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Dirac Notation Help

  1. Sep 11, 2004 #1
    We are working on Dirac notation in my quantum class, and for the most part I see that it is a very easy way to do problems. But I am still getting stuck on how to deal with a few things on my current homework assignment. These come out of chapter 2 in the Cohen-Tannoudji book if you want to look them up.

    #1. |[tex]\phi_{n}[/tex]> are eigenstates of a Hermitian operator H and they form a discrete orthonormal basis. The operator U(m,n) is defined by U(m,n)=|[tex]\phi_{m}[/tex]><[tex]\phi_{n}[/tex]|.

    b. Calculate the commutator [H,U(m,n)].

    I'm not really sure how to deal with this in such a general case. I get to the first step: [tex]H|\phi_{m}><\phi_{n}|-|\phi_{m}><\phi_{n}|H[/tex]

    but I don't know where to go from there.

    e. Let A be an operator, with matrix elements [tex]A_{mn}=<\phi_{m}|A|\phi_{n}>[/tex]

    Prove the relation:[tex]A=\Sigma A_{mn}U(m,n)[/tex]

    If I start with [tex]A_{mn}=<\phi_{m}|A|\phi_{n}>[/tex], is it legal to do this:

    [tex]A_{mn}|\phi_{m}><\phi_{n}|=<\phi_{m}|\phi_{m}> A<\phi_{m}|\phi_{n}>[/tex]

    then, since the states are orthonormal:
    [tex]<\phi_{m}|\phi_{m}>=<\phi_{n}|\phi_{n}>=1[/tex]

    If I can do that I get: [tex]A=A_{mn}|\phi_{m}><\phi_{n}|[/tex]

    but I'm not sure where the summation comes in.

    #4. Let K be the operator defined by [tex]K=|\phi><\psi|[/tex] where [tex]|\phi>, |\psi>[/tex] are two vectors of the state space.

    c. show that K can always be written in the form [tex]K=\lambda P_{1}P_{2} [/tex] where [tex]\lambda[/tex] is a constant to be calculated and [tex]P_{1}, P_{2}[/tex] are projectors.

    I'm not really sure where to get started on this one. Any hints would be appreciatted, especially since this is due monday morning and I won't have time to talk to my professor before hand.
     
    Last edited: Sep 11, 2004
  2. jcsd
  3. Sep 11, 2004 #2
    #1
    b. If [tex]|\phi_{n}>[/tex] is a eigenket of H, then what is the action of H on [tex]|\phi_{n}>[/tex]? Similarly, what is [tex]<\phi_{n}|H[/tex]? Also note that the communtator between two operators is in general an operator.

    e. The identity operator is [tex]\sum_{n}|\phi_{n}><\phi_{n}|[/tex]. Presumably you want to find A=?. Try to insert the identity operator in front and after A and exchange terms to see what you get.

    #4

    What is a projector? [tex]\frac{|\phi><\phi|}{<\phi|\phi>}[/tex] and [tex]\frac{|\psi><\psi|}{<\psi|\psi>}[/tex] would be projectors.
     
    Last edited: Sep 11, 2004
  4. Sep 12, 2004 #3
    So I figured out the first two questions, but I'm still stuck on that last one. I get to this point: [tex]K=\lambda \frac{|\phi><\psi|}{<\phi|\phi>} \frac{|\phi><\psi|}{<\psi|\psi>} [/tex]

    How do I get from that to [tex]K=\lambda \frac{|\phi><\phi|}{<\phi|\phi>} \frac{|\psi><\psi|}{<\psi|\psi>} [/tex]?
     
  5. Sep 12, 2004 #4

    Hurkyl

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    Remember that [itex]<\psi|\phi>[/itex] is just a number...



    Actually, how did you get to

    [tex]K=\lambda \frac{|\phi><\psi|}{<\phi|\phi>} \frac{|\phi><\psi|}{<\psi|\psi>} [/tex]

    ?

    It seems to me that if you did things slightly different, you'd get the answer you seek.
     
    Last edited: Sep 12, 2004
  6. Sep 12, 2004 #5
    I started with [tex] K=|\phi><\psi|[/tex] and multiplied by [tex]\frac{<\phi|\phi>}{<\phi|\phi>}[/tex] and [tex]\frac{<\psi|\psi>}{<\psi|\psi>}[/tex]. That is how I got the [tex]\lambda=<\phi|\psi>[/tex] term out front. Although, now that I look over it again, I'm not sure if I can rearrange the terms like that.
     
  7. Sep 12, 2004 #6

    Hurkyl

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    It's fine; numbers can always be moved around to wherever you want. Why did you opt to pull the [itex]<\phi|\psi>[/itex] term out instead of the [itex]<\psi|\phi>[/itex] term?
     
  8. Sep 12, 2004 #7
    I can't believe I didn't see that. Thanks for the help.
     
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