Quantum Mechanics II
Problem Set #1
Due: 16/11/2015
Path Integrals
1. Expectation value of operators
Consider the following Lagrangian with a linear potential
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L = mẋ2 (t) − F x(t)
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where F is some constant force. Using Path integral methods with discrete
spacetime, compute the expectation values of x(t1 ) and ẋ(t1 ) at some intermediate time 0 < t1 < T for a particle emitted from (x, t) = (x0 , 0) and measured
at (x, t) = (xf , T ).
(Hint: Try to think what is the best order in which you want to evaluate the
integrals).
Compare your results to the classical trajectory and explain your results.
2. The Aharonov-Bohm Effect on a Circle
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Consider a particle living on a circle, in the presence of a vector potential A.
The action is
Z
Z
1 2
S=
φ̇ dt − Aφ (φ)dφ .
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On the circle the magnetic field vanishes, but there is an overall magnetic flux
Φ passing through the interior of the circle.
(a) Warmup: Suppose we place the particle near φ = 0 (i.e. its wavefunction
is concentrated near this point), wait for some time T , and then measure
its position. Assume for simplicity that the particle cannot make a 2π
roundtrip on the circle, so there are just two classical paths that can take
it from φ = 0 to φ = π.
Using the path integral approach, calculate the probability to find the
particle near φ = π after time T . Express it in terms of the appropriate free
theory propagator GF (φf , T ; φi , 0), and the flux Φ. Explain, qualitatively,
how changing the flux affects the probability.
Note that the result depends on the flux, even though the magnetic field
is zero on the circle. This is the Aharonov-Bohm effect, and it is a purely
quantum effect.
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(b) Calculate the propagator of this theory, G(φf , T ; φi , 0).
(c) Express the propagator in terms of the theta function,
θ(ν, τ ) =
n=∞
X
exp(πin2 τ + 2πinν) .
n=−∞
Use the identity
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1
θ(ν/τ, − ) = (−iτ ) 2 exp(πiν 2 /τ )θ(ν, τ )
τ
and calculate the spectrum of this theory. Express it in terms of the flux
Φ. The fact that the spectrum depends on Φ is another aspect of the
Aharonov-Bohm effect.
(d) If we measure the spectrum of the particle, what can we say about Φ?
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