Conditions for a density matrix; constructing a density matrix

[Juqon]

Homework Statement

What are the conditions so that the matrix $\hat{p}$ describes the density operator of a pure state?

Homework Equations

[PLAIN]http://img846.imageshack.us/img846/2835/densitymatrix.png
$p=\sum p_{j}|\psi_{j}><\psi_{j}$

The Attempt at a Solution

I know that tr($\rho$)=1 for pure states.
But I do not know how to construct the density matrix. Or is it $\hat{p}$ already?
In my book, there is the example "hermitian, tr=1, [PLAIN]http://img89.imageshack.us/img89/4346/densitymatrixexample.png", but I do not know how they constructed that.

Maybe I can just say a+d+e=1; c,b arbitrary?

 

[vela]

http://en.wikipedia.org/wiki/Density_matrix#Pure_and_mixed_states

 

[Juqon]

What do you think about that?

[PLAIN]http://img155.imageshack.us/img155/7663/densitymatrixexamplep2.png
a²+bc=a (a$\neq$0) => $a+\frac{bc}{a}=1$
ab+bd=b (b$\neq$0) => a+d=1 <=> a= 1-d <=> d=1-a
ca+dc=c (c$\neq$0) => a+d = 1
cb+d²=2 (d$\neq$0) => $d+\frac{cb}{d}=1$
e=e²

a=0 v a= 1 - d
b € |R
c € |R
d=0 v d= 1 - a
e=0 v 1

 

[vela]

You might find this page helpful too, just to learn about the density matrix in general.

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/DensityMatrix.htm

You should also use the requirements tr(p)=1 and the fact that the matrix is Hermitian to further restrict the variables.

 

[Juqon]

Ok, thanks, I have read that.

with the condition for the trace:
a+d+e=1 => e=0: a+d=1;
e=1: a=d=0

and since the matrix is hermitian and p^T=p:
b = c, c € |R

On the page you linked to it says only one element on the diagonale can be different from zero when you chose the right basis, but I think I can not change the basis here. So I think I can not further restrict it from that.
Right?

 

[vela]

Ok, thanks, I have read that.

with the condition for the trace:
a+d+e=1 => e=0: a+d=1;
e=1: a=d=0

For e=1, you can say a+d=0, which isn't quite the same as saying a=d=0.

and since the matrix is hermitian and p^T=p:
b = c, c € |R

Both b and c can be complex. At best you can say c=b^*.

On the page you linked to it says only one element on the diagonale can be different from zero when you chose the right basis, but I think I can not change the basis here. So I think I can not further restrict it from that.
Right?

Yes, I agree. I think the problem wants you to find a general form for the density matrix.