The Schrödinger Equation

Equivalent forms

i\hbar\,\partial_t\Psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\Psi

\hat{H}\psi = E\psi

One linear equation generates atoms, chemistry, semiconductors and lasers. The 'i' on the left is not decoration — it forces Ψ to be complex, and that complexness is exactly what lets probability waves interfere.

Symbol	Name	SI unit	Dimension	Range
$\Psi$	Wavefunctionoutput Complex probability amplitude over space and time	m^{-3/2}	L^-3/2	—
$\hat{H}$	Hamiltonian operator Total-energy operator: kinetic + potential	J	ML^2T^-2	—
$\hbar$	Reduced Planck constant h / (2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$V$	Potential energy External potential the particle moves in	J	ML^2T^-2	—
$m$	Particle mass	kg	M	1e-31 – 1e-26

Where it holds

Non-relativistic quantum mechanics (v << c). For relativistic particles use the Dirac or Klein–Gordon equations; for many-body field theory use QFT. Exact only for closed systems — open systems require density matrices and the Lindblad equation.

Dimensional analysis

\hbar \cdot \partial \Psi /\partial t \to [M*L^2*T^-1]\cdot [T^-1]\cdot [\Psi ] = [M*L^2*T^-2]\cdot [\Psi ]

Ĥ

\cdot \Psi \to [

energy

]\cdot [\Psi ] = [M*L^2*T^-2]\cdot [\Psi ]

Both sides are energy

\times \Psi

.

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Discovery

Erwin Schrödinger · 1926

Over a 1925 Christmas holiday in the Swiss village of Arosa — companion not his wife — Schrödinger took de Broglie's matter-wave hypothesis seriously and asked what wave equation those waves obey. Within weeks he had reproduced the hydrogen spectrum exactly. He first tried a relativistic version (now the Klein–Gordon equation), got the fine structure wrong, and published the non-relativistic equation instead. Within months Schrödinger proved his wave mechanics was mathematically identical to Heisenberg's matrix mechanics — two utterly different pictures of the same reality.

Research status: stable

Real-world applications

Predicting molecular bonding and reaction rates (quantum chemistry)
Designing semiconductor band structures and transistors
Modeling tunneling in flash memory and STM imaging
Quantum dots and the color of nanocrystals

Common misconceptions

$\Psi$ itself is not observable or even real-valued; only | $\Psi |^{2}$ (a probability density) is measurable
The equation is fully deterministic — randomness enters only at measurement, via the Born rule
It is a first-order equation in time but second-order in space; this asymmetry is why it is non-relativistic

Experimental verification

The equation predicts the hydrogen energy levels (Rydberg spectrum), the periodic table's shell structure, tunneling rates in scanning tunneling microscopes, and the band gaps of every semiconductor — all confirmed to high precision.

Derivation

Postulate a plane wave

\Psi = \exp (i(kx - \omega t))

.

De Broglie sets

p = \hbar k

and Planck sets

E = \hbar \omega

.

Then

\partial \Psi /\partial t = -i\omega \Psi

⟹

i\hbar \partial \Psi /\partial t = \hbar \omega \Psi = E \Psi

,

and \partial ^{2}\Psi /\partial x^{2} = -k^{2}\Psi

⟹

-(\hbar ^{2}/2m)\partial ^{2}\Psi /\partial x^{2} = (p^{2}/2m)\Psi

.

For a free particle

E = p^{2}/2m

, so

i\hbar \partial \Psi /\partial t = -(\hbar ^{2}/2m)\partial ^{2}\Psi /\partial x^{2}

.

Adding a potential V(x) promotes this to

i\hbar \partial \Psi /\partial t = [-(\hbar ^{2}/2m)\nabla ^{2} + V]\Psi

= Ĥ

\Psi

.

Separating variables

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

yields the time-independent equation Ĥ

\psi = E\psi

.

What if…

What if there were no factor of i?

You'd get the diffusion (heat) equation — $\Psi$ would decay instead of oscillate, and interference would vanish. The 'i' is what makes quantum mechanics wave-like.

What if

V = 0

everywhere?

Free-particle solutions: plane waves with $E = p^{2}/2m$ , a continuous spectrum and no quantization.

What if the potential is time-dependent?

No stationary states exist; you must solve the full time-dependent equation, and transitions between levels become possible (the basis of spectroscopy and lasers).

1

Time evolution of an energy eigenstate

Given ·

\psi (x)

is an eigenstate of Ĥ with energy E

Find ·

\Psi (x

, t)

Steps

Separate variables: $\Psi (x$ , $t) = \psi (x)\cdot T(t)$
$i\hbar \psi T$ ' = (Ĥ $\psi ) T = E \psi T$ ⟹ $T'/T = -iE/\hbar$
$T(t) = e^{-iEt/\hbar }$ , so probability | $\Psi |^{2} = |\psi |^{2}$ does not change in time

Result ·

\Psi (x

,

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

; |

\Psi |^{2} is

time-independent (a stationary state)

2

Phase rotation rate of a 2 eV state

Given ·

E:: $2\,\mathrm{eV} = 3.2e-19\,\mathrm{J}$

Find · angular frequency

\omega of

the phase factor

Steps

$\omega = E/\hbar = 3.2e-19 / 1.0546e-34$
$\omega \approx 3.04e15\,\mathrm{rad/s}$

Result ·

\omega = E/\hbar \approx 3.04e15\,\mathrm{rad/s}

(period

\approx 2.1\,\mathrm{fs})

Time-Dependent Schrödinger Equation→Time-Independent Schrödinger Equation→Born Rule→Canonical Commutation Relation→Path-Integral Formulation→