Questions: The Adiabatic Theorem and Berry's Phase

Question 1

A particle sits in the ground state of an infinite square well. The well's width is slowly doubled. What is the final state?

• A. The ground state of the wide well, with one quarter of the original energy
• B. The original wavefunction, unchanged
• C. An equal mixture of all states of the wide well
• D. The first excited state of the wide well

Answer: (A)

By the adiabatic theorem the particle stays in the ground state of the changing well. Since E₁ ∝ 1/L², doubling L cuts the energy to one quarter. A sudden change would instead leave the wavefunction unchanged.

Question 2

For a closed loop in parameter space traversed adiabatically, Berry's phase depends on the geometry of the loop and not on how fast the loop is traversed.

• T. True
• F. False

Answer: True

The geometric phase γ = i∮⟨n|∇_R n⟩·dR depends only on the path. The dynamic phase depends on the time taken; the geometric phase does not.

Question 3

What condition must hold for the adiabatic approximation to apply?

• A. The Hamiltonian must be time-independent
• B. The time over which H changes must be long compared with ħ/ΔE, where ΔE is the gap to neighbouring levels
• C. The temperature must be zero
• D. The energy must be conserved

Answer: (B)

Transitions to other levels are suppressed when the change is slow compared with the internal time scale ħ/ΔE. A level crossing, where ΔE goes to zero, breaks the approximation.

Question 4

A spin-½ particle with spin aligned along a magnetic field follows the field as its direction slowly traces out a closed cone that subtends solid angle Ω. What is the Berry phase?

Model answer: γ = −Ω/2

The Berry phase for the aligned spin-½ state equals minus one half of the solid angle traced out by the field direction. The anti-aligned state has +Ω/2.