Theme 3

Postulates and Applications

Choose an exercise to open its statement and solution.

Exercise 1—Successive measurements on a three-level system

Keywords: postulates · Born rule · wave-function collapse · degenerate eigenvalue · expectation value · standard deviation · incompatible observables

Exercise 2—Probability current

Keywords: probability current · continuity equation · plane wave · standing wave · evanescent wave · velocity

Exercise 3—Uncertainty relation for a spin one-half particle

Keywords: uncertainty relation · Robertson inequality · spin one-half · Pauli matrices · standard deviation

Exercise 4—Commutators, translations, and the virial theorem

Keywords: canonical commutator · translation operator · Ehrenfest theorem · virial theorem · stationary state

Exercise 5—The ammonia molecule: a two-level system

Keywords: two-level system · ammonia · tunnelling · oscillations · beats · Bohr frequency · time evolution

Exercise 6—Position and momentum in the infinite well

Keywords: infinite well · expectation value · standard deviation · uncertainty relation · classical limit

Exercise 7—Precession of a spin in a magnetic field

Keywords: spin one-half · magnetic field · Larmor precession · Bloch sphere · pi pulse · Ehrenfest theorem

Exercise 8—Angular momentum of a 2p orbital

Keywords: hydrogen atom · 2p orbital · spherical harmonics · angular momentum · measurement of Lz · normalisation

Exercise 9—Expectation values in the hydrogen ground state

Keywords: hydrogen atom · 1s orbital · radial density · virial theorem · classically forbidden region · order of magnitude

Exercise 10—Fourier transform of an exponential wave function

Keywords: Fourier transform · momentum-space representation · Lorentzian distribution · uncertainty relation · Plancherel theorem

Exercise 11—Wave packet in free fall

Keywords: Ehrenfest theorem · free fall · linear potential · wave-packet spreading · covariance

Exercise 12—Parabolic initial state in the infinite well

Keywords: infinite well · expansion in an eigenbasis · energy probabilities · mean energy · time evolution · revival period

Exercise 13—Breathing of a wave packet in the harmonic oscillator

Keywords: harmonic oscillator · ladder operators · superposition · expectation values · time evolution · Bohr frequencies

Exercise 14—Hückel model of the allyl radical

Keywords: Hückel model · quantum chemistry · diagonalisation · molecular orbitals · delocalisation energy · electron populations

Exercise 15—Delta well: bound state and scattering

Keywords: delta potential · matching condition · bound state · reflection and transmission coefficients · probability current

Exercise 16—Charged oscillator in an electric field

Keywords: harmonic oscillator · electric field · translation · polarisability · sudden approximation · wave-function overlap