Postulate 4: Quantum Mechanics

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The time development of the state functions of an isolated quantum system is governed by the time-dependent Schrödinger equation

\[ \hat {H} (r , t) \psi (r , t) = i \hbar \dfrac {\partial}{\partial t} \Psi (r , t ) \tag {4.1}\]

where \(H=T+V\) is the Hamiltonian of the system.

Discussion

The time evolution or time dependence of a state is found by solving the time-dependent Schrödinger equation (Eq. 4.1). For the case where \(\hat{H}\) is independent of time, the time dependent part of the wavefunction is \(e^{-i\omega t}\) where \(\omega = \frac {E}{ħ}\) or equivalently \(\nu = \frac {E}{h}\), which shows that the energy-frequency relation used by Planck, Einstein, and Bohr results from the time-dependent Schrödinger equation. This oscillatory time dependence of the probability amplitude does not affect the probability density or the observable properties because in the calculation of these quantities, the imaginary part cancels in multiplication by the complex conjugate.