Ladder Operators - I need help.

Pmb

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Apr 2009
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Boston's North Shore
I can't see his justification of Eq. [2.63] on page 47. I know its true but long ago forgot why and I can't see Griffith writing anything to justify it. Help/thoughts?
 
Oct 2017
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Glasgow
I can't see his justification of Eq. [2.63] on page 47. I know its true but long ago forgot why and I can't see Griffith writing anything to justify it. Help/thoughts?
I think it's considered as a consequence of Eq. 2.61, which shows that the next/previous wave function in the ladder is obtained by applying the operator and renormalising. Therefore, the proportionality constant, \(\displaystyle c_n\), is just the inverse of \(\displaystyle A_n\), repurposed for the algebraic method. Considering how \(\displaystyle c_n\) is linked to \(\displaystyle A_n\), it is unsurprising, then, that the result is a square root of a factorial.