$The integral is taken overall space.
- The average or expectation value of an observable quantity $\alpha$ with which an operator $\hat{\alpha}$ is associated is defined by

$\left< \alpha \right> = \int_{-\infty}^{+\infty} \psi^{*} \hat{\alpha} \psi d\tau$