University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 10
Please read and work on the following problems in groups of 3 to 4.
Solutions will be posted on the course webpage.
Announcements
• The 130C web site is:
http://physics.ucsd.edu/∼mcgreevy/w14/ .
Please check it regularly! It contains relevant course information!
• This is going to be the last worksheet of the quarter so let’s finish with fun things.
• It’s been wonderful teaching you all! If you have the chance please fill out the course
evaluation.
Problems
1. Supersymmetry?
Warning this problem may corrupt your mind. Look away if you wish.
Consider a spin- 12 particle on a line and in a magnetic field. The Hamiltonian is given
by:
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1
Ĥ = (− ∂x2 + V (x))1 + B(x)σ z
2
2
The wavefunction of this system would have two components called a spinor.
Suppose your wrote your V (x) and B(x) as the following:
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V (x) = (∂x W )2
2
B(x) = ∂x2 W
Where W (x) is an arbitrary function known as the superpotential. Now define the
following operators:
Q = (P̂x − i∂x W )σ +
Q† = (P̂x + i∂x W )σ −
Where recall σ ± = σ x ± iσ y are raising and lowering operators. Q and Q† are known
as supercharges.
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(a) Show that Q2 = 0 = (Q† )2
(b) Show that {Q, Q† } = 2Ĥ where recall {A, B} = AB + BA
(c) Show that [Q, Ĥ] = 0 = [Q† , Ĥ]
Now what does supersymmetry do? Consider an eigenstate |ψi with energy E.
(d) Compute hψ|Ĥ|ψi. What does this tell us about the ground state of the system?
(e) Now also suppose that W (x) → ∞ as x → ±∞. Using
above constraint on
the ψ+ (x)
the ground state construct the wavefunction ψ0 (x) =
in terms of W (x).
ψ− (x)
Hint: It may be helpful to write:
0
0
0 P̂x − iW 0 (x)
†
Q=
Q =
0
0
P̂x + iW 0 (x) 0
2. Topological Order
Consider a system of qubits defined on the links of the square lattice with periodic
boundary conditions. The Hamiltonian for the system is given by:
X
X
Bp
As −
Ĥ = −
s
p
where
As ≡ Πl∈v(s) Zl
Bp ≡ Πl∈∂p Xl
The first term is a sum over sites for an operator which is a product of legs sticking
out of the site and the second term is a sum over plaquettes for an operator which is
a product of legs which surround the plaquette.
This is easily pictured as:
(a) The Hamiltonian above is solvable. Show explicitly that [As , Bp ] = 0 for a site
and plaquette which neighbor each other.
(b) Let’s understand the ground state of the system. For a given site, which possible
configuration of spins produces the lowest energy? Diagonalize the As operator
for a given site recalling that Z| ↑i = | ↑i and Z| ↓i = −| ↓i
(c) Now suppose we took a marker and colored all the down spins. A continuous line
down spins now looks like a string! Interpret the above in terms of this string
language.
(d) Now consider the action of Bp on a plaquette. For the ground state |ψ0 i we should
take that Bp |ψ0 i = |ψ0 i analogously to the above. What would this do to a given
spin/string configuration on the plaquette?
Putting these two conditions together we arrive at a description of the ground
state in words as ”an equal superposition of all closed string configurations”.
But how many ground states are there? Recall we’ve chosen periodic boundary
conditions which places our lattice model on the torus. Think Pac-Man.
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Figure 1: The toric code visualized
(e) Convince yourself that configuration of spins with a single loop going around the
torus cannot be related to a state with only trivial (not going completely around,
think a single plaquette) loops but has the same energy. How many ground states
are there? This is an example of topological degeneracy.
There’s obviously a lot more to say about this model, its generalizations, and it’s
role both in condensed matter and quantum error correction but I’ve ranted long
enough. Feel free to ask more!
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