# Ladder operators to find Hamiltonian of harmonic oscillator

1. Mar 4, 2015

### Maylis

Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book glanced over this part)

The derivation starts at about 8:00

First defining
$$\hat {a}_{\pm} = \frac {1}{\sqrt{2 \hbar m \omega}}( \mp i \hat {p} + m \omega \hat {x})$$

$$\hat {a}_{-} \hat {a}_{+} = \frac {1}{2 \hbar m \omega}(i \hat {p} + m \omega \hat {x})(-i \hat {p} + m \omega \hat {x})$$
$$= \frac {1}{2 \hbar m \omega}( \hat {p}^{2} + i m \omega \hat {p} \hat {x} - i m \omega \hat {x} \hat {p} + m^{2} \omega^{2} \hat {x}^{2})$$
$$= \frac {1}{2 \hbar m \omega}( \hat {p}^{2} - i m \omega (\hat {x} \hat {p} - \hat {p} \hat {x}) + m^{2} \omega^{2} \hat {x}^{2})$$
And I know the commutator $[\hat {x}, \hat {p}] = i \hbar$
$$= \frac {1}{2 \hbar m \omega}( \hat {p}^{2} - i m \omega ( i \hbar) + m^{2} \omega^{2} \hat {x}^{2} )$$
At this point both the video and Griffiths stop, although each do something different
I have no idea how Griffiths goes from this

To this

I will continue with what I was doing,
$$= \frac {1}{2 \hbar m \omega}( \hat {p}^{2} + m \omega \hbar + m^{2} \omega^{2} \hat {x}^{2})$$
$$= \frac {1}{2 \hbar m \omega}\hat {p}^{2} + \frac {1}{2 \hbar^{2} m^{2} \omega^{2}} + \frac {m \omega \hat {x}^{2}}{2 \hbar}$$
And from here I am not sure how both the video and Griffiths conclude

How do I get to the answer from my steps? Since I am not skipping..

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2. Mar 4, 2015

### matteo137

There is an i too much in the term in the middle

The term in the middle is again wrong...

Basically the last two equations you wrote before the Griffiths insert are wrong.
Once you have them right, you know that the Hamiltonian is
$$H=\dfrac{p^2}{2 m} + \dfrac{1}{2} m \omega^2 x^2$$

3. Mar 4, 2015

### Maylis

Woops, I caught my error and now I am able to see how it was done. Thanks.