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SunGod87
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[SOLVED] Quantum Mechanics - Spin
Problem is attached.
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?
Homework Statement
Problem is attached.
Homework Equations
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?