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The relationship between quantum mechanics and operators has helped to present the concepts in the form of some postulates. These Postulates are:
1. The state of a system is described by a wave function \({ \Psi (x, y, z, t) }\).
$$ \widehat { A } \Psi = a\Psi $$
2. Every Observable physical property of a system can be characterised by a linear operator. e.g. Position (A) is characterised by operator \(\widehat { A }\).
3. The possible value of any physical quantity of a system are given by Eigen value (a) in the operator equation.
4. When an operator \(\widehat { A }\) is operated on a function \({ \Psi }\), an average result obtained is given by,
$$ \left< A \right> =\frac { \int _{ -\infty }^{ +\infty }{ { \Psi }^{ * }.A\Psi .d\tau } }{ \int _{ -\infty }^{ +\infty }{ { \Psi }^{ * }.\Psi .d\tau } } $$
5. As all the wave functions are time dependent. i.e. \({ \Psi (x, y, z, t) }\) then its subsequent behaviour should be described by time-dependent Schrödinger equation. Which is,
$$ -\frac { ih }{ 2\pi } .\frac { \partial \Psi \left( x, y, z, t \right) }{ d\tau } = \widehat { H } \Psi \left( x, y, z, t \right) $$
Where \(\widehat { H }\) is known as Hamiltonian Operator of the system.
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