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fatema
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hi, do the translation operator commute with parity operator?
I was thinking since it's easy to show that the two operators do not commute, why not push it a little further to know the exact relation between ##TP## and ##PT##.hilbert2 said:It's enough to find a single example of a translation ##T\psi (x) = \psi (x+\Delta x)## and a function ##\psi (x)## for which ##TP\psi (x)## and ##PT\psi (x)## don't have the same value at some point ##x##.
On coordinate basis [itex]|x \rangle[/itex], the action of translation operator [itex]T_{a} = e^{- i a p}[/itex] is given by [tex]T_{a} | x \rangle = | x + a \rangle \ .[/tex] And in the same basis, the parity operator is given by [tex]\pi = \int dy \ |-y \rangle \langle y | \ .[/tex] Now it is an easy exercise to show that [tex]T_{a} \ \pi = \int dy \ |y \rangle \langle - y + a | \ ,[/tex] [tex]\pi \ T_{a} = \int dy \ |y \rangle \langle - y - a |\ .[/tex] So, in general they do not commute. This becomes clear if you use the above two equations to evaluate the action on the wave function [tex]\left( T_{a} \ \pi \Psi \right) ( - x) = \Psi (x + a) \ ,[/tex] [tex]\left( \pi \ T_{a} \Psi \right) ( - x) = \Psi ( x - a) \ .[/tex]fatema said:hi, do the translation operator commute with parity operator?
A commutation relation is a mathematical relationship between two operators that represent physical observables. It describes how these operators behave when applied to a system in terms of their order of operations.
The parity operator is an operator that determines the symmetry of a system under spatial inversion. It changes the sign of all spatial coordinates, indicating whether the system is symmetrical (even) or anti-symmetrical (odd) under inversion.
The commutation relation between the parity operator and another operator describes how these two operators behave when applied to a system. For example, the commutation relation between the parity operator and the position operator determines whether the position operator is even or odd under spatial inversion.
Commutation relations are important in quantum mechanics because they determine the fundamental properties and behavior of physical systems. They allow us to make predictions about the outcomes of measurements and understand the relationship between different physical quantities.
The uncertainty principle states that certain pairs of physical quantities, such as position and momentum, cannot be measured with arbitrary precision simultaneously. This is due to the commutation relations between these operators, which dictate that their uncertainties cannot be reduced to zero at the same time.