Problem 6: Dirac-Hamilton-Operator in a magnetic field

Relativistic Quantum Mechanics

F755, academic year 2009 — Prof. M. Kastner

Problem sheet 3 Submission deadline: August 12, 2009

Problem 5: Zitterbewegung

Consider a system whose time evolution is governed by the free Dirac-Hamiltonian H=α·P +βm₀.

(a) Compute the time derivative ˙X(t) = i[H,X(t)] of the position operatorX(t) in the Heisenberg picture. Is this a physically reasonable velocity operator?

(b) Show

X(t) = 2iHF¨ (t) where F(t)≡X(t)˙ −PH⁻¹.

(c) Show that [F(0), H]₊ = 0 and that the vanishing of this anti-commutator implies F(t) = e^2iHtF(0). Now integrate the operator ˙X from (a) with respect to time.

(Result:X(t) =X(0) +PH⁻¹t−₂ⁱH⁻¹(e^2iHt−1)F(0).

(d) The oscillations occurring in the result of (c) are termed Zitterbewegung. Give an estimate for the amplitude and frequency of these oscillations.

(e) What is the expectation valuehE|F|EiofF with respect to an eigenstate|EiofH?

Problem 6: Dirac-Hamilton-Operator in a magnetic field

Let B(r) =∇×A(r) be a magnetic field resulting from a stationary vector potentialA(r). For a spin-1/2 particle of rest massm₀ and chargeein a magnetic field the Dirac-Hamiltonian is given by

H_D=α·[P −eA(r)] +βm₀. (a) ExpressHDin terms of the operator Q=^√1_m

0[P−eA(r)]·σ.

(b) Using the Lorentz gauge∇·A= 0, show that

H_P := 1

2Q²= 1 2m0

[P−eA(r)]²− e 2m0

σ·B(r),

i.e.,HP is the Pauli-Hamiltonian of anonrelativistic spin-1/2 particle in a magnetic field.

(c) Show the relation

H_D² =m²₀ 12+^2H_m^P

0 0

0 12+^2H_m^P

0

! .

What does this imply for the eigenvalues ofH_Dand H_P? (d) Expandp

H_D² for large rest energiesm₀up to order 1/m₀.