Deriving the Commutator of Exchange Operator and Hamiltonian

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.
  • #1
Samama Fahim
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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
 
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  • #2
$$P_{12}H(1,2)\psi(12) = P_{12}H(1,2){P_{12}}^\dagger P_{12}\psi(1,2) = H(2,1)\psi(2,1)$$
using ##P_{12}^2 = I## and ##{P_{12}}^\dagger = P_{12}##.
 
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FAQ: Deriving the Commutator of Exchange Operator and Hamiltonian

What is the commutator of an operator?

The commutator of two operators A and B, denoted as [A, B], is defined as [A, B] = AB - BA. It measures the extent to which the two operators fail to commute, meaning that the order in which they are applied affects the outcome. In quantum mechanics, the commutator is crucial for understanding the relationships between observables and their corresponding operators.

What is the exchange operator in quantum mechanics?

The exchange operator is a specific type of operator used in quantum mechanics, particularly in the context of identical particles. It swaps the states of two particles. For example, if we have two particles described by states |ψ₁⟩ and |ψ₂⟩, the exchange operator, denoted as P, acts on these states such that P|ψ₁⟩|ψ₂⟩ = |ψ₂⟩|ψ₁⟩. This is essential for understanding the symmetrization postulate for bosons and the antisymmetrization for fermions.

What is the Hamiltonian in quantum mechanics?

The Hamiltonian is an operator that represents the total energy of a quantum system, encompassing both kinetic and potential energy. It is central to the Schrödinger equation, which governs the time evolution of quantum states. The Hamiltonian can be expressed in various forms depending on the system, but it typically includes terms that account for interactions between particles and external potentials.

Why is it important to derive the commutator of the exchange operator and the Hamiltonian?

Deriving the commutator of the exchange operator and the Hamiltonian is important because it provides insights into the symmetries of the system and the behavior of identical particles. This commutation relation can reveal whether the exchange of particles affects the energy of the system, which is crucial for understanding phenomena such as Bose-Einstein condensation and Fermi-Dirac statistics. It also aids in determining the conserved quantities in the system.

How do you derive the commutator of the exchange operator and the Hamiltonian?

To derive the commutator [P, H], where P is the exchange operator and H is the Hamiltonian, you apply both operators in sequence to a state and analyze the resulting expressions. You typically start with the action of the Hamiltonian on a two-particle state, and then apply the exchange operator. By carefully expanding the expressions and simplifying, you can show how the Hamiltonian interacts with the exchange operator. The result often reveals whether the two operators commute or not, which has implications for the system's physical properties.

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