Exploring Quantum Physics - Week 6 Question 5, 6

Question:

A state $\Psi\left(x_1,x_2,x_3\right)$ is said to have even parity if $\Psi\left(x_1,x_2,x_3\right) = \Psi\left(-x_1,-x_2,-x_3\right)$ and odd parity if $\Psi\left(x_1,x_2,x_3\right) = -\Psi\left(-x_1,-x_2,-x_3\right)$.

5) What is the parity of the wave function $ |n_1\rangle|n_2\rangle|n_3\rangle $

6) Is it true that same energy implies same parity.

Solution:

This one is simple. For the basic quantum harmonic oscillator solution, even energy is even function, odd energy is odd function.

Together with the basic facts

Product of two even functions is a even function.
Product of two odd functions is a even function.
Product of an even function with an odd function is an odd function.

We deduce the the wavefunction is even parity if $ n_1 + n_2 + n_3 $ is even, and it is odd parity if $ n_1 + n_2 + n_3 $ is odd.

Now question 6 is obvious - by question 4 energy depends on sum of indices - and by question 5 parity depends on sum of indices. So if we have the same energy, we must also have the same parity!