Consider an operator A so that [A A†] 1.

(a) Evaluate the commutators [A†A A] and [A†A A†].

( b ) I f t h e a c t i o n s o f A a n d A † o n t h e s t a t e s a a r e g i v e n b y A a a a 1 a n d

A † a a 1 a 1 a n d i f a a a a , c a l c u l a t e a A a 1 , a 1 A † a a n d a A † A a a n d a A A † a .

(c) CalculateaAA†2 aandaAA†2 a.

Consider a two-dimensional space where a Hermitian operator A is defined by A 1 1 a n d A 2 2 ; 1 a n d 2 a r e o r t h o n o r m a l .

(a) Do the states 1 and 2 form a basis?

(b) Consider the operator B 1 2 . Is B Hermitian? Show that B 2 0.

(c) Show that the products B B † and B † B are projection operators.

(d) Show that the operator B B † B † B is unitary.

(e)Consider C BB†B†B. Show that C 11andC 22.