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.(adsbygoogle = window.adsbygoogle || []).push({});

#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.

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# Homework Help: Dirac Notation Help

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