# Thread: Harmonic Oscillator Finding <x^2>

1. ## Harmonic Oscillator Finding <x^2>

Dear Experts,

I am having difficulty undersanding the pattern when working with operators which is part of the work to evaluate <x^2>. See attached jpg.

Likewise the same with evaluating $\displaystyle a_{+}a_{-}\psi_{n}(x)$

Thanks

2. Originally Posted by bugatti79
Dear Experts,

I am having difficulty undersanding the pattern when working with operators which is part of the work to evaluate <x^2>. See attached jpg.

Likewise the same with evaluating $\displaystyle a_{+}a_{-}\psi_{n}(x)$

Thanks
$\displaystyle a_{+} \sqrt{n+1} \, \psi_{n+1} = \sqrt{n+1} \, a_{+} \psi_{n+1}$ and $\displaystyle a_{+} \psi_{n+1} = \sqrt{n+2} \, \psi_{n+2}$ by definition of the raising operator.

3. On that basis,

I calculate $\displaystyle a_{+}a_{-} \psi_{n}=\sqrt{n} a_{+} \psi_{n-1}=\sqrt{n} \sqrt{n+1} \psi_{n}$

$\displaystyle a_{-} a_{+}\psi_{n}=\sqrt{n+1} a_{-} \psi_{n+1}=\sqrt{n+1} \sqrt{n} \psi_{n}$

$\displaystyle a_{-}^2 \psi_{n}=\sqrt{n} a_{-} \psi_{n-1}=\sqrt{n} \sqrt{n-1} \psi_{n-2}$

The first 2 calculations are wrong by my lectures notes, so obviously dont understand the definition.

For example, it there is an adding operator, I will sum it with n. Ie, adding a_+ operator to n+1 gives n+2, subtracting a_- to n gives n-1 etc.....

4. Originally Posted by bugatti79
On that basis,

I calculate $\displaystyle a_{+}a_{-} \psi_{n}=\sqrt{n} a_{+} \psi_{n-1}=\sqrt{n} \sqrt{n+1} \psi_{n}$

$\displaystyle a_{-} a_{+}\psi_{n}=\sqrt{n+1} a_{-} \psi_{n+1}=\sqrt{n+1} \sqrt{n} \psi_{n}$

$\displaystyle a_{-}^2 \psi_{n}=\sqrt{n} a_{-} \psi_{n-1}=\sqrt{n} \sqrt{n-1} \psi_{n-2}$

The first 2 calculations are wrong by my lectures notes, so obviously dont understand the definition.

For example, it there is an adding operator, I will sum it with n. Ie, adding a_+ operator to n+1 gives n+2, subtracting a_- to n gives n-1 etc.....
Use the definitions more carefully. eg. $\displaystyle a_{+} \psi_{n-1} \neq \sqrt{n+1} \psi_{n}$ etc.

5. Originally Posted by mr fantastic
Use the definitions more carefully. eg. $\displaystyle a_{+} \psi_{n-1} \neq \sqrt{n+1} \psi_{n}$ etc.
I have it now, when I compare
$\displaystyle a_{+}\psi_{n}=\sqrt{n+1} \psi_{n+1}$ and $\displaystyle a_{+} \psi_{n-1}$ together I know hows it derived.

Cheers