Parity Operator and odd potential function.

• sudipmaity
In summary, the conversation discusses how to determine the parity of a wave function in a potential that is either even or odd. It is mentioned that the parity operator must commute with the Hamiltonian for the wave function to have even or odd parity. However, in the case of an odd parity potential, the Hamiltonian does not commute with the parity operator, indicating that the wave function may not have even or odd parity. The question of a possible typographical error in the homework is also brought up, and an example of a one-dimensional potential with an even potential is provided for further clarification.
sudipmaity

Homework Statement

This is a university annual exam question: Show that for a potential V (-r)=-V (r) the wave function is either even or odd parity.

The Attempt at a Solution

We can determine whether a wavefunctions' parity is time independent based on if the parity operator commutes with the hamiltonian.As far as I know for even parity potential V (-r)=V (r), hamiltonian and parity will commute and we can show that wave function to have even or odd parity.But for odd parity potential as given by the question the hamiltonian won't commute with the parity as it no longer remains invariant under parity operation.So wavefuntion should not have even or odd parity.I talked with some friends and some of them think that there might be some printing mistake in the question. What do you guys have to say.

To help see if the problem might have a typographical error, consider the one dimensional potential V(x) shown below. Note, V(-x) = -V(x).

Imagine a particle "trapped" in the region -a < x < 0. Consider roughly what the wavefunction for the ground state would look like. Would the wavefunction be even, odd, or neither?

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1. What is a parity operator?

A parity operator is a mathematical operator that determines the symmetry of a physical system. It is used to determine if a system is symmetric under a spatial inversion, meaning that it remains unchanged when its coordinates are reversed.

2. How does a parity operator work?

A parity operator works by flipping the sign of the coordinates in a given system. If the system remains unchanged after this transformation, it is considered to have parity symmetry. If it changes, it does not have parity symmetry.

3. What is an odd potential function?

An odd potential function is a mathematical function that is symmetric under a spatial inversion. This means that the potential function remains unchanged when its coordinates are reversed, and the sign of the function is also flipped.

4. How does an odd potential function relate to parity?

An odd potential function is closely related to parity because it has the same symmetry properties as a parity operator. This means that a system with an odd potential function will have parity symmetry, and vice versa.

5. What are some examples of systems with parity symmetry?

Some examples of systems with parity symmetry include a particle in a uniform magnetic field, a simple harmonic oscillator, and a free particle in one dimension. These systems have odd potential functions and are symmetric under spatial inversion.

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