Schrodinger Equation — Mathematics Knowledge System

Overview

The Schrodinger equation is the fundamental equation of non-relativistic quantum mechanics, governing how a quantum state evolves in time. The time-dependent form is a first-order linear PDE in time; the time-independent (stationary) form is an eigenvalue problem for the Hamiltonian operator. Solutions give wavefunctions whose squared modulus yields probability densities. The equation was derived by Erwin Schrodinger in 1926 by analogy with wave optics and de Broglie's matter waves.

Intuition

The Schrodinger equation plays the role in quantum mechanics that Newton's second law plays in classical mechanics: given the state now, it predicts the state at all future times. The wavefunction psi(x, t) is a probability amplitude: |psi|² gives the probability density of finding the particle at position x at time t. The time-independent equation determines the allowed energy levels (eigenvalues) of a system.

Formal Definition

Definition

The time-dependent Schrodinger equation (TDSE) governs the unitary evolution of a quantum state; the time-independent Schrodinger equation (TISE) is the eigenvalue problem for the Hamiltonian.

i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle

Time-dependent Schrodinger equation (TDSE, abstract form)

i\hbar \frac{\partial \psi}{\partial t}(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}(x,t) + V(x)\psi(x,t)

TDSE in position representation (1D)

\hat{H}\psi_n(x) = E_n\psi_n(x)

Time-independent Schrodinger equation (TISE): energy eigenvalue problem

\psi(x,t) = \sum_n c_n \psi_n(x)\, e^{-iE_n t/\hbar},\quad c_n = \langle\psi_n|\psi(x,0)\rangle

General solution via energy eigenstates (stationary states)

|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle

Formal solution: unitary time evolution operator

Notation

Notation	Meaning
$\psi(x,t)$	Wavefunction (position-space representation of the state)
$\|\psi(t)\rangle$	State vector (abstract Hilbert space)
$E_n$	Energy eigenvalue (energy level)
$\psi_n$	Energy eigenstate (stationary state)
$U(t) = e^{-i\hat{H}t/\hbar}$	Unitary time evolution operator

Theorems

Theorem 1: Conservation of Probability

\frac{\partial}{\partial t}|\psi(x,t)|^2 + \nabla \cdot \mathbf{J} = 0,\quad \mathbf{J} = \frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*)

Theorem 2: Stationary States

\psi_n(x,t) = \psi_n(x)\,e^{-iE_n t/\hbar} \implies |\psi_n(x,t)|^2 = |\psi_n(x)|^2 \text{ (time-independent)}

Theorem 3: Energy-Time Uncertainty

\Delta E \cdot \Delta t \geq \frac{\hbar}{2}

Worked Examples

1
V = 0 inside (0, L), V = infinity outside. TISE: −(ℏ²/2m)ψ'' = Eψ.
$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi,\quad \psi(0)=\psi(L)=0$
2
General solution: ψ = A sin(kx) + B cos(kx); BC at x=0 forces B=0.
$\psi(x) = A\sin(kx),\quad k = \sqrt{2mE}/\hbar$
3
BC at x=L: sin(kL) = 0 → kL = nπ.
$k_n = \frac{n\pi}{L},\quad n = 1,2,3,\ldots$
4
Energy eigenvalues:
$E_n = \frac{\hbar^2 k_n^2}{2m} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$

✓ Answer

Energy levels: Eₙ = n²π²ℏ²/(2mL²), n = 1, 2, 3, …

Practice Problems

Hardfree response

Show that if H is time-independent, the solution to the TDSE is |ψ(t)⟩ = e^{−iHt/ℏ}|ψ(0)⟩, and verify it satisfies the equation.

Common Mistakes

Common Mistake

The wavefunction ψ is directly measurable

Only |ψ|² is physically observable (probability density). The wavefunction itself is not directly measurable and can have complex values; global phase is unphysical.

Common Mistake

Stationary states are states where nothing changes

Stationary states have time-independent probability densities, but the state still evolves: ψₙ(x,t) = ψₙ(x) e^{−iEₙt/ℏ} — the phase changes in time, which matters for superpositions.

Quiz

The time-independent Schrodinger equation is:

A first-order ODE for the wavefunctionAn eigenvalue equation H psi = E psi for the HamiltonianA conservation law for probabilityA boundary condition on the wavefunction

The squared modulus |ψ(x,t)|² gives:

The energy of the stateThe probability density of finding the particle at x at time tThe potential energy at xThe speed of the wave packet

Historical Background

Erwin Schrodinger published his wave equation in a series of four papers in Annalen der Physik in 1926. He was motivated by de Broglie's hypothesis that matter has wave-like properties and by Hamilton's analogy between classical mechanics and optics. Schrodinger showed his equation reproduced the hydrogen atom energy levels exactly. He also demonstrated the equivalence of his formulation with Heisenberg's matrix mechanics. The 1926 papers earned Schrodinger the 1933 Nobel Prize in Physics.

1924
de Broglie proposes matter waves: lambda = h/p
Louis de Broglie
1926
Schrodinger publishes the wave equation in four Annalen der Physik papers
Erwin Schrodinger
1926
Born provides the probabilistic interpretation (Born rule) of the wavefunction
Max Born
1933
Schrodinger and Dirac share the Nobel Prize in Physics
Erwin Schrodinger, Paul Dirac

Summary

The TDSE iℏ ∂|ψ⟩/∂t = H|ψ⟩ governs quantum time evolution; for time-independent H, |ψ(t)⟩ = e^{−iHt/ℏ}|ψ(0)⟩.
The TISE Hψₙ = Eₙψₙ determines energy eigenvalues (allowed energy levels) and stationary states.
|ψ(x,t)|² is the probability density; the continuity equation ensures total probability is conserved.
A general state is a superposition: ψ = Σₙ cₙψₙ e^{−iEₙt/ℏ}, with |cₙ|² = probability of energy Eₙ.
For the infinite square well, Eₙ = n²π²ℏ²/(2mL²), illustrating quantisation from boundary conditions.

References

BookSchrodinger, E. — Quantisierung als Eigenwertproblem, Annalen der Physik 79, 1926, pp. 361–376
BookGriffiths, D.J. & Schroeter, D.F. — Introduction to Quantum Mechanics, 3rd ed. (2018), Cambridge University Press
BookCohen-Tannoudji, C., Diu, B. & Laloe, F. — Quantum Mechanics, Vol. 1 (1977), Wiley
WebsiteWikipedia — Schrodinger equation
WebsiteNIST DLMF — Schrodinger Equation