 
Summary: Physics 633 Exam 2
Please show all significant steps clearly in all problems.
1. Consider a onedimensional system of noninteracting fermions having the Lagrangian
density
L0 = i¯h
(x, t)
(x, t)
t

(x, t)T (x) (x, t) . (1)
Here T (x) operates on functions of x, but not on state vectors in the occupation number
representation. When the field (x, t) is quantized, it is replaced by the Heisenberg field
operator (x, t), which can be written in terms of the usual Heisenberg destruction operators
ck(t) and a complete orthonormal set of functions k (x):
(x, t) =
k
k (x) ck (t). (2)
Since there are both positive energy states labeled by k+, with k+ > 0, and negative
energy states labeled by k, with k < 0, this can be written more explicitly as
(x, t) =
