- #1
guitarphysics
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Homework Statement
In Zettili's QM textbook, we are asked to find the trace of an operator [itex] |\psi><\chi| [/itex]. Where the kets [itex]|\psi> [/itex] and [itex]|\chi> [/itex] are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets.
Homework Equations
[itex] Tr(\hat{A}\hat{B})=Tr(\hat{B}\hat{A}) [/itex]
The Attempt at a Solution
The solution is given in the textbook. Zettili says the following:
Since [itex] Tr(\hat{A}\hat{B})=Tr(\hat{B}\hat{A}) [/itex], we know that [itex] Tr(|\psi><\chi|)=Tr(<\chi|\psi>) [/itex].
From then on calculating the trace is no problem.
My question is if this is a valid argument. It seems to me that the equation Zettili is starting from talks about the trace of two operators not caring about the order the operators are multiplied in. However, the trace we want to find in this problem is the trace of a single operator, namely [itex] |\psi><\chi| [/itex]. The reversal of the product from ket-bra to bra-ket doesn't seem to be the same concept as changing the order of multiplication of two operators. Can anybody help me out here?
(If anyone's curious, this is Problem 2.1 c on page 133-134 of the second edition of Quantum Mechanics: Concepts and Applications.)