In classical physics, Newton’s laws guide everything. In quantum mechanics, it’s the postulates – they define how systems exist, how we measure them, and how they evolve. Here are the five postulates:
State of the System: At any time t, the quantum system is described by a wave function. It contains all the information about the system.
Observables and Operators: Every measurable quantity (position, momentum, energy, etc.) is represented by a mathematical operator. To every physical observable in classical mechanics there corresponds a Hermitian operator in quantum mechanics.
Measurements and eigenvalues of operators: When we measure an observable, the only possible result is one of the eigenvalues of its operator.
If \( A\psi = a \psi \), then \( a \) is the measurement outcome.
Time Evolution: The time evolution of wave function of a system governed by time-dependent Schrödinger equation:
\[ i\hbar \frac{\partial \psi(x,t)}{\partial t} = H \psi(x,t) \]
The Hamiltonian operator \( H \) corresponding to the total energy of the system
Expectation value: The average value (expectation value) of an observable (M) of a system with a state function is given by
\[ < M > = \frac{\int \psi^*M\psi d\tau}{\int \psi*\psi d\tau} \]