Schrödinger Equation and Time Evolution

Summary

The Schrödinger equation governs how a quantum state evolves in time. Time evolution is deterministic and unitary, driven by the Hamiltonian. Key solved systems include the free particle, the particle in a box, and the quantum harmonic oscillator — these serve as building blocks for quantum field theory.

Overview

Between measurements, a quantum state $\psi (t)$ evolves continuously and deterministically according to the Schrödinger equation. The Hamiltonian $H$, representing total energy, is the generator of this evolution. Because $H$ is Hermitian, evolution is unitary — probabilities are conserved. The Schrödinger equation is the quantum analogue of Newton’s second law.

The Schrödinger Equation

Time-Dependent Schrödinger Equation

The time evolution of a quantum state $\psi (t)$ in a Hilbert space $\mathcal{H}$ is governed by:

$$i\hslash \frac{\partial }{\partial t}\psi (t)=H\psi (t)$$

where $H$ is the Hamiltonian operator (total energy) and $\hslash$ is the reduced Planck constant.

Solution: Time-Evolution Operator

The formal solution is:

$$\psi (t)=e^{-iHt/\hslash }\psi (0)\equiv U(t)\psi (0)$$

Time-Evolution Operator

$U(t)=e^{-iHt/\hslash }$ is unitary: $U^{†}U=I$. This ensures the norm of $\psi$ is preserved, i.e., probabilities sum to 1 at all times.

Conservation Laws

Any observable $A$ that commutes with $H$ (i.e. $[A,H]=0$) is conserved: its expectation value $⟨A⟩$ does not change in time. This is the quantum version of Noether’s theorem: symmetries of the Hamiltonian correspond to conservation laws.

Stationary States

Eigenstates of $H$ are stationary states: their probability distributions are time-independent:

$$H{\varphi }_n=E_n{\varphi }_n⟹\psi (t)=e^{-i{E}_{n}t/\hslash }{\varphi }_n$$

The time-dependent phase factor $e^{-i{E}_{n}t/\hslash }$ does not affect any observable.

The time-independent Schrödinger equation is:

$$H\psi =E\psi$$

Worked Examples

Free Particle (1D)

Setup: A particle with no external forces; $H=P^2/(2m)=-({\hslash }^2/2m)d^2/d{x}^2$.

Solution: General solution is a superposition of plane waves:

$$\psi (x,t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\hat{\psi }(k,0)e^{i(kx-\frac{\hslash k^2}{2m}t)}dk$$

where $\hat{\psi }(k,0)$ is the Fourier transform of the initial state and $p=\hslash k$.

Key result: A Gaussian wave packet moves at constant velocity but spreads over time — position uncertainty grows while momentum uncertainty stays constant. This illustrates the uncertainty principle.

Particle in a Box (1D)

Setup: Zero potential inside $[0,L]$, infinite walls outside. Time-independent Schrödinger eq.:

$$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi$$

Boundary conditions: $\psi (0)=\psi (L)=0$.

Solution: ${\psi }_n(x)=\sqrt{2/L}\sin (n\pi x/L)$ with quantized energies:

$$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2},n=1,2,3,\dots$$

Interpretation: Energy is quantized — only discrete values are allowed. The lowest energy (ground state, $n=1$) is non-zero, a consequence of the uncertainty principle.

Quantum Harmonic Oscillator

Setup: $H=P^2/(2m)+\frac{1}{2}m{\omega }^2{X}^2$.

Solution via ladder operators: Define $\hat{a}=\sqrt{m\omega /2\hslash }(\hat{X}+i\hat{P}/m\omega )$ and ${\hat{a}}^{†}$ (its adjoint). Then:

$$H=\hslash \omega \left({\hat{a}}^{†}\hat{a}+\frac{1}{2}\right)$$

Energy levels: $E_n=\hslash \omega (n+1/2)$, $n=0,1,2,\dots$

Interpretation: The zero-point energy $\hslash \omega /2$ is non-zero even in the ground state. This exact same algebra reappears in quantum field theory, where $\hat{a}$ and ${\hat{a}}^{†}$ become particle annihilation and creation operators.

Connections

• Wave Function and Hilbert Space: $\psi$ lives in $\mathcal{H}$; the Hamiltonian is a self-adjoint operator on $\mathcal{H}$.
• Uncertainty Principle: The free-particle Gaussian wave packet directly demonstrates position-momentum uncertainty.
• Canonical Quantization of Fields: The quantum harmonic oscillator is the prototype — QFT quantizes fields mode-by-mode as independent harmonic oscillators.

See Also

• Wave Function and Hilbert Space — the state $\psi$ that evolves
• Uncertainty Principle — consequences of the non-commuting position and momentum operators
• Canonical Quantization of Fields — the harmonic oscillator generalized to fields