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Proving the following properties

  • #1
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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?
 
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Answers and Replies

  • #2
DrClaude
Mentor
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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.
 

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