Relativistic Quantum Mechanics
F755, academic year 2009 — Prof. M. Kastner
Problem sheet 2 Submission deadline: August 03, 2009
Problem 3: Wave packets as solutions of the Klein-Gordon equation
Consider the two-component form of the free Klein-Gordon equation, i∂
∂tξ=Hξ withH = 1 2m0
−1 −1
1 1
∆ +m0
1 0 0 −1
,
where~=c= 1.
(a) For a given momentum p∈R3, determine plane-wavesξ±p = ϑ±p
χ±p
which are solutions of the free Klein-Gordon equation and subject to the (improper) orthonormality condition
i 2m0
Z
d3r j0= Z
d3r
ϑ±p∗ϑ±p0−χ±p∗χ±p0
=±δ(p−p0).
We denote byW+ andW− the sets of (not necessarily normalized) wave packets obtained by super- position of plane waves normalized to +δ(p−p0) and−δ(p−p0), respectively.
(b) Use the dispersion relation E(p) to express the wave packets Ξ± ∈ W± as superpositions of plane waves with expansion coefficientsg±(p). Determine the condition ong± to guarantee that the wave packets Ξ±(t,r) =
ϑ±(t,r) χ±(t,r)
obey the (proper) normalization condition Z
d3r |ϑ(t,r)|2− |χ(t,r)|2
=±1.
(c) Show that the components Ri of the position operator Rhave the property RiW± 6⊂W± for i= 1,2,3.
Problem 4: Klein-Gordon particle in a Coulomb potential
Consider the Klein-Gordon equation in an external potential, h(i∂t−eϕ(r))2−P2−m20i
ψ(t,r) = 0.
(a) Use the ansatzψ(t,r) = e−iEtψ(r) to derive the stationary Klein-Gordon equation.
(b) Show from the definitionL=R×P thatP2=Pr2+r−2L2wherer=|r|andPr=−i(∂r+r−1).
Use this identity to write the stationary Klein-Gordon equation in the form h
(E−eϕ(r))2−Pr2−m20−r−2L2i
ψ(r) = 0.
(c) Consider now a central symmetric potential,ϕ=ϕ(r). Since the eigenfunctionsYlm of L2 are known, it is reasonable to make the separation ansatzψ(r, ϑ, ϕ) =R(r)Ylm(ϑ, ϕ) for the wave function in spherical coordinates. Derive the radial part of the Klein-Gordon equation, i.e., the differential equation forR.
(d) Rewrite this radial equation for the Coulomb potential eϕ(r) = −Zα/r with fine-structure constantα=e2 and atomic numberZ. Show that the resulting equation is formally equivalent to the radial part of the corresponding nonrelativistic problem (and therefore can be solved analogously).
