Download Problem set VI Problem 6.1 Problem 6.2 Problem 6.3 Problem 6.4

Quantum Engineering [FKA132] 2014
Problem set VI
Problem 6.1
Consider a beam of spin 12 particles in a Stern-Gerlach experiment, having spin aligned in the positive x
direction, i.e. |Sx , +i. When this beam goes through a Stern-Gerlach apparatus with an inhomogeneous
magnetic field in the z-direction (SGz), it splits into two beams of equal intensity, i.e.
| h+| Sx , +i |2 = | h−| Sx , +i |2 =
1
2
(1)
where |±i ≡ |Sz , ±i are the eigenstates of Sz . Using only this knowledge as well as similar knowledge of
what happens for beams and apparatus’ oriented in other directions, derive expressions for |Sx , ±i and
|Sy , ±i and the operators Sx and Sy in terms of the eigenstates of Sz . At this point the expressions may
contain no more than two relevant, undetermined phase factors. Discuss how they can be determined.
What commutation relations do the operators Sx , Sy and Sz obey?
Problem 6.2
A beam of spin
1
2
atoms goes through a series of Stern-Gerlach-type measurements as follows:
a) The first measurement accepts sz = h̄/2 atoms and rejects sz = −h̄/2 atoms.
b) The second measurement accepts sn = h̄/2 atoms and rejects sn = −h̄/2 atoms, where sn is the
eigenvalue of the operator S · n̂, with n̂ making an angle β in the xz-plane with respect to the
z-axis.
c) The third measurement accepts sz = −h̄/2 atoms and rejects sz = h̄/2 atoms.
What is the intensity of the final sz = −h̄/2 beam when the sz = h̄/2 beam surviving the first
measurement is normalized to unity? How must we orient the second measuring apparatus if we are
to maxim’ize the intensity of the final sz = −h̄/2 beam?
Problem 6.3
An electron is initially in its spin-up state (relative to the z-direction) in zero magnetic field. At t = 0,
~ = B0 (1, 1, 1) is applied.
a magnetic field B
a) Describe the time evolution of the spin state of the electron.
b) Calculate the probability P↑ (t) for finding the electron in the spin-up state, and the probability
P↓ (t) for finding the electron in the spin-down state, after a time t.
Problem 6.4
A two-level system is governed by the Hamiltonian
H = a(|1ih1| − |2ih2| + |1ih2| + |2ih1|),
where a is a number (having a dimension of energy). Calculate the eigenenergies and the corresponding
eigenfunctions (eigenkets) as a linear combination of the basis states |1i and |2i.
1
Problem 6.5
A two-level system is governed by the Hamiltonian H0 , defined by
1 0
H0 =
0 2
~ = B0 x̂ is applied (in the
where 1 and 2 are the eigenenergies of the system. If a magnetic field B
x-direction), calculate the new eigenenergies.
2