Math 896
Homework # 1 Solutions
Throughout H denotes a nite dimensional Hilbert space and L(H ) the set of linear
operators on H .
1. Prove that any nite dimensional inner product space is complete. (Use that C is
complete.)
Solution: Let V be a nite dime
Math 896
Homework # 3 Solutions
1. Prove that (| |) = | |.
Solution: Let A = | | and B = | |. Let |u , |v H . Also, let |w =
B |u = | |u . Then w| = |u | = u | |. Thus B |u , |v = |w , |v =
w |v = u | |v = u| A |v = |u , A |v . By denition of the adjoint
Math 896
Homework # 4 Solutions
1. Let C be an observable of a quantum system which is in the state . Show that the
expected value of a measurement of C is tr(C).
Solution: Let C = i i |i i| be a spectral decomposition for C . The outcome i
is obtained wi
Math 896
Homework # 5 Solutions
Throughout, H denotes a nite dimensional Hilbert space and L(H ) the Hilbert space
of operators on H .
1. Let f, g : L(H ) L(H ) be continuous maps. Prove that the map f g : L(H ) L(H )
given by (f g )(A) = f (A)g (A) is co