Relativistic Quantum Mechanics
F755, academic year 2011 — Prof. M. Kastner
Problem sheet 2 Submission deadline: August 08, 2011
Problem 3: Wave packet solutions of the Klein-Gordon equation (written) 5 points
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-waves ξ±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).
Here the superscript±labels plane waves of positive and negative energies, respectively. We denote by W+ andW− the sets of (not necessarily normalized) wave packets obtained by superposition of plane waves normalized to +δ(p−p0) and−δ(p−p0), respectively.
(b) Express the wave packets Ξ± ∈W± as superpositions of plane waves with expansion coefficients g±(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 R have the property RiW± 6⊂ W± for i= 1,2,3.
Problem 4: Klein paradox 5 points
Consider a relativistic, spinless particle of rest massm0>0 and energyE > m0in one spatial dimension, incident from the left onto a potential step
V(z) :=
(0 forz <0, V0 forz>0,
withV0>0. This physical situation is described by the Klein-Gordon equation (i∂t−V(z))2+∂z2−m20
ψ(t, z) = 0.
(a) Use a separation ansatz to derive the corresponding stationary Klein-Gordon equation.
(b) Use an ansatz consisting of an incoming, a reflected, and a transmitted plane wave to obtain solutions of the stationary Klein-Gordon equation with the given step potential.
Hint: Since the Klein-Gordon equation is a second order differential equation, both the wave function and its first derivative (with respect toz) have to be continuous atz= 0.
(c) Compute from this solution the charge density eρ(t, z) := e
2m0[ψ∗(i∂t−V(z))ψ−ψ(i∂t+V(z))ψ∗]
forz >0, wheree >0 is the charge of the Klein-Gordon particle. Furthermore, compute the group velocityvg:=∂E(p)/∂pof the wave (wherepis the momentum or wave vector of the plane wave solution) as well as the reflection coefficientR :=|Ar/Ai|2, where Ai and Ar are the amplitudes of the incoming and reflected waves.
(d) Use the results of (c) to discuss the solution for the cases (i) E−m0> V0,
(ii) E > V0> E−m0, (iii) E+m0> V0> E, (iv) V0> E+m0.
In particular, discuss whether eρand vg are imaginary or real, and, if real, whether positive or negative. Check whetherRis smaller than, equal to, or larger than 1.
