Schrödinger Equation (Time-Independent)

Equivalent forms

\hat{H}\psi = E\psi

A single linear PDE describes atoms, molecules, semiconductors, and chemistry itself.

Symbol	Name	SI unit	Dimension	Range
$E$	Energy eigenvalueoutput Allowed total energy of the quantum state	J	M·L²·T⁻²	-1e-17 – 1e-17
$m$	Particle mass Mass of the quantum particle	kg	M	1e-31 – 1e-25
$V$	Potential energy External potential energy field	J	M·L²·T⁻²	-1e-17 – 1e-17

Unit systems

Symbol	SI	CGS	Imperial
$E$	J	erg	eV
$m$	kg	g	lb
$V$	J	erg	eV

Where it holds

Non-relativistic single-particle quantum mechanics. For v approaching c, replace with the Dirac or Klein–Gordon equation. Many-body systems require the full N-particle Schrödinger equation in 3N-dimensional configuration space.

Dimensional analysis

$[\hbar ^{2}/(2m)\cdot \nabla ^{2}\psi ] = (M\cdot L^{2}\cdot T^{-1})^{2}/(M\cdot L^{2}) \cdot \psi = M\cdot L^{2}\cdot T^{-2}\cdot \psi = [E\cdot \psi ] \checkmark$

Discovery

Erwin Schrödinger · 1926

Schrödinger wrote down his wave equation during a holiday in the Alps, founding wave mechanics.

Try this

What equation governs the ghostly wavefunction of every electron in the universe?

Find the ground-state energy of an electron trapped in a 1 nm infinite square well.

Research status: stable

Real-world applications

All of computational chemistry (DFT, Hartree–Fock, coupled cluster).
Semiconductor band structure and transistor design.
Quantum dots, lasers, LEDs, and photovoltaic devices.
MRI and NMR rely on quantum spin states governed by Ĥ.

Common misconceptions

$\psi$ itself is not directly observable — only | $\psi |^{2}$ gives probability density.
The equation is deterministic; randomness enters only at measurement (Born rule).
It is non-relativistic — invalid for fast electrons in heavy atoms; use Dirac equation there.

Experimental verification

The hydrogen spectrum, predicted exactly by solving the Schrödinger equation in a Coulomb potential, matches observation

to \sim 5

decimal places (with QED corrections beyond). Quantum tunneling rates

in \alpha

-decay (Gamow), STM, and Josephson junctions all confirm the equation's predictions.

Derivation

Starting from de Broglie's matter waves and demanding linearity + plane-wave solutions

e^{i(kx - \omega t)}

that respect

E = \hbar \omega and p = \hbar k

, one constructs an operator equation

i\hbar \partial \psi /\partial t

= Ĥ

\psi

with Ĥ

= -\hbar ^{2}\nabla ^{2}/(2m) + V

.

Separating variables

\psi (x

,

t) = \psi (x)e^{-iEt/\hbar }

yields the time-independent form Ĥ

\psi = E\psi

.

Limiting cases

V = 0

(free particle)⟶

\psi \propto e^{ikx}

,

E = \hbar ^{2}k^{2}/(2m)

Plane-wave solutions form a continuum of energies — no quantization without confinement.

V \to \infty

outside region⟶ Discrete bound statesConfinement quantizes the energy spectrum (e.g., particle in a box:

E_n = n^{2}\pi ^{2}\hbar ^{2}/(2mL^{2}))

.

\hbar \to 0

⟶ Classical Hamilton–Jacobi equationQuantum mechanics smoothly reduces to classical mechanics in the small-

\hbar

limit (WKB approximation).

What if…

What if the well were

10\times

wider?

$E_1 \propto 1/L^{2}$ → energy drops by $100\times$ . Larger boxes (quantum dots) emit redder light — basis of size-tuned LEDs.

What if V were a delta function attractive potential?

Exactly one bound state exists in 1D, with $E = -m\alpha ^{2}/(2\hbar ^{2})$ . Used as a toy model for impurity states in solids.

What if we used the Schrödinger equation for a relativistic electron in hydrogen?

Predictions miss fine structure $(\sim 10^{-4}$ corrections), Lamb shift, and antiparticles. Dirac's equation fixes these.

1

Electron in a 1 nm infinite well

Given ·

L:: 1e-9
m e:: 9.109e-31
n:: 1

Find · ground-state energy E_1

Steps

Solution to TISE in box: $E_n = n^{2}\pi ^{2}\hbar ^{2}/(2m_e L^{2})$
$\hbar ^{2} = (1.055\times 10^{-34})^{2} \approx 1.113\times 10^{-68}$
$E_1 = \pi ^{2} \times 1.113\times 10^{-68} / (2 \times 9.109\times 10^{-31} \times 10^{-18}) \approx 6.0\times 10^{-20} J$
Convert: $6.0\times 10^{-20} / 1.602\times 10^{-19} \approx 0.376\,\mathrm{eV}$

Result ·

E_1 = \pi ^{2}\hbar ^{2}/(2m_e L^{2}) \approx 6.0\times 10^{-20} J \approx 0.376\,\mathrm{eV}

2

Harmonic oscillator ground-state energy

Given ·

ω:: 100000000000000

Find · E_0

Steps

TISE for $V = \tfrac{1}{2}m\omega ^{2}x^{2}$ gives $E_n = (n + \tfrac{1}{2})\hbar \omega$
$E_0 = \tfrac{1}{2} \times 1.055\times 10^{-34} \times 10^{14}$
$E_0 \approx 5.27\times 10^{-21} J$ — the famous zero-point energy.

Result ·

E_0 = \hbar \omega /2 \approx 5.27\times 10^{-21} J

Heisenberg Uncertainty Principle→De Broglie Wavelength→