(adsbygoogle = window.adsbygoogle || []).push({}); [QM] Hamiltonian and symmetries

1. The problem statement, all variables and given/known data

Let there be the hamiltonian:

[tex]H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2(x^2+y^2+z^2)+kxyz+\frac{k^2}{\hbar \omega}x^2y^2z^2[/tex]

Find the expectation value of the three components of [tex]\vec r[/tex] in the ground state using ONLY the symmetry properties of the hamiltonian.

2. Relevant equations

3. The attempt at a solution

I define this parity:

[tex]\Pi_{xy}: x\rightarrow-x\ \ \ \ y\rightarrow-y[/tex]

Then the hamiltonian commutes with this parity: [tex][H, \Pi_{xy}]=0[/tex]

The ground state is not degenerate, so it has a definite parity with respect to [tex]\Pi_{xy}[/tex]:

[tex]<gs|x|gs>=<gs|\Pi_{xy}\Pi_{xy}x\Pi_{xy}\Pi_{xy}|gs>=-<gs|\Pi_{xy}x\Pi_{xy}|gs>=-<gs|x|gs>[/tex]

So <gs|x|gs>=0;

Is it right?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Hamiltonian and symmetries

**Physics Forums | Science Articles, Homework Help, Discussion**