Commuting set of operators (misunderstanding)

  • #1

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I don’t see how the definition of |an> transmorphs into the statement involving the kroneck delta functions.
 

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  • #2
PeterDonis
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I don’t see how the definition of |an> transmorphs into the statement involving the kroneck delta functions.
What definition and what statement? Please give specific references.
 
  • #4
stevendaryl
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So we have two different complete bases:

##|a_n\rangle##

##|b_m\rangle##

If we let ##C_{nm} = \langle b_m|a_n\rangle##, then you can write:

##|a_n\rangle = \sum_m C_{nm} |b_m\rangle##

At this point, they are just defining ##|(a_n) b\rangle## to be ##\sum_m C_{nm} \ \delta_{b, b_m}|b_m\rangle##. The point of the ##\delta_{b, b_m}## is to include only those terms such that ##b_m = b##. It's just a fact that:

##\sum_m C_{nm} |b_m\rangle = \sum_b \sum_m C_{nm} \ \delta_{b, b_m}|b_m\rangle = \sum_b |(a_n) b\rangle##

So:

##|a_n\rangle = \sum_b |(a_n) b\rangle##
 

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