Departement Physik, Universit¨
at Basel
Prof. C. Bruder (Zimmer 4.2, Tel.: 267-3692, [email protected])
H¨
ohere Quantenmechanik, Fru
¨ hjahrsemester 2015
Blatt 9
Abgabe: 21.5.15, 12:00H (Treppenhaus 4. Stock)
Tutor: Rakesh Tiwari, Zi. 4.16
Schriftlicher Test: Freitag, 22. Mai 2015, 10.15 - 12 Uhr
Hilfsmittel: Ein handbeschriebenes A4 Blatt.
(1) Ward identity
From [SF (p)]−1 = p/ − m and [SF (p)][SF (p)]−1 = 1, derive
(5 Punkte)
∂SF (p)
= −SF (p)γ µ SF (p) .
∂pµ
Hence, derive the following exact relation between self energy Σ(p) and the vertex
function Λµ (p0 , p)
∂Σ(p)
−
= Λµ (p, p) .
∂pµ
(2) Measurement operators
(5 Punkte)
Consider a two-dimensional Hilbert space with the basis |0i, |1i. Consider a continuous
measurement result φ that can
√ take values between 0 and 2π. We define the measurement operators Mφ = |φihφ|/ π, where
√
|φi = (|0i + exp(iφ)|1i)/ 2
In this case the effects are Eφ = π1 |φihφ|.
R 2π
(a) Show that 0 dφ Eφ = 1.
(b) Is Eφ a projection operator?
(c) Show that different effects are not orthogonal in general.
(d) If the system is originally in state |φi, what is the probability to measure φ˜ 6= φ?
(3) Decoherence and relaxation
(5 Punkte)
Using the density operator we can describe processes that correspond to a non-unitary
time evolution, e.g., the influence of the environment. The density operator ρˆ is mapped
on a new density operator ρˆ0 according to
X
ˆ k ρˆM
ˆ† ,
ρˆ 7−→ ρˆ0 =
M
(1)
k
k
ˆ k are the measurement operators (or Kraus operators) that characterize the
where M
influence of the environment.
(a) Explain why (1) can be used to model the influence of the environment.
(b) Show that ρˆ0 is hermitian and fulfills tr(ˆ
ρ0 ) = 1.
(c) We consider ρˆ for a single spin 1/2 {|↑i ≡ 10 , |↓i ≡ 01 } under the influence of
the Kraus operators
√
p
p
0
0
0
ˆ1 =
ˆ0 = 1 − p 1
ˆ2 =
√
M
M
M
p
0 0
0
P ˆ† ˆ
where 0 ≤ p ≤ 1 is a probability. Check that k M
ˆ0 for
k Mk = 1 and calculate ρ
a general density matrix ρˆ with matrix elements ρi,j .
(d) We now assume that the process (1) describes
i.e., during ∆t
P a ˆtime evolution,
†
ˆ
the density matrix changes like ρˆ(t + ∆t) = k Mk ρˆ(t)Mk , where p = Γ∆t 1
and Γ > 0 is a rate. What is the time evolution of the matrix elements ρi,j in the
continuum limit ∆t → 0 ? Calculate ρˆ(t) for t → ∞.
(e) Repeat (c) and (d) for the Kraus operators
√
0 0
1−p 0
ˆ
ˆ
M1 = √
.
M0 =
p 0
0
1
Which physical processes are described by the Kraus operators in (c) and (e)?
(4) Singlet state
Consider two spin 1/2 particles in a singlet state:
(5 Punkte)
1
|Ψi = √ (|↑z i1 |↓z i2 − |↓z i1 |↑z i2 ).
2
(a) Show that the singlet state is isotropic, i.e., that its representation is independent
of the choice of the quantization axis. For this, express |Ψi in the σ
ˆx and in the
σ
ˆy -eigenbasis. Here, σ
ˆx , σ
ˆy , σ
ˆz are the Pauli matrices, which are related to the spin
ˆ
~
ˆ
~ = ~σ .
operators via S
2
(b) Consider now a measurement where the spin 1 is measured along an arbitrary
direction ~n1 and the spin 2 is measured along a different direction ~n2 . Calculate
the probability P↑↑ , for obtaining the result “spin up” for both spins. (Hint: Define
an operator that gives the eigenvalue +1 when acting on a spin 1/2 that points
along ~n and 0 for acting on a spin that points in the opposite direction). How does
the isotropy of the singlet state manifest itself in P↑↑ ?