Proving the following properties

[Abhishek11235]

Mentor note: Member warned that an attempt must be shown.
1. Homework Statement

This question is from book Afken Weber, Mathematics for Physicist.
An operator $T(t + ε,t)$ describes the change in the wave function from t to t + $\epsilon$ . For $\epsilon$ real and small enough so that $\epsilon^{2}$ may be neglected,

$$T(t+\epsilon, t)= 1 - \frac{i * \epsilon* \text H(t)}{h} $$

WHERE H(t) is hamiltonian, i is complex number $i = \sqrt-1$, h is constant. Prove that if

• T is unitary matrix, H is hermitian
• H is hermitian, T is Unitary

How do I prove this Property. Also what is structure of right hand side of the equation i.e how do I visualise R.H.S of equation?

 

[DrClaude]

How do I prove this Property.

Start from the initial condition: "if T is a unitary matrix," start with the assumption that T is unitary, write the condition for unitarity of a matrix (applied to T), and continue the derivation until you reach a condition on H. For the second case, you need to do it the other way around.

Also what is structure of right hand side of the equation i.e how do I visualise R.H.S of equation?

It is a matrix. 1 will be the identity matrix, and the second term on the RHS is the Hamiltonian matrix multiplied by some scalars.