Quantum Dynamics §

From the classical Hamiltonian to the quantum one §

In classical mechanics, the state evolves via Hamilton’s equations: $\dot{q} = \partial H/\partial p$, $\dot{p} = -\partial H/\partial q$. In quantum mechanics, the state is a vector in Hilbert space (see Quantum states) and evolves via the Schrödinger equation. The classical Hamiltonian function $H(q,p)$ becomes an operator $\hat{H}$ by the correspondence $q \to \hat{q}$, $p \to \hat{p} = -i\hbar\,\partial/\partial x$.

Step 1: Time-independent case.

If $\hat{H}$ does not depend on time, the evolution operator is $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$. If the initial state is an energy eigenstate $\hat{H}|n\rangle = E_n|n\rangle$, then $|\psi(t)\rangle = e^{-iE_n t/\hbar}|n\rangle$ — the state picks up only a global phase, and all measurable probabilities are constant. Superpositions of energy eigenstates, however, oscillate: $|\psi(t)\rangle = c_1 e^{-iE_1 t/\hbar}|1\rangle + c_2 e^{-iE_2 t/\hbar}|2\rangle$ has a time-dependent relative phase $e^{-i(E_2-E_1)t/\hbar}$ that produces oscillating probabilities.

Step 2: Time-dependent case.

When $\hat{H}$ depends on time (e.g., a time-varying electromagnetic field), the evolution operator involves a time-ordering $\mathcal{T}$. There is no simple exponential form, and one must use perturbation theory or numerical methods.

Example: a two-level system in a resonant field.

A qubit with energy splitting $\hbar\omega_0$ is driven by an oscillating field of frequency $\omega \approx \omega_0$. In the rotating-wave approximation, the effective Hamiltonian in the interaction picture is $\hat{H}_{\text{eff}} = \frac{\hbar}{2}\begin{pmatrix} \Delta & \Omega \\ \Omega & -\Delta \end{pmatrix}$ where $\Delta = \omega - \omega_0$ is the detuning and $\Omega$ is the Rabi frequency. Starting in $|0\rangle$, the probability of finding the system in $|1\rangle$ oscillates as $P_1(t) = \frac{\Omega^2}{\Omega^2 + \Delta^2}\sin^2\left(\frac{\sqrt{\Omega^2+\Delta^2}\,t}{2}\right)$. At resonance ($\Delta = 0$), full population transfer occurs at $t = \pi/\Omega$ (a $\pi$-pulse).

Example problem.

When the Hamiltonian changes slowly §

Many physical situations involve a small time-dependent perturbation: an atom in a weak electromagnetic wave, a particle encountering a short potential barrier, a qubit subjected to a control pulse. Exact solutions are rare; perturbation theory expands in powers of $\hat{V}$.

Step 1: First-order amplitude.

Assume the system starts in an eigenstate $|i\rangle$ of $\hat{H}_0$ at $t = -\infty$. The first-order correction to the amplitude of finding it in state $|f\rangle$ at time $t$ is an integral of the matrix element of $\hat{V}$ weighted by the phase factor $e^{i\omega_{fi}t'}$. The integral is large when $\omega_{fi} \approx 0$ (energy conservation) — this is the resonance condition.

Step 2: Fermi’s golden rule.

For a perturbation that is on for a long time $T$, the transition probability grows as $T^2\,\text{sinc}^2(\omega_{fi}T/2)$. The rate (probability per unit time) in the limit $T \to \infty$ becomes proportional to $2\pi\delta(E_f - E_i)$, which for a continuum of final states gives the density of states $\rho(E_f)$. This is Fermi’s golden rule.

Example: spontaneous emission.

An excited atom coupled to the electromagnetic vacuum (a time-dependent perturbation from the quantized field) decays with rate $\Gamma = \frac{\omega_0^3|\mathbf{d}|^2}{3\pi\epsilon_0\hbar c^3}$ where $\mathbf{d}$ is the dipole matrix element. This is a first-order process in the field coupling.

Example problem.

From wave packets to plane waves §

A scattering experiment sends a particle toward a target and measures where it ends up. Quantum mechanically, the incoming state is a wave packet (or, in the idealized limit, a plane wave). The outgoing state consists of an undistorted forward component plus a spherical outgoing wave $f(\theta,\phi)\,e^{ikr}/r$.

Step 1: The scattering amplitude.

For a spherically symmetric potential $V(r)$, the scattering amplitude in the partial-wave expansion is:

$f(\theta) = \frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta)$

where $\delta_\ell$ are the phase shifts and $P_\ell$ are Legendre polynomials. Each partial wave $\ell$ contributes a term weighted by $(2\ell+1)$ and the phase shift $\delta_\ell$.

Step 2: Low-energy (s-wave) scattering.

At low energies, only $\ell = 0$ contributes. The cross-section becomes isotropic: $\sigma = 4\pi a^2$ where $a = -\tan\delta_0/k$ is the scattering length. This single parameter characterizes low-energy scattering and is central to ultracold atom physics and neutron scattering.

Connection to the Standard Model.

In relativistic quantum field theory (see Quantum field theory), scattering is described by Feynman diagrams and the S-matrix elements are computed from loop integrals. The non-relativistic scattering theory above is the low-energy limit of the full QFT calculation.

Example problem.

A third formulation of quantum mechanics §

Alongside matrix mechanics (Heisenberg) and wave mechanics (Schrödinger), the path integral (Feynman, 1948) provides a conceptually distinct formulation. Instead of evolving a state vector, one sums over all possible histories weighted by $e^{iS/\hbar}$.

Step 1: The propagator as a sum over paths.

The amplitude for a particle to go from $(x_i, t_i)$ to $(x_f, t_f)$ is:

$K = \lim_{N\to\infty}\left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2}\int\prod_{n=1}^{N-1}dx_n\,\exp\left[\frac{i}{\hbar}\sum_{n=0}^{N-1}\left(\frac{m(x_{n+1}-x_n)^2}{2\epsilon} - \epsilon V(x_n)\right)\right]$

where $\epsilon = (t_f-t_i)/N$. Each term in the sum is the Lagrangian evaluated along a discrete path.

Step 2: The classical limit.

When the action $S \gg \hbar$, paths near the classical trajectory (where $\delta S = 0$) contribute constructively (their phases align), while paths far from the classical one cancel by destructive interference. The result is that the particle follows the classical trajectory — the correspondence principle connecting to Mechanics.

Step 3: Why path integrals matter beyond computation.

Path integrals are not merely a calculational tool for QFT. They provide:

• A natural framework for quantum gravity (sum over geometries).
• A direct connection between quantum and statistical mechanics (Wick rotation $t \to -i\hbar\beta$ turns the path integral into a partition function; see Thermodynamics).
• A geometric picture of quantum interference (the double-slit experiment is a sum over paths through both slits).

Example: the free-particle propagator.

For $V = 0$, the path integral can be evaluated exactly. The result is:

$K(x_f,t_f;x_i,t_i) = \sqrt{\frac{m}{2\pi i\hbar(t_f-t_i)}}\exp\left[\frac{im(x_f-x_i)^2}{2\hbar(t_f-t_i)}\right]$

This is a Gaussian in $(x_f - x_i)$ with a complex phase — the quantum analogue of the classical free-particle trajectory.

Example problem.

Problem. A qubit with Hamiltonian $\hat{H} = \frac{\hbar\omega}{2}\sigma_z$ is initially in state $|0\rangle = |\uparrow_z\rangle$. At time $t = 0$, a sudden perturbation $\hat{V} = \hbar\Omega\,\sigma_x$ is applied for a duration $\tau = \pi/(2\Omega)$ (a $\pi/2$-pulse). Find the state after the pulse (to first order in $\Omega/\omega$, assuming $\Omega \ll \omega$).