Raising and lowering operators

[mkhurana]

Homework Statement

Can anyone please explain why any term which is a product of 4 raising and lowering operators with a lowering operator on the extreme right (eg. A-A+A+A-) has zero expectation value in the ground state of a harmonic oscillator?

Homework Equations

The Attempt at a Solution

 

[Cyosis]

What happens if you let the lowering operator work on $\psi_1$? What do you get when you let the lowering operator work on $\psi_0$?

Calculate $a \psi_1$ and $a \psi_0$. Does $a \psi_0$ exist?

 

[mkhurana]

I get it now... thank you. When the lowering operator operates on v1, I get v0. When it operates on v0, I get zero. A- v0 doesn't exist so the rest of the operators in the product have nothing to operate on.

Cheers.