Sakurai Exercise 2.17 (Harmonic Oscillator, Ladder operators) [closed]

Question

I the field of the harmonic oscillator and ladder operators I am trying to solve exercise 2.17 from Sakurai and want to proof the following relation $$ \langle x^{2n} \rangle = (2n - 1)!! \langle x^2 \rangle^n $$ as a useful identity should be $$ \int_{-\infty}^\infty dx x^{2n} e^{-cx^2} = \left(-\frac{d}{dc}\right)^n \int_{-\infty}^\infty dx e^{cx^2} $$

Until now I tried to plug in the wave function of the harmonic oscillator but with no prosperous results.

Does anybody has a clue how to proof the first relation?

Answer 1

I'm sorry but it's a easy problem just using the ladder operator.You need to let x applied on the wavefunction one by one and the rest is easy.

Applying the following property which can easily derived from the ladder operator (or just using the wavefunction,either way is ok,the specific procedure can be found in Griffiths' book):

$\displaystyle x\psi_n(x)=\frac{1}{\alpha}[\sqrt{\frac{n}{2}}\psi_{n-1}(x)+\sqrt{\frac{n+1}{2}}\psi_{n+1}(x)]$

where $\displaystyle\alpha\equiv\sqrt{\frac{m\omega}{\hbar}}$.

And I suppose the rest you can do it on your own.Just simply apply x on the bra and the ket and make sure they're in the same state.