SHO Operator Expressions

The position ($\hat{X}$) and momentum ($\hat{P}$) operators for a Simple Harmonic Oscillator (SHO) can be expressed using annihilation ($a$) and creation ($a^\dagger$) operators:

• $\hat{X} = \sqrt{\frac{\hbar}{2m\omega}}(a + a^\dagger)$
• $\hat{P} = i\sqrt{\frac{m\omega\hbar}{2}}(a^\dagger - a)$

The product operator $\hat{P}\hat{X}$ is then:

$\hat{P}\hat{X} = \left(i\sqrt{\frac{m\omega\hbar}{2}}(a^\dagger - a)\right) \left(\sqrt{\frac{\hbar}{2m\omega}}(a + a^\dagger)\right) = i\frac{\hbar}{2} (a^\dagger - a)(a + a^\dagger)$

Expanding and using the commutation relation $a a^\dagger = a^\dagger a + 1$ yields:

$\hat{P}\hat{X} = i\frac{\hbar}{2} (a^\dagger a + a^\dagger a^\dagger - a a - a a^\dagger) = i\frac{\hbar}{2} (2a^\dagger a + a^\dagger a^\dagger - a a - 1)$

Matrix Element P X Calculation

We evaluate the matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ using the properties of ladder operators on SHO energy eigenstates $|n\rangle$:

• $a|n\rangle = \sqrt{n}|n-1\rangle$
• $a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$

The matrix elements $\langle m|\hat{O}|n\rangle$ involving the terms in $\hat{P}\hat{X}$ have specific selection rules:

• $\langle m|a^\dagger a|n\rangle \neq 0$ only if $m = n$.
• $\langle m|a^\dagger a^\dagger|n\rangle \neq 0$ only if $m = n+2$.
• $\langle m|a a|n\rangle \neq 0$ only if $m = n-2$.
• $\langle m|1|n\rangle \neq 0$ only if $m = n$.

Non-Zero Conditions Derived

The matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is a sum of terms derived from $\hat{P}\hat{X} = i\frac{\hbar}{2} (2a^\dagger a + a^\dagger a^\dagger - a a - 1)$. For the total matrix element to be non-zero, at least one of its constituent terms must be non-zero.

This requires the quantum number $n$ to satisfy one of the following conditions derived from the operators:

• $m = n$ (from $2a^\dagger a$ and $-1$ terms)
• $m = n+2$ (from $a^\dagger a^\dagger$ term)
• $m = n-2$ (from $-a a$ term)

Therefore, the matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is non-zero when $m = n$ or $m = n \pm 2$.