# Deriving the Commutator of Exchange Operator and Hamiltonian

• I
• Samama Fahim
In summary, the conversation discusses the use of the exchange operator and the Hamiltonian in the boxed equation. It explains how to obtain the right hand side from the left hand side and how the application of the operators affects the result. It also mentions the properties of the exchange operator.
Samama Fahim

In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange operator and ##H(1,2)## is the hamiltonian.

Source: https://books.google.com.pk/books?i...inction as to which particle is which&f=false

Last edited:
$$P_{12}H(1,2)\psi(12) = P_{12}H(1,2){P_{12}}^\dagger P_{12}\psi(1,2) = H(2,1)\psi(2,1)$$
using ##P_{12}^2 = I## and ##{P_{12}}^\dagger = P_{12}##.

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