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1. The problem statement, all variables and given/known data

Problem is attached.

2. Relevant equations

3. The attempt at a solution

The first part is seemingly straight forward. Measurements are +/- hbar/2, both with probability (1/sqrt[2])^2 = 1/2 of being observed.

For the next part I have written the operator S_x as:

S_x = 1/2 [S+ + S-]

Where S+ and S- are the raising and lowering operators respectively.

ie. S+ = S_x + i S_y and S- = S_x - i S_y

Then using:

S+ | s,m > = [s(s+1 - m(m+1)]^(1/2) hbar | s,m+1 >

S- | s,m > = [s(s+1 - m(m-1)]^(1/2) hbar | s,m+1 >

With s = 1/2 (this is clear from the first part, since we have the eigenvalues of S_z (m) = -1/2 and 1/2 and -s <= m <= s in integer steps.

I obtain

S+ | 1/2 > = 0

S- | 1/2 > = hbar | -1/2 >

S+ | -1/2 > = hbar | 1/2 >

S- | - 1/2 > = 0

So

S_x | 1/2 > = hbar/2 | -1/2 >

S_x | -1/2 > = hbar/2 | 1/2 >

and

S_x | psi > = hbar/2 1/sqrt[2] ( | 1/2 > + | -1/2 > )

So the measurement is simply hbar/2

For the final part (where I become stuck!)

S_y | 1/2 > = 1/2i (S+ - S-) | 1/2 > = ihbar/2 | -1/2 >

S_y | -1/2 > = 1/2i (S+ - S-) | 1/2 > = -ihbar/2 | 1/2 >

So I am required to find a state vector psi such that:

S_y | psi > = -hbar/2 | psi >

1/2i (S+ - S-) | psi > = -hbar/2 | psi >

(S+ - S-) | psi > = -ihbar | psi >

But is it even possible to construct a state vector out of the spin-up and spin-down eigenvectors to give this result? I can't seem to do it???

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# Homework Help: Quantum Mechanics - Spin

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