Determining the state vector and density matrix in quantum mechanics?

Question

(a)Consider a pure ensemble of identically prepared spin 1/2 systems. Suppose the expectation values and and the sign of are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of ?

(b) Consider a mixed ensemble of spin 1/2 systems. Suppose the ensemble averages [Sx], [Sy], and [Sz] are all known. Show how we may construct the 2x2 density matrix that characterizes the ensemble.

Answer 1

Sx = Sz = Sy = Sa + Sb

Answer 2

Dear Mitijah,
If you are taking an MSc Course in this stuff - I suggest uyou get hold of the recomended books. which will be available somewhere in you Maths or Physics Library.

PART A(I haven't finished PARTB yet)

As we are dealing only with the Spin part of the state vector we can use the Ket and label it ¦s>

If ¦xj> are unit directional vectors then we can resolve in the 3 spacial directions by using the UNIT OPERATOR
SUM over j ¦xj >
So the State Vector is: ¦s> = Sum[ ¦xj > ]
¦s> = ¦x > ] + ¦y > + ¦z >

But is ; ; ;

So the State Vector is:
¦s> = ¦x > + ¦y > + ¦z>
where x y z are all unit vectors (usually written i j k).
The spin vectors and direction vectors are Orthonormal Sets
So:
= 1 = ^2 + ^2 +^2

OR:
1 = sx^2 + sy^2 +sz^2

So sy^2 = 1 - (sx^2 +sz^2)

and if we know the sign of y then we also know y.

Then sz = 1 implies sx = 0 and sy = 0
and sz = 0 and sx = 0 implies sy = 1 plus/minus
and sz = 0 and sx = 1 implies sy = 0


And we can use these values to write the State Vector:

¦s> = ¦i> ] + ¦j > + ¦k >


This is really difficult to write out as I do not have the Greek Script!!!!!!

I THINK this first easy part is just to get you into the BRA-KET mood for doing the second part. It all looks like something off an MSc Examination Paper!

I hope this has been of some help!



PART B:
The method of showing that you multiply the Function Matrix by the Density Matrix and use that as your OPERATOR i as shown in:
Tolman The Principles of Statistical MechanicsSection 78:
The density Matrix in Quantum Statistical Mechanics pp327

I got 1/2 way through putting it into Bra-ket Notation but putting it on here is a pain!!!!!!!!!!

Write it out as he does - and then muddle it into the newer notation.