The discussion revolves around the commutation relations in quantum mechanics, specifically focusing on the operators \(N = a^\dagger a\) and \(K_r = \frac{a^\dagger^r a^r}{r!}\). Participants are attempting to demonstrate that these operators commute.
Participants are actively engaging with the problem, sharing insights on the commutation relations and exploring the implications of their findings. There is a mix of attempts to clarify mathematical expressions and to understand the requirements for proving certain identities.
Some participants question the sufficiency of demonstrating a specific summation involving \(K_r\) to establish a broader identity related to the vacuum state. There is an indication of confusion regarding the implications of their calculations.
Tangent87 said:I'm trying to show that [tex]N=a^\dagger a[/tex] and [tex]K_r=\frac{a^\dagger^r a^r}{r!}[/tex] commute. So basically I need to show [tex][a^\dagger^r a^r,a^\dagger a]=0[/tex]. I'm not quite sure what to do, I've tried using [tex][a,a^\dagger][/tex] in a few places but so far haven't had much success.
latentcorpse said:What's [itex][AB,CD][/itex] equal to?
Tangent87 said:Ah, thanks latentcorpse!
[tex][AB,CD]=A[B,CD]+[A,CD]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B[/tex]
Also, if I want to show [tex]\sum_{r=0}^\infty (-1)^r K_r=|0><0|[/tex] is it sufficient to show [tex]\sum_{r=0}^\infty (-1)^r K_r|n>=|0><0|n>=0[/tex]?