I encountered this part in Griffith's Introduction to Quantum Mechanics that I have been unable to figure out. It is probably obvious, but I am not seeing it. I probably need more practice with operators in order to have it fully understood.(adsbygoogle = window.adsbygoogle || []).push({});

Equation 2.64 in the second edition states:

[itex]\int_{-\infty}^{\infty}{f^*}(a_{\pm}g)dx = \int_{-\infty}^{\infty}(a_{\mp}f)^*gdx[/itex]

It starts out by making the substitution where [itex]a_\pm = \frac{1}{\sqrt{2\hbar m\omega}}(\mp\hbar\frac{d}{dx}+m\omega x)[/itex]...

[itex]\int_{-\infty}^{\infty}f^*(a_\pm g)dx=\frac{1}{\sqrt{2\hbar m\omega}}\int_{-\infty}^{\infty}f^*(\mp\hbar\frac{d}{dx}+m\omega x)gdx[/itex]

From there, it states that the integrals must exist, which means that f(x) and g(x) must go to zero. This makes sense, since what is being done is normalizing, and a normalizable function must have values of 0 at the extrema.

What I don't understand is the next step, whereby it states that integration by parts takes [itex]\int f^*(\frac{dg}{dx})dx[/itex] to [itex]-\int(\frac{df}{dx})^*gdx[/itex]

How did they arrive at this? I tried writing it out, but I think I am not dealing with the operators correctly. What are the steps to arrive at this conclusion? It says that it has to do with the fact that the function values at the extrema disappear, but I am not seeing it when I write it out. It just ends up getting messy. The book skips over these steps, but I always include the explicit steps in my notes.

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# Additional understanding needed on proof involving Hermitian conjugates

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