Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: the interaction picture, first-order transition amplitudes, constant and periodic perturbations, the golden rule, and the two extremes of a changing Hamiltonian with the geometric phase the slow one leaves. No worked example sits above the answers.

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V^_{I} (t) = e^{\frac{i H ^ _{0} t}{ℏ}} V^e^{- \frac{i H ^ _{0} t}{ℏ}}

P_{i \to f} = ∣ c_{f} (t) ∣^{2}

ω_{f i} = \frac{E _{f} - E _{i}}{ℏ}

c_{f} (t) \propto 0 \int t V_{f i} (t^{'}) e^{i ω_{f i} t^{'}} d t^{'}

Chapter navigation

Intuition

Bring the chapter together: the interaction picture, first-order transition amplitudes, constant and periodic perturbations, the golden rule, and the two extremes of a changing Hamiltonian with the geometric phase the slow one leaves. No worked example sits above the answers.

A pilot reads the weather as gusts, steady winds and slow drifts, and handles each with its own rule. This chapter’s methods are those rules for a Hamiltonian that changes in time.

Before you practise

First order: $c_{f}=-\frac{i}{\hbar}\int V_{fi}e^{i\omega_{fi}t}dt$ . Resonance at $E_{f}=E_{i}\pm\hbar\omega$ . Into a continuum, $\Gamma=\frac{2\pi}{\hbar}|V_{fi}|^{2}\rho$ , lifetime $1/\Gamma$ and width $\hbar\Gamma$ .

P_{i\to f}(t)=\frac{4|V_{fi}|^{2}}{\hbar^{2}\omega_{fi}^{2}}\sin^{2}\frac{\omega_{fi}t}{2}

Reminders

Sudden change: the state is unchanged, and $P_{n'}=|\braket{n'|\psi}|^{2}$ .
Slow change: the state follows its level, with a dynamical and a geometric phase.
A spin along a field that encloses the solid angle $\Omega$ gains $\gamma=-\Omega/2$ .

Applications

Practice

Try it

Two levels of equal energy are coupled by a constant $V_{fi}=0.002$ eV. What is the first-order probability of the jump after a time with $t/\hbar=25$ eV $^{-1}$ ?