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Question 5 [2 + 6 + 5 = 13 marks ] A beam of spin-1/2 particles (travelling in the y direction) is sent through a Stern-Gerlach apparatus which is aligned to measure in the z direction and divides the incident beam into two beams with m = ±1/2. The m = +1/2 beam is allowed to enter a second Stern-Gerlach apparatus aligned to measure along the direction $vec{e} = sin hetahat{x} + cos hetahat{z}$. a) For the net spin operator $S = (hbar/2)vec{sigma}cdotvec{e}$ (where $vec{sigma}$ are the Pauli spin matrices) write out the operator S in matrix representation (i.e. in the $S_z$ spinor basis). Data: $sigma_x = egin{pmatrix} 0 & 1\ 1 & 0 end{pmatrix}$, $sigma_y = egin{pmatrix} 0 & -i\ i & 0 end{pmatrix}$, $sigma_z = egin{pmatrix} 1 & 0\ 0 & -1 end{pmatrix}$ b) Determine the eigenvalues of S and the corresponding eigenvectors (up to an overall normalisation constant). c) Calculate the ratio of the intensities (probabilities) of the two beams which emerge from the second Stern- Gerlach apparatus as a function of $ heta$.

          Question 5 [2 + 6 + 5 = 13 marks ]
A beam of spin-1/2 particles (travelling in the y direction) is sent through a Stern-Gerlach apparatus which
is aligned to measure in the z direction and divides the incident beam into two beams with m = ±1/2. The
m = +1/2 beam is allowed to enter a second Stern-Gerlach apparatus aligned to measure along the direction
$vec{e} = sin	hetahat{x} + cos	hetahat{z}$.
a) For the net spin operator $S = (hbar/2)vec{sigma}cdotvec{e}$ (where $vec{sigma}$ are the Pauli spin matrices) write out the operator S in
matrix representation (i.e. in the $S_z$ spinor basis).
Data:
$sigma_x = egin{pmatrix} 0 & 1\ 1 & 0 end{pmatrix}$, $sigma_y = egin{pmatrix} 0 & -i\ i & 0 end{pmatrix}$, $sigma_z = egin{pmatrix} 1 & 0\ 0 & -1 end{pmatrix}$
b) Determine the eigenvalues of S and the corresponding eigenvectors (up to an overall normalisation constant).
c) Calculate the ratio of the intensities (probabilities) of the two beams which emerge from the second Stern-
Gerlach apparatus as a function of $	heta$.

Question 5 [2 + 6 + 5 = 13 marks ]
A beam of spin-1/2 particles (travelling in the y direction) is sent through a Stern-Gerlach apparatus which
is aligned to measure in the z direction and divides the incident beam into two beams with m = ±1/2. The
m = +1/2 beam is allowed to enter a second Stern-Gerlach apparatus aligned to measure along the direction
vece = sin hetahatx + cos hetahatz.
a) For the net spin operator S = (hbar/2)vecsigmacdotvece (where vecsigma are the Pauli spin matrices) write out the operator S in
matrix representation (i.e. in the Sz spinor basis).
Data:
sigmax = eginpmatrix 0 1 1 0 endpmatrix, sigmay = eginpmatrix 0 -i i 0 endpmatrix, sigmaz = eginpmatrix 1 0 0 -1 endpmatrix
b) Determine the eigenvalues of S and the corresponding eigenvectors (up to an overall normalisation constant).
c) Calculate the ratio of the intensities (probabilities) of the two beams which emerge from the second Stern-
Gerlach apparatus as a function of heta.

Added by Daniel G.

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