Time-Dependent Schrödinger Equation

Equivalent forms

i\hbar \partial_t \Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi

A single linear PDE underlies all of non-relativistic quantum mechanics.

Symbol	Name	SI unit	Dimension	Range
$Psi$	Wavefunctionoutput Complex probability amplitude, function of position and time	m^-3/2	L^-3/2	0 – 1
$hbar$	Reduced Planck constant h / (2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$H_hat$	Hamiltonian operator Total energy operator: kinetic + potential	J	ML^2T^-2	0 – 1e-15

Unit systems

Symbol	SI	CGS	Imperial
$Psi$	m^-3/2	cm^-3/2	ft^-3/2
$hbar$	J*s	erg*s	ftlbfs
$H_hat$	J	erg	ft*lbf

Where it holds

Non-relativistic quantum mechanics (v << c). For relativistic systems use Dirac or Klein-Gordon equations. Fails for high-energy pair production.

Dimensional analysis

[M*L^2*T^-1]*[T^-1]*[L^-3/2] \to [M*L^-1/2*T^-2]

[M*L^2*T^-2]*[L^-3/2] \to [M*L^-1/2*T^-2]

Both sides match (energy density^1/2 per unit volume).

\checkmark

Discovery

Erwin Schrödinger · 1926

Inspired by de Broglie's matter waves, Schrödinger wrote down the wave equation governing non-relativistic quantum mechanics.

Try this

What is the 'F = ma' of quantum mechanics?

The wavefunction evolves under this PDE. How does it generalize Newton's second law for quantum systems?

Research status: stable

Real-world applications

Quantum chemistry (DFT, Hartree-Fock)
Semiconductor band structure calculations
Quantum computing simulations
Nuclear reaction modeling

Common misconceptions

Psi is not directly observable — only |Psi|^2 is a probability density
The 'i' is essential — the equation is first-order in time but complex, not second-order real
It is NOT a classical wave equation — dispersion is quadratic, not linear

Experimental verification

Every quantum experiment ever done — atomic spectra, electron diffraction, tunneling, STM, quantum dots, Bose-Einstein condensates. Accuracy: QED extends it to parts per trillion for the electron g-2.

Derivation

Postulate a wave description Psi(x,t) with de Broglie

p =

hbar*k and Einstein

E =

hbar*omega.

A plane wave exp(i(kx - omega*t)) satisfies p -> -i*hbar*d/dx and E -> i*hbar*d/dt when acted upon by these derivative operators.

Substituting into the classical

E = p^2/(2m) + V

gives i*hbar*dPsi/

dt = [

-hbar^

2/(2m)*d^2/dx^2 + V] \Psi = H*\Psi

.

Limiting cases

V = 0

(free particle)⟶ Plane wave solutions exp(i(kx - omega t))With dispersion

\omega =

hbar*k^2/(2m) — quadratic, unlike classical waves.

Stationary state⟶ Psi(x,

t) = \psi (x)*\exp (-i*E*t

/hbar)Time dependence factors out, reducing to the time-independent Schrödinger equation.

Classical limit (hbar

\to 0)

⟶ Recovers Hamilton-Jacobi equationWKB approximation connects quantum amplitudes to classical trajectories.

What if…

What if Psi were real?

The equation would force dPsi/ $dt = 0$ — nothing evolves. Complex amplitudes are essential.

What if H depended on time?

Non-autonomous evolution — solve with time-ordered exponential $U(t) = T*\exp (-i$ *integral(H dt')/hbar).

What if we measure Psi?

Collapse postulate: Psi projects onto an eigenstate of the measured observable (in Copenhagen interpretation).

1

Free particle plane wave

Given ·

V:: 0

Find · dispersion relation

Steps

Assume $\Psi = \exp (i(kx - \omega t))$
LHS: i*hbar* $(-i*\omega )*\Psi =$ hbar*omega*Psi
RHS: -hbar^ $2/(2m)*(-k^2)*\Psi =$ hbar^2*k^2/(2m)*Psi
Equate: $\omega =$ hbar*k^2/(2m)

Result ·

\omega =

hbar*k^2/(2m)

2

Stationary state separation

Given ·

V:: V(x)

Find · time-independent form

Steps

Ansatz: Psi(x, $t) = \psi (x)*T(t)$
Divide both sides by $\psi *T \to i$ *hbar* $T'/T = H*\psi /\psi = E$
$T(t) = \exp (-i*E*t$ /hbar), psi satisfies $H*\psi = E*\psi$

Result ·

H*\psi (x) = E*\psi (x)

Heisenberg Uncertainty Principle→Particle in a 1D Box→Bohr Hydrogen Energy Levels→Dirac Equation→