# Spin matrix

1. Dec 31, 2008

### Rajini

Hi everyone,
I now able to understand spin matrix (if i am correct in other words Pauli matrix).
For e.g.,
for S=5/2 systems the spin matrix (say for SX) is given by:

Sx= 1/2[a 6X6 matrix]

I hope members will know what is this 6X6 matrix! Since i don't know how to type matrix in this forum]..
Actually i wanted to know how this 6X6 matrix is obtained??

thanks and wish u a very happy new yr '09.
Rajini

2. Dec 31, 2008

### malawi_glenn

"just" use the definition of the rotation operator:

$$D^{(s)}_{M,m} = <s,M|e^{-i\vec{S}\cdot \vec{n} \phi / \hbar} |s,m>$$

where $$\vec{S} = (S_x, S_y, S_z)$$

See Sakurai, Modern Quantum mechanics, chapter 3 or similar textbooks.

3. Dec 31, 2008

### malawi_glenn

or wait a minute, what i just wrote is crap, it is not what you are looking for.

You want matrix representation of operators S_x , S_y and S_z

Just rewrite S_x and S_y in terms of ladder operators and work out each entry in the matrix :-)

Easy but time-consuming!

4. Dec 31, 2008

### Rajini

Hi thanks..
but how?
Can you please make one calculation..for one (Sx or Sy or Sz? for a S=5/2..
thanks

5. Dec 31, 2008

### malawi_glenn

i mean, this is easy if you are a researcher, the ladder operators are defined as:

$$J_{\pm} \equiv J_x \pm i J_y$$

And then use:

$$J_{\pm}|j,m> = \hbar \sqrt{j(j+1) - m(m\pm 1 )}|j,m \pm 1 >$$

That's all you need man

(Here J is the standard symbol for angular momentum)

6. Jan 3, 2009

### Rajini

i really dont understand of..how to operate for S=5/2
I am a expert..but started learning slowly...
or if possible can u explain each steps found in this page (they did for 1/2)
http://quantummechanics.ucsd.edu/ph130a/130_notes/node278.html
i hope if you explain me i can derive for S=5/2,MS> state
thanks (if u dont find time..i hope i can succeed..but takes at least 1 week)
rajini

7. Jan 3, 2009

### malawi_glenn

1 week?

Why dont you just try the matrix element

< S = 5/2, M = 5/2 | S_y | S = 5/2, M = 3/2>

Write S_y = (S(+) - S(-))/(2i)

And use the formula in my latest post. It is a really straightforward calculation, good luck

8. Jan 3, 2009

### malawi_glenn

a question, do you know how matrix representation of operators work at all? I am having a problem to understand where your lack of understanding is. You only say "explain each step" but it is really hard to know what you know and what you don't know.

Why don't you tell what part you don't understand?

9. Jan 3, 2009

### Rajini

really i dont know how matrix rep. of operators work!
< S = 5/2, M = 5/2 | S_y | S = 5/2, M = 3/2>
Okay how to rep. the above relation as a matrix.also why m=3/2?
thanks

10. Jan 3, 2009

### malawi_glenn

I just gave a particular matrix element to begin with.

Now look at the spin 1/2 matrix, lets look at S_z :

The first entry in the upper left corner (1) is this one: < S = 1/2, M = 1/2 |S_z|S = 1/2, M = 1/2> = 1 (I pull out the factor hbar/2 ...) Make sure you can do this!!

Then the entry in the upper right corner (0) is: < S = 1/2, M = 1/2 |S_z|S = 1/2, M = -1/2> = 0

The entry in the lower left is < S = 1/2, M = -1/2 |S_z|S = 1/2, M = 1/2> = 0

and

The entry in the lower right corner is < S = 1/2, M = -1/2 |S_z|S = 1/2, M = -1/2> ...

So what one does is to put the states with decreasing value of M as rows and coloums, then work out each entry in the matrix (that's why one calls <A| H | B> matrix element, it is an element in the matrix representing the operator H)

Now make sure you can do the S_z for spin 1/2 and also the S_x and S_y for spin 1/2. Then try to do spin 5/2, which is same procedure, but many many more elements to calculate.

11. Jan 3, 2009

### CPL.Luke

also remember that the s_z matrix is just a diagonal matrix with the eigenvalues going down the diagonal.

ie for spin 5/2 along the diagonal the values are 1,1,1,-1,-1,-1

12. Jan 3, 2009

### malawi_glenn

yes, so in reality, only the S_x and S_y are "tricky" to work out.
but the diagonal values should be 5/2, 3/2, 1/2, -1/2, -3/2, -5/2 ...

Last edited: Jan 3, 2009
13. Jan 4, 2009

### Rajini

I think i have to read a bit!. Since u wrote 'M' decreases (by 1) in rows and columns...Then for S=5/2 (2S+1=6 i.e., a 6X6 matrix) one will not get a diagonal matrix..(but this is not true)
or i dont understand properly the <A|H|B> methods..

14. Jan 4, 2009

### malawi_glenn

Yes you will obtain a diagonal matrix for S_z, it is trivial to see that.

15. Jan 4, 2009

### Rajini

One question. How you say its 1. There are four 1/2 so which u take? (i understand u take 1/2 outside the matrix so its 1...but which 1/2 u take)
thanks

16. Jan 4, 2009

### malawi_glenn

I said "upper left corner"...

17. Jan 4, 2009

### Rajini

Then here upper right corner means i assume u r talking upper right corner M=-1/2 (minus 1/2)...so decreasing means it should be -1/4..but u say zero!

18. Jan 4, 2009

### malawi_glenn

eh? can you evaluate < S = 1/2, M = 1/2 |S_z|S = 1/2, M = -1/2>
for me and show how it could be zero...

19. Jan 4, 2009

### Rajini

actually i dont know so i asked you how it could be zero! but you are asking me the same question..anyway i dont want to disturb you now..normally i like QM but only without lesser than, greater than and pipe symbols (i wont say it bra-ket)..since i dont understand all these properly...also i guess there is lots of books which deals with all these lesser than, greater than and pipe symbols but NOT WITH EXAMPLES (or only for 1/2=S)...Any way i think i can manage to search for some books (tomorrow) and hope to find the solution..
anyway thanks
rajini

20. Jan 4, 2009

### malawi_glenn

But the bra-ket notation is as simple as wavefunctions. Look:

wave function:

$$S_z \chi _+ = +\frac{\hbar}{2}\chi _+$$

Bra-ket

$$S_z |+> = +\frac{\hbar}{2}|+>$$

Wave function:

$$\int d\vec{x} \chi _+ ^*\chi _+ = 1$$
$$\int d\vec{x} \chi _- ^*\chi _+ = 0$$

Bra-Ket:
$$<+|+> = 1$$
$$<-|+> = 0$$

I also mean, why do you want matrix representation of operators if you don't even know what it is or how bra-ket works???