PX262 - C8 - postulate 5

the time dependence of the wavefunction

time dependence is given by the schrödinger equation:

\begin{matrix} - \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} ψ + V (\vec{r}, t) ψ = i ℏ \frac{\partial ψ}{\partial t} \\ H ψ = i ℏ \frac{\partial ψ}{\partial t} \end{matrix}

another time dependence is when measurement is done
at the time of the measurement, the wavefunction collapses to one of the eigenfunctions of the corresponding order

postulate 5

between measurements, the development of the wavefunction with time is governed by the time-dependent schrödinger equation

how it looks for a general wavefunction: $ψ (\vec{r}, t) = \sum_{n} a_{n} (t) ϕ_{n} (\vec{r})$

i ℏ \sum_{n} \frac{d a_{n} (t)}{d t} ϕ_{n} (\vec{r}) = \sum_{n} a_{n} (t) \hat{H} ϕ_{n} = \sum_{n} a_{n} (t) E_{n} ϕ_{n} (\vec{r})

where, $ϕ_{n}$ are eigenfunctions of the hamiltonian operator

\sum_{n} (i ℏ \frac{d a_{n}}{d t} - a_{n} E_{n}) ϕ_{n} = 0

to hold in general for all cases:

\begin{aligned} i ℏ \frac{d a_{n}}{d t} - a_{n} E_{n} & = 0 \\ i ℏ \frac{d a_{n}}{d t} & = a_{n} E_{n} \\ a_{n} (t) & = a_{n} (0) \exp (- \frac{i E_{n} t}{ℏ}) \end{aligned}

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