Quantum Mechanics · Lesson 07

Chapter practice

Bring the chapter together: the variational principle and its upper bounds, trial states and the Rayleigh–Ritz matrix, the WKB approximation, turning points and connection formulas, Bohr–Sommerfeld quantisation and tunnelling through smooth barriers. No worked example sits above the answers.

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E [ψ] \geq E_{0}

E [ψ] = \frac{⟨ ψ ∣ H ^ ∣ ψ ⟩}{⟨ ψ ∣ ψ ⟩}

Chapter navigation

Intuition

Bring the chapter together: the variational principle and its upper bounds, trial states and the Rayleigh–Ritz matrix, the WKB approximation, turning points and connection formulas, Bohr–Sommerfeld quantisation and tunnelling through smooth barriers. No worked example sits above the answers.

A navigator without satellites still crosses the ocean with a compass, a clock and good estimates. Approximation methods are the navigator’s tools of quantum mechanics.

Before you practise

Variational: $E[\psi]\ge E_{0}$ for every trial state, with errors of second order. WKB: , valid where varies slowly.

p (x) = 2 m (E - V (x))

ψ_{WKB} \propto \frac{e ^{\frac{i}{ℏ} \int p d x}}{p}

[

ψ

]

≥

E_{0}

\psi\approx\frac{C}{\sqrt p}e^{\pm\frac i\hbar\int\limits p\,dx}

p

\begin{gathered}\oint\limits p\,dx=2\pi\hbar\bigl(n+\tfrac12\bigr)\\[0.35em] T\approx\exp\Bigl(-\frac{2}{\hbar}\int\limits_{a}^{b}|p|\,dx\Bigr)\end{gathered}

Reminders

Rayleigh–Ritz: the lowest eigenvalue of $H_{ij}=\bra{\phi_{i}}\hat{H}\ket{\phi_{j}}$ bounds $E_{0}$ from above.
Smooth turning points contribute $\pi/4$ each; hard walls $\pi/2$ .
$|\psi|^{2}\propto1/p$ : the density follows the classical time spent.
Bohr–Sommerfeld is exact for the oscillator and best for large $n$ .

Applications

Practice

Try it

A trial state gives average energies 3.2, 2.9 and 3.1 for three values of its parameter. What is the best upper bound on $E_{0}$ they give?