Relativistic Quantum Mechanics
F755, academic year 2010 — Prof. M. Kastner
Problem sheet 4 Submission deadline: August 24, 2010
Problem 8: Zitterbewegung 5 points
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) = e2iHtF(0). Now integrate the operator ˙X from (a) with respect to time.
(Result:X(t) =X(0) +PH−1t−2iH−1(e2iHt−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 9: Foldy-Wouthuysen transformation (written) 5 points
(a) As preliminary work, consider the operator
S=−iβα·P
|P| s(P) and show the identities
{S, H}= 0 and e2iS= cos(2s) +βα·P
|P| sin(2s), whereH =α·P+βm0 is the free Dirac-Hamiltonian.
(b) Independently of the method used in the lecture, the Foldy-Wouthuysen transformation provides another way to diagonalizeH: We are looking for a unitary operator eiS, such that
H0= eiSHe−iS
is diagonal. Use the above ansatz forS and determine the functions:R3→Rsuch thatH0 is diagonal.
