Quantum Fundamentals: NoTerm-2022 HW 5 : Due Day 13 W 2/23
Spin Matrix
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The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are
defined by:
\[\sigma_x=
\begin{pmatrix}
0&1\\ 1&0\\
\end{pmatrix}
\hspace{2em}
\sigma_y=
\begin{pmatrix}
0&-i\\ i&0\\
\end{pmatrix}
\hspace{2em}
\sigma_z=
\begin{pmatrix}
1&0\\ 0&-1\\
\end{pmatrix}
\]
These matrices are related to angular momentum in
quantum mechanics.
By drawing pictures, convince yourself that the arbitrary unit
vector \(\hat n\) can be written as:
\[\hat n=\sin\theta\cos\phi\, \hat x +\sin\theta\sin\phi\,\hat y+\cos\theta\,\hat z\]
where \(\theta\) and \(\phi\) are the parameters used to describe
spherical coordinates.
Find the entries of the matrix \(\hat n\cdot\vec \sigma\) where the
“matrix-valued-vector” \(\vec \sigma\) is given in terms of the
Pauli spin matrices by
\[\vec\sigma=\sigma_x\, \hat x + \sigma_y\, \hat y+\sigma_z\, \hat z\]
and \(\hat n\) is given in part (a) above.
Eigenvectors of Pauli Matrices
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Find the eigenvalues and normalized eigenvectors of the Pauli
matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) (see the Spins Reference Sheet posted on the course website).
Eigen Spin Challenge
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Consider the arbitrary Pauli matrix \(\sigma_n=\hat n\cdot\vec
\sigma\) where \(\hat n\) is the unit vector pointing in an arbitrary
direction.
Find the eigenvalues and normalized eigenvectors for \(\sigma_n\).
The answer is:
\[
\begin{pmatrix}
\cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\
\end{pmatrix}
\begin{pmatrix}
-\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\
\end{pmatrix}
\]
It is not sufficient to show that this answer is correct by plugging
into the eigenvalue equation. Rather, you should do all the steps
of finding the eigenvalues and eigenvectors as if you don't know the
answer. Hint: \(\sin\theta=\sqrt{1-\cos^2\theta}\).
Show that the eigenvectors from part (a) above are orthogonal.
Simplify your results from part (a) above by considering the three separate special cases: \(\hat n=\hat\imath\), \(\hat
n=\hat\jmath\), \(\hat n=\hat k\). In this way, find the eigenvectors and eigenvalues of \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\).
Diagonalization
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Let
\[|\alpha\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ 1
\end{pmatrix}
\qquad \rm{and} \qquad
|\beta\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ -1
\end{pmatrix}\]
Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal.
(If a pair of vectors is orthonormal, that suggests that
they might make a good basis.)
Consider the matrix
\[C\doteq
\begin{pmatrix}
3 & 1 \\ 1 & 3
\end{pmatrix}
\]
Show that the vectors
\(|\alpha\rangle\) and
\(|\beta\rangle\) are
eigenvectors of C and find the eigenvalues.
(Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
A operator is always represented by a diagonal matrix if it is written in terms of
the basis of its own eigenvectors. What does this mean? Find the matrix elements for a
new matrix \(E\) that
corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\),
\(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and
\(\langle\beta|C|\beta\rangle\)
and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix
elements in the order that you did.
Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?
Diagonalization Part II
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First complete the problem Diagonalization. In that notation:
Find the matrix \(S\) whose columns are \(|\alpha\rangle\) and \(|\beta\rangle\).
Show that \(S^{\dagger}=S^{-1}\) by calculating \(S^{\dagger}\) and multiplying it by \(S\). (Does the order of multiplication matter?)
Calculate \(B=S^{-1} C S\). How is the matrix \(E\) related to \(B\) and \(C\)? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.