Dirac Notation Help: Solve H, A, K Problems

[Hypnotoad]

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. |\phi_{n}> 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)=|\phi_{m}><\phi_{n}|.

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: H|\phi_{m}&gt;&lt;\phi_{n}|-|\phi_{m}&gt;&lt;\phi_{n}|H

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

e. Let A be an operator, with matrix elements A_{mn}=&lt;\phi_{m}|A|\phi_{n}&gt;

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

If I start with A_{mn}=&lt;\phi_{m}|A|\phi_{n}&gt;, is it legal to do this:

A_{mn}|\phi_{m}&gt;&lt;\phi_{n}|=&lt;\phi_{m}|\phi_{m}&gt; A&lt;\phi_{m}|\phi_{n}&gt;

then, since the states are orthonormal:
&lt;\phi_{m}|\phi_{m}&gt;=&lt;\phi_{n}|\phi_{n}&gt;=1

If I can do that I get: A=A_{mn}|\phi_{m}&gt;&lt;\phi_{n}|

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

#4. Let K be the operator defined by K=|\phi&gt;&lt;\psi| where |\phi&gt;, |\psi&gt; are two vectors of the state space.

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

 

[Wong]

#1
b. If |\phi_{n}&gt; is a eigenket of H, then what is the action of H on |\phi_{n}&gt;? Similarly, what is &lt;\phi_{n}|H? Also note that the communtator between two operators is in general an operator.

e. The identity operator is \sum_{n}|\phi_{n}&gt;&lt;\phi_{n}|. 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? \frac{|\phi&gt;&lt;\phi|}{&lt;\phi|\phi&gt;} and \frac{|\psi&gt;&lt;\psi|}{&lt;\psi|\psi&gt;} would be projectors.

 

[Hypnotoad]

So I figured out the first two questions, but I'm still stuck on that last one. I get to this point: K=\lambda \frac{|\phi&gt;&lt;\psi|}{&lt;\phi|\phi&gt;} \frac{|\phi&gt;&lt;\psi|}{&lt;\psi|\psi&gt;}

How do I get from that to K=\lambda \frac{|\phi&gt;&lt;\phi|}{&lt;\phi|\phi&gt;} \frac{|\psi&gt;&lt;\psi|}{&lt;\psi|\psi&gt;}?

 

[Hurkyl]

Remember that &lt;\psi|\phi&gt; is just a number...



Actually, how did you get to

K=\lambda \frac{|\phi&gt;&lt;\psi|}{&lt;\phi|\phi&gt;} \frac{|\phi&gt;&lt;\psi|}{&lt;\psi|\psi&gt;}

?

It seems to me that if you did things slightly different, you'd get the answer you seek.

 

[Hypnotoad]

Remember that &lt;\psi|\phi&gt; is just a number...



Actually, how did you get to

K=\lambda \frac{|\phi&gt;&lt;\psi|}{&lt;\phi|\phi&gt;} \frac{|\phi&gt;&lt;\psi|}{&lt;\psi|\psi&gt;}

?

It seems to me that if you did things slightly different, you'd get the answer you seek.

I started with K=|\phi&gt;&lt;\psi| and multiplied by \frac{&lt;\phi|\phi&gt;}{&lt;\phi|\phi&gt;} and \frac{&lt;\psi|\psi&gt;}{&lt;\psi|\psi&gt;}. That is how I got the \lambda=&lt;\phi|\psi&gt; term out front. Although, now that I look over it again, I'm not sure if I can rearrange the terms like that.

 

[Hurkyl]

It's fine; numbers can always be moved around to wherever you want. Why did you opt to pull the &lt;\phi|\psi&gt; term out instead of the &lt;\psi|\phi&gt; term?

 

[Hypnotoad]

It's fine; numbers can always be moved around to wherever you want. Why did you opt to pull the &lt;\phi|\psi&gt; term out instead of the &lt;\psi|\phi&gt; term?

I can't believe I didn't see that. Thanks for the help.