Commutation Relation: Hi Parity Operator?

In summary, the translation operator and the parity operator do not commute. On coordinate basis, the translation operator is defined as T_{a} = e^{- i a p} and the parity operator is given by \pi = \int dy \ |-y \rangle \langle y |. It can be shown that T_{a} \ \pi \neq \pi \ T_{a} in general, as demonstrated by the different actions on a wave function.
  • #1
fatema
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hi, do the translation operator commute with parity operator?
 
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  • #2
Parity operator ##\hat\pi## is defined such that when acting on a position eigenvector ##|x\rangle## to be ##\hat\pi|x\rangle = |-x\rangle##. Start from
$$\hat x \hat\pi = \int dx' \ \hat x \hat\pi |x'\rangle \langle x'|$$
and with the help of the eigenvalue relation ##\hat x|x'\rangle = x'|x'\rangle##.
 
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  • #3
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##.
 
  • #4
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##.
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##.
 
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  • #5
fatema said:
hi, do the translation operator commute with parity operator?
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]
 
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1. What is a commutation relation?

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.

2. What is the parity operator?

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.

3. How is the parity operator related to commutation relations?

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.

4. What is the significance of commutation relations in quantum mechanics?

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.

5. How do commutation relations affect the uncertainty principle?

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.

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