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[SOLVED] Quantum Mechanics  Spin
1. Homework Statement
Problem is attached.
2. Homework 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(m1)]^(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 spinup and spindown eigenvectors to give this result? I can't seem to do it???
1. Homework Statement
Problem is attached.
2. Homework 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(m1)]^(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 spinup and spindown eigenvectors to give this result? I can't seem to do it???
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