Schrodinger equation

The Schrödinger equation is a fundamental equation in Quantum Mechanics that describes how the quantum state of a system changes over time.

See the derivation of Schrodinger equation from the postulates of QM.

Time-dependent Schrödinger equation (TDSE):

\begin{matrix} (1) & i ℏ \frac{\partial}{\partial t} | ψ (t) ⟩ = \hat{H} | ψ (t) ⟩ \end{matrix}

Where:

$i$ is the imaginary unit.
$ℏ$ is the reduced Planck constant, equal to the Planck constant $h$ divided by $2 π$ .
$\frac{\partial}{\partial t}$ is the partial derivative with respect to time.
$| ψ (t) ⟩$ is the state vector (or wave function) of the system, which is a function of time.
$\hat{H}$ is the Hamiltonian operator, which represents the total energy of the system.

Time-independent Schrödinger equation (TISE):

When the Hamiltonian operator does not explicitly depend on time, we can consider the time-independent Schrödinger equation:

\begin{matrix} (2) & \hat{H} ψ (x) = E ψ (x) \end{matrix}

Where:

$ψ (x)$ is the spatial part of the wave function (state vector), which does not depend on time.
$E$ are the energy eigenvalues of the system.

The function $ψ (x)$ is called a stationary state or an eigenstate of the Hamiltonian $\hat{H}$ . A stationary state is a state in which the probability distribution of the particle's position (or other observables) does not change with time. This does not mean that the wavefunction, which is indeed

| ψ (t) ⟩ = e^{- i E t / ℏ} ψ (x),

doesn't change with time, but rather than its absolute square, $| ψ (x) |^{2}$ , which gives the probability density of finding the particle at position $x$ , remains unchanged.

In quantum mechanics, any state $| ψ (t) ⟩$ can be expressed as a superposition of these stationary states (eigenstates of the Hamiltonian), which is a direct consequence of the linearity of the Schrödinger equation.

Relation between TDSE and TISE:
Suppose we have a solution $ψ_{E} (x)$ for equation (2) for every $E \in R$ . Suppose also that the set ${ψ_{E}}_{E \in R}$ spans the space of function with which we are dealing. Then, for every $t$ :

| ψ (t) ⟩ = \int c^{E} (t) ψ_{E} (x) d E

Substituting into (1):

\int c_{E}^{'} (t) ψ_{E} (x) d E = \int c^{E} (t) E ψ_{E} (x) d E

and therefore it should be:

(c^{E})^{'} (t) = c^{E} (t) E .

So $c_{E} (t) = K_{E} e^{E t}$ , with $K_{E}$ constants to be determined by the initial data. For example, suppose we are given

| ψ (0) ⟩ = f (x) = \int c_{0}^{E} ψ_{E} (x) d E .

Then $c_{0}^{E} = c_{E} (0) = K_{E} e^{E 0} = K_{E}$ , and finally

| ψ (t) ⟩ = \int c^{E} (t) ψ_{E} (x) d E = \int c_{0}^{E} e^{E t} ψ_{E} (x) d E

Related: Heisenberg vs Schrodinger picture
Related: Fourier transform
Related: To introduce electromagnetic field interaction, we must go into Dirac equation.