The time-dependent Schrödinger equation

The central equation

In nonrelativistic quantum mechanics, the state of a particle in one dimension is described by a wavefunction $psi(x,t)$. The wavefunction is generally complex and changes in time according to the time-dependent Schrödinger equation:

ihbarrac{partial psi}{partial t}=hat{H}psi

Here $hbar=h/(2pi)$, $i$ is the imaginary unit, and $hat{H}$ is the Hamiltonian operator, representing total energy.

For a particle of mass $m$ in potential energy $V(x,t)$, the Hamiltonian is

hat{H}=-rac{hbar^2}{2m}rac{partial^2}{partial x^2}+V(x,t)

so the equation becomes

ihbarrac{partial psi}{partial t}= -rac{hbar^2}{2m}rac{partial^2psi}{partial x^2}+V(x,t)psi

Why an equation for waves?

de Broglie suggested that matter has wave properties. Schrödinger found an equation whose solutions behave like matter waves and reproduce known energy quantization. The equation is linear, which means superpositions of solutions are also solutions.

Linearity is essential for interference. If $psi_1$ and $psi_2$ are possible states, then

$psi=apsi_1+bpsi_2$

is also a possible state, for complex constants $a$ and $b$.

Free particle solutions

For a free particle with $V=0$, plane wave solutions have the form

$psi(x,t)=Ae^{i(kx-omega t)}$

The de Broglie and Planck relations are

$p=hbar k$

and

$E=hbaromega$

Substituting a plane wave into the free-particle Schrödinger equation gives

E=rac{p^2}{2m}

which is the nonrelativistic kinetic energy.

Operators and energy

In wave mechanics, momentum and energy are represented by differential operators:

hat{p}=-ihbarrac{partial}{partial x}

hat{E}=ihbarrac{partial}{partial t}

The Hamiltonian operator gives the energy associated with the state. The Schrödinger equation can be read as a statement that the energy operator acting on the wavefunction equals the Hamiltonian acting on it.

Probability conservation

The Schrödinger equation is designed so that total probability is conserved. If the wavefunction is normalized at one time, it remains normalized under time evolution for a suitable Hermitian Hamiltonian.

This is crucial because $|psi(x,t)|^2$ is interpreted as probability density.

Stationary states

If the potential does not depend on time, solutions can often be separated as

$psi(x,t)=phi(x)e^{-iEt/hbar}$

Substitution leads to the time-independent Schrödinger equation:

$hat{H}phi=Ephi$

Stationary states have definite energy. Their probability density is time-independent because the phase factor cancels in $|psi|^2$.

The big idea

The time-dependent Schrödinger equation is the dynamical law of nonrelativistic quantum mechanics. It replaces Newton's trajectory equation with wavefunction evolution. The Hamiltonian operator determines how the quantum state changes, while superposition, complex phases, and probability conservation make quantum behavior fundamentally wave-like.