Time-Independent Schrödinger Equation

Equivalent forms

-\frac{\hbar^2}{2m}\nabla^2\psi + V(\mathbf{r})\psi = E\psi

\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)\right]\psi(x) = E\psi(x)

An eigenvalue equation — the same structure as a vibrating drum — produces the entire periodic table.

Symbol	Name	SI unit	Dimension	Range
$E$	Energy eigenvalueoutput Allowed total energy of the stationary state	J	ML^2T^-2	-1e-17 – 1e-17
$hbar$	Reduced Planck constant h/(2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$m$	Particle mass Mass of the quantum particle	kg	M	1e-31 – 1e-25
$V$	Potential energy Potential energy at position r	J	ML^2T^-2	-1e-17 – 1e-17

Unit systems

Symbol	SI	CGS	Imperial
$E$	J	erg	ft*lbf
$hbar$	J*s	erg*s	ftlbfs
$m$	kg	g	lb
$V$	J	erg	ft*lbf

Where it holds

Non-relativistic, spinless. Requires V time-independent. Breaks down at

v \sim c (use

Dirac equation) and at strong gravity.

Dimensional analysis

H*\psi \to [M*L^2*T^-2] * [\psi ]

E*\psi \to [M*L^2*T^-2] * [\psi ]

Both sides carry units of energy times wavefunction.

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Discovery

Erwin Schrödinger · 1926

Schrödinger sought a wave equation matching de Broglie's matter waves. Within months of publishing the time-dependent form, he separated variables to obtain the stationary equation that reproduced hydrogen's spectrum exactly.

Try this

What equation tells you where an electron is allowed to live?

Stationary states of any quantum system — atoms, molecules, quantum dots — are governed by one eigenvalue equation. What does it look like, and why is energy quantized?

Research status: stable

Real-world applications

Semiconductor band structure $(k\cdot p$ theory)
Molecular orbital calculations (Hartree-Fock, DFT)
Quantum dot laser design
Vibrational spectra of molecules

Common misconceptions

psi is not directly observable — only |psi|^2 gives a probability density
E is not the kinetic energy — it includes the potential
Stationary does not mean motionless — observables have time-independent expectation values

Experimental verification

Hydrogen line spectrum (Balmer, Lyman, Paschen series) matches predicted

E_n = -13.6\,\mathrm{eV} / n^2

to one part in 10^4. Quantum dot spectroscopy, scanning tunneling spectroscopy, and atom-trap experiments all confirm discrete eigenvalues.

Derivation

Start from the time-dependent Schrödinger equation i*hbar*

d_t*\Psi = H*\Psi

.

Separate variables Psi(r,

t) = \psi (r)*\varphi (t)

.

Substitution gives

\varphi (t) = \exp (-i*E*t

/hbar) and

H*\psi = E*\psi

— the time-independent equation.

The Hamiltonian

H = p^2/(2*m) + V

becomes -(hbar^2)/(2*m)*nabla^2 + V on substituting

p \to -i

*hbar*nabla.

Limiting cases

V = 0

everywhere⟶

\psi = \exp (i*k*x)

,

E =

hbar^2*k^2/(2*m)Free particle — continuous energy spectrum, plane-wave solutions.

V \to \infty

outside a box⟶

E_n = n^2 * pi^2

* hbar^2 / (2*m*L^2)Infinite square well — discrete energies labeled by integer n.

V(x) = (1/2)*m*\omega ^2*x^2

⟶

E_n =

hbar*omega*(n + 1/2)Harmonic oscillator — equally spaced ladder of levels.

What if…

What if hbar were larger?

Energy spacings between bound states would widen — quantization would become visible at macroscopic scales.

What if V(r) were time-dependent?

No stationary states exist — must solve the full time-dependent equation, leading to transitions (Fermi's Golden Rule).

What if the particle is relativistic?

Schrödinger equation fails — replace with Klein–Gordon (spin 0) or Dirac (spin 1/2) equation.

1

Ground state of infinite well (electron in 1 nm box)

Given ·

m:: 9.1093837015e-31
L:: 1e-9
n:: 1

Find · E_1

Steps

$E_n = n^2 * pi^2$ * hbar^2 / (2*m*L^2)
$= 1 * 9.8696 * (1.054571817e-34)^2 / (2 * 9.1093837015e-31 * 1e-18)$
$\approx 6.02e-20\,\mathrm{J} = 0.376\,\mathrm{eV}$

Result ·

E_1 = pi^2

* hbar^

2 / (2*m*L^2) \approx 6.02e-20\,\mathrm{J} \approx 0.376\,\mathrm{eV}

2

Harmonic oscillator ground-state energy

Given ·

omega:: 100000000000000

Find · E_0

Steps

$E_0 = (1/2) * 1.054571817e-34 * 1e14$
$\approx 5.27e-21\,\mathrm{J}$
$\approx 0.033\,\mathrm{eV}$ — zero-point energy is nonzero

Result ·

E_0 = (1/2)

*hbar*

\omega \approx 5.27e-21\,\mathrm{J} \approx 0.033\,\mathrm{eV}

Time-Dependent Schrödinger Equation→particle in boxQuantum Harmonic Oscillator→Heisenberg Uncertainty Principle→