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Quantum Mechanics - Ladder Operators

  • Thread starter Tangent87
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  • #1
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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.
 

Answers and Replies

  • #2
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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.
What's [itex][AB,CD][/itex] equal to?
 
  • #3
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What's [itex][AB,CD][/itex] equal to?
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]?
 
  • #4
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redundant
 
  • #5
148
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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]?
No wait of course it's not, what am I thinking!
 

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