The Parity Operator: Find the average value of the parity.

[latnoa]

Homework Statement

A particle of mass m moves in the potential energy V(x)= $\frac{1}{2}$ mω²x²
. The ground-state wave function is
$\psi$₀(x)=($\frac{a}{π}$)^1/4e^-ax2/2
and the first excited-state wave function is
$\psi$₁(x)=($\frac{4a^3}{π}$)^1/4e^-ax2/2
where a = mω/$\hbar$

What is the average value of the parity for the state

ψ(x)=$\frac{\sqrt{3}}{2}$$\psi$₀(x)+ $\frac{1-i}{2\sqrt{2}}$$\psi$₁(x)

Homework Equations

∏$\psi$(x)=$\psi$(-x)
∏$\psi$_λ(x) = $\psi$_λ(x)

The Attempt at a Solution

First off I'm extremely confused on how to approach this to the point of where I don't know what I'm solving for so I'm someone can help me understand the problem and what it's asking me to do.

I just finished reading the parity operator section and all I understand was that ∏ inverts the coordinates of the wave function. I also got that ∏^2 is the identity vector which means that the eigenvalues have to be ±1 and that an even eigenfunction corresponds to 1 while an odd function corresponds to -1 and The eigenfunction of the parity operator form a complete set. That's it from the book but it shows no examples or anything remotely close to what this questions asks.

Am I trying to solve for ∏? Why are the ground state and first state included in this problem? Please help. Anything will be helpful.

 

[vela]

Homework Statement

A particle of mass m moves in the potential energy V(x)= $\frac{1}{2}$ mω²x²
. The ground-state wave function is
$\psi$₀(x)=($\frac{a}{π}$)^1/4e^-ax2/2
and the first excited-state wave function is
$\psi$₁(x)=($\frac{4a^3}{π}$)^1/4e^-ax2/2
where a = mω/$\hbar$

What is the average value of the parity for the state

ψ(x)=$\frac{\sqrt{3}}{2}$$\psi$₀(x)+ $\frac{1-i}{2\sqrt{2}}$$\psi$₁(x)

Your expression for $\psi_1(x)$ is incorrect. There should be a factor of x in there.

First off I'm extremely confused on how to approach this to the point of where I don't know what I'm solving for so I'm someone can help me understand the problem and what it's asking me to do.

I just finished reading the parity operator section and all I understand was that ∏ inverts the coordinates of the wave function. I also got that ∏^2 is the identity vector which means that the eigenvalues have to be ±1 and that an even eigenfunction corresponds to 1 while an odd function corresponds to -1 and The eigenfunction of the parity operator form a complete set. That's it from the book but it shows no examples or anything remotely close to what this questions asks.

Am I trying to solve for ∏? Why are the ground state and first state included in this problem? Please help. Anything will be helpful.

You're being asked to find $\langle \psi | \hat{\Pi} | \psi \rangle$.

Think about how you'd calculate the average energy $\langle \psi | \hat{H} | \psi \rangle$ of the state. You're being asked to do the same sort of calculation except this time with the parity operator instead of the Hamiltonian.

Hint: The energy eigenstates of the harmonic oscillator are also parity eigenstates.

 

[chessmeister]

Use Expansion Coefficients

First off, $\psi$₀ and $\psi$₁ are eigenfunctions of $\Psi$. You can see their expansion coefficients given in the equation for $\Psi$, I will refer to these as c_n

The average value of the parity, or <$\Pi$> will be Ʃ|c_n|²a_n, where the a's are the eigenvalues of the given eigenfunctions.

To find the eigenvalues use $\prod\psi = a\psi$

This should simplify the problem quite a bit -- it's actually very straightforward if you approach it this way, and you don't have to deal with any messy integrals.