Quantum Physics

An operator

%5Chat%7BO%7D

is linear if:

%5Chat%7BO%7D%20(a%20%5Cpsi%20%2B%20b%20%5Cphi)%20%3D%20a%20%5Chat%7BO%7D%20%5Cpsi%20%2B%20b%20%5Chat%7BO%7D%20%5Cphi

where

a

and

b

are scalars.
-

x%5E3

is a linear function multiplication operator.
- Parity

%5Chat%7BP%7D

is a linear operator since it transforms wavefunctions as

%5Cpsi(x)%20%5Cto%20%5Cpsi(-x)

.
-

i%5Chat%7Bp%7D

is the momentum operator, which is linear.
However, time reversal

%5Chat%7BT%7D

is an anti-unitary operator, meaning:

%5Chat%7BT%7D%20(a%20%5Cpsi%20%2B%20b%20%5Cphi)%20%5Cneq%20a%20%5Chat%7BT%7D%20%5Cpsi%20%2B%20b%20%5Chat%7BT%7D%20%5Cphi

Chapter Questions
1 MCQs