Question

Which one of the following is \textit{not a linear operator?}

• A. \( x^3 \)
• B. Parity
• C. Time reversal
• D. \( i\hat{p} \)

Hint:

Time-reversal symmetry is often broken in quantum systems with dissipation or magnetic fields.

Answer 1

The Correct Option is C

An operator \( \hat{O} \) is linear if:
\[ \hat{O} (a \psi + b \phi) = a \hat{O} \psi + b \hat{O} \phi \] where \( a \) and \( b \) are scalars.
- \( x^3 \) is a linear function multiplication operator.
- Parity \( \hat{P} \) is a linear operator since it transforms wavefunctions as \( \psi(x) \to \psi(-x) \).
- \( i\hat{p} \) is the momentum operator, which is linear.
However, time reversal \( \hat{T} \) is an anti-unitary operator, meaning:
\[ \hat{T} (a \psi + b \phi) \neq a \hat{T} \psi + b \hat{T} \phi \]