I Deriving the Commutator of Exchange Operator and Hamiltonian

Samama Fahim
Messages
52
Reaction score
4
exchange operator.JPG


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:
Physics news on Phys.org
$$P_{12}H(1,2)\psi(12) = P_{12}H(1,2){P_{12}}^\dagger P_{12}\psi(1,2) = H(2,1)\psi(2,1)$$
using ##P_{12}^2 = I## and ##{P_{12}}^\dagger = P_{12}##.
 
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...

Similar threads

Back
Top