Quantum Fundamentals: Winter-2026
HW 7: Due W4 D3
- Spin One Interferometer Brief
Consider a spin 1 interferometer which prepares the state as \(| 1\rangle\), then sends this state through an \(S_x\) apparatus and then an \(S_z\) apparatus. For the four possible cases where a pair of beams or all three beams from the \(S_x\) Stern-Gernach analyzer are used, calculate the probabilities that a particle entering the last Stern-Gerlach device will be measured to have each possible value of \(S_z\). Compare your theoretical calculations to results of the simulation. Make sure that you explicitly discuss your choice of projection operators.
Note: You do not need to do the first case, as we have done it in class.
- Spin One Eigenvectors
The operator \(\hat{S}_x\) for spin-1 particles, can be written in matrix form in the \(S_z\) basis as:
\[\hat{S}_x=\frac{\hbar}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\ 1&0&1 \\ 0&1&0 \\
\end{pmatrix}
\]
- Find the eigenvalues of this matrix.
- Find the eigenvectors corresponding to each eigenvalue.
- Finding Matrix Elements
Carry out the following matrix calculations.
\[ \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \begin{pmatrix}a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\cr\end{pmatrix} \begin{pmatrix}1\cr0\cr0\cr\end{pmatrix}\] and
\[\begin{pmatrix} 0 & 1 & 0\end{pmatrix} \begin{pmatrix} a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\end{pmatrix} \begin{pmatrix}0\cr1\cr0\cr\end{pmatrix} \]
What matrix multiplication would you do if you wanted the answer to be \(a_{31}\)?
In the first question above, the bra/ket representations for the calulations are:
\[\left\langle {2}\right| A\left|{1}\right\rangle = ? \quad \hbox{and} \quad \left\langle {2}\right| A\left|{2}\right\rangle = ?\]
Write the second question in bra/ket notation.
- Spin Three Halves Operators
If a beam of spin-3/2 particles is input to a Stern-Gerlach analyzer, there are four output beams whose deflections are consistent with magnetic moments arising from spin angular momentum components of \(\frac{3}{2}\hbar\), \(\frac{1}{2}\hbar\), \(-\frac{1}{2}\hbar\), and \(-\frac{3}{2}\hbar\). For a spin-3/2 system:
- Write down the eigenvalue equations for the \(\hat S_z\) operator.
- Write down the matrix representation of the \(\hat S_z\) eigenstates in the \(S_z\) basis.
- Write down the matrix representation of the \(\hat S_z\) operator in the \(S_z\) basis.
- Write down the eigenvalue equations for the \(\hat {S^2}\) operator. (The eigenvalues of the \(S^2\) are \(\hbar^2s(s+1)\), where \(s\) is the spin quantum number. \(\hat {S^2}=(\hat S_x)^2+(\hat S_y)^2+(\hat S_z)^2\), which is proportional to the identify operator. For spin-3/2 system, \(s=\frac{3}{2}\))
- Write down the matrix representation of the \(\hat {S^2}\) operator in the \(S_z\) basis. Check Beasts: Is your operator proportional to the identity operator?