Introduction to Quantum Mechanics, Spring 2015
Problem Set 14
Due Tuesday, April 21
Problem 1: Show that, assuming the standard Poisson brackets
{φ(x), π(x0 )} = δ(x − x0 ), {φ(x), φ(x0 )} = {π(x), π(x0 )} = 0
the Hamilton’s equations for the Hamiltonian
Z
1 2
(π + (∇φ)2 + m2 φ2 )d3 x
h=
2
3
R
are equivalent to the Klein-Gordon equation for a classical field φ.
Problem 2: For non-relativistic quantum field theory of a free particle in three
dimensions, find the quadratic functions Lj on the phase space of complex
valued functions on R3 corresponding to the angular momentum operators on
the single-particle space H1 . Show that these functions Poisson-commute with
the Hamiltonian function. Find the corresponding quantized operators Lˆj , show
that these commute with the Hamiltonian operator, and satisfy the Lie algebra
commutation relation
[L̂1 , L̂2 ] = iL̂3
Problem 3: For the complex-valued scalar field theory, show that the charge
b has the following commutators with the fields
operator Q
b = −φ,
b [Q,
b φ]
b φb† ] = φb†
[Q,
and thus that φb on charge eigenstates reduces the charge eigenvalue by 1,
whereas φb† increases the charge eigenvalue by 1.
Problem 4: In class we showed that taking two real free scalar fields, one could
make a theory with SO(2) symmetry, and we found the charge operator Q that
gives the action of the Lie algebra of SO(2) on the state space of this the ory.
Instead, consider two complex free scalar fields, and show that this theory has
a U (2) symmetry. Find the four operators that give the Lie algebra action for
this symmetry on the state space, in terms of a basis for the Lie algebra of U (2).
Note that this is the field content and symmetry of the Higgs sector of the standard model (where the difference is that the theory is not free, but interacting,
and has a lowest energy state not invariant under the symmetry).
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