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Department of Physics
Examination paper for TFY4205 Quantum Mechanics II Examination date: 28.11.20
Examination time (from-to): 0900-1300
Permitted examination support material: A / All support material is allowed Academic contact during examination: Jacob Linder
Phone: 951 73 515
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Problem 1 [5 points]
Consider a spin 1/2 particle with zero electric charge in a magnetic field. The field is fixed at a polar angle with respect to the -axis, but is rotating slowly in the -direction. Upon completing one full rotation, the tip of the magnetic field has thus traced out a circle.
When a Hamilton-operator has a time-dependence represented by a vector of parameters , the Berry- connection is defined as
a) Show that the Berry potential is a purely real quantity.
b) Compute the Berry potential for the spin-up state with respect to the instantaneous magnetic field direction.
Hint: with a magnetic field oriented at an angle with respect to the -axis, the state vector for a spin-state parallell to the magnetic field direction is:
where is the state vector for when the spin is oriented along the axis.
c) Compute the Berry-phase defined by
where is the closed contour describing one completed rotation of the magnetic field tip.
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Maximum marks: 5
Assume is a solution of the Schrödinger equation
and that it has the following asymptotic behavior for large :
Above, is the scattering potential and is the reduced mass.
Let the Green function be the solution to the equation .
a) What is the general solution of the Schrödinger equation given above when ?
b) Write down the formal, exact solution for when which satisfies the boundary condition (asymptotic behavior) given initially in the problem. Explain how you arrived at this result.
(c) Consider a particle with mass scattered by a potential where Give an
expression for the differential scattering cross section to lowest order in by using the Born approximation.
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Maximum marks: 4
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Problem 3 [4 points]
Consider an electron at rest. A magnetic field is present which has a constant amplitude, but a time-dependent direction:
It thus rotates with an angular velocity . The Hamilton-operator for the system describes the interaction between the spin and the magnetic field (anomalous Zeeman effect) as follows:
Here, and we defined
a) What are the normalized eigenspinors and eigenvalues that satisfy ? b) The exact solution of the time-dependent Schrödinger equation is given as
where . Here, we have assumed that the electron at is in a spin-up
state along :
Compute the transition probability to a spin-down state with respect to the
instantaneous magnetic field at a given time for an arbitrary angular velocity. Write down this result. Then consider two different limits:
Comment on the physical interpretation of your result in both limits.
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Consider scattering by a potential where are both positive, real constants. The incident particle has very low energy so that we may set where is the wavevector of the incident particle.
For such low energies, only the zeroth partial wave should contribute to scattering. Define where is the scattering phase.
It can then be shown that the solution for the wavefunction in the region may be written as:
where A is a coefficient to be determined by matching the above wavefunction to the wavefunction in the region . In this inner region, the general solution of the wavefunction is
.
Here, are two additional coefficients to determined.
a) The quantity determines the differential scattering cross section in the low-energy limit. Derive an explicit expression for for the system described. Note that the expression may be simplified by using
.
b) Compute the total scattering cross section in the low-energy limit. Comment on the result you get when . In particular, which physical scenario is it equivalent to?
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Maximum marks: 5
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Problem 5 [4 points]
a) Consider the state
where .
Using the qubit-states and as basis vectors, the Pauli matrices in this representation are written in the standard way, for instance . Compute the reduced density matrix (operator, to be strict)
using the qubit-states as basis vectors. Is the state entangled? Justify your answer.
b) Consider now a system described by a density operator
where is given in a). Compute the expectation value of the the observable quantity where is the identity operator.
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Maximum marks: 4
Miscellaneous topics. The answer to each question doesn't have to be more than, at most, 4-5 lines.
a) Depending on which gauge you choose for the vector potential in the Landau level problem (free electrons in a magnetic field), you get different forms of the wavefunction. These wavefunctions do not have the same absolute value square, which seems to suggest that the physical probability density is affected by our choice of gauge. Resolve this apparent problem.
b) Explain in a consise manner why a 2-particle system with spin-independent interactions nevertheless can give rise to an energetically preferred spin-configuration of the particles.
c) Which classical system can in some sense be thought of as analogous to coherent photon states?
How can coherent photon states be produced experimentally?
How does the uncertainty in the expectation value of the electric field in such a state scale with the amplitude of the state (i.e. the eigenvalue of the coherent state produced as the annihilation operator acts on it)?
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Maximum marks: 4.5