Quantum Fundamentals: Winter-2024 Matrix Practice : Due Day 10 F 2/23 Math Bits
Matrix Refresher
S0 4962S
Calculate the following quantities for the matrices:
\[A\doteq
\begin{pmatrix}
1&0&0\\ 0&0&1\\ 0&-1&0\\
\end{pmatrix}
\hspace{2em}
B\doteq
\begin{pmatrix}
a&b&c\\ d&e&f\\ g&h&j\\
\end{pmatrix}
\hspace{2em}
C\doteq
\begin{pmatrix}
\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\\
\end{pmatrix}
\]
and the vectors:
\[\left|D\right\rangle\doteq
\begin{pmatrix}
1\\ i\\ -1\\
\end{pmatrix}
\hspace{2em}
\left|E\right\rangle\doteq
\begin{pmatrix}
1\\ i\\
\end{pmatrix}
\hspace{2em}
\left|F\right\rangle\doteq
\begin{pmatrix}
1\\ -1\\
\end{pmatrix}
\]
\(AB\)
\({\rm tr} (AB)\)
\(C^{-1}\)
\(A\vert D\rangle\)
\(\det(\lambda{\cal I}-A)\) where \(\lambda\) is a scalar.
Pauli Practice
S0 4962S
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. Prove, and become familiar with, the identities
listed below.
Show that each of the Pauli matrices is hermitian. (A matrix is
hermitian if it is equal to its hermitian adjoint.)
Show that the determinant of each of the Pauli matrices is \(-1\).
Show that \(\sigma_i^2={\cal I}\) for each of the Pauli matrices,
i.e. for \(i\in\left\{x,y,z\right\}\).
Hermitian Adjoints
S0 4962S
Calculate the following quantities for the matrices:
\[A\doteq
\begin{pmatrix}
1&0&0\\ 0&0&1\\ 0&-1&0\\
\end{pmatrix}
\hspace{2em}
B\doteq
\begin{pmatrix}
a&b&c\\ d&e&f\\ g&h&j\\
\end{pmatrix}
\hspace{2em}
C\doteq
\begin{pmatrix}
\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\\
\end{pmatrix}
\]
and the vectors:
\[\left|D\right\rangle\doteq
\begin{pmatrix}
1\\ i\\ -1\\
\end{pmatrix}
\hspace{2em}
\left|E\right\rangle\doteq
\begin{pmatrix}
1\\ i\\
\end{pmatrix}
\hspace{2em}
\left|F\right\rangle\doteq
\begin{pmatrix}
1\\ -1\\
\end{pmatrix}
\]
\(A^{\dagger}\)
\(\vert E\rangle^{\dagger}\equiv\langle E\vert\)
\(\langle D\vert A\vert D\rangle\)
\(\left(A\vert D\rangle\right)^{\dagger}\)
Using explicit matrix multiplication (without using a theorem)
verify that \(\left(A\vert D\rangle\right)^{\dagger}
=\langle D\vert
A^{\dagger}\)
Pauli
S0 4962S
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. Prove, and become familiar with, the identities
listed below.
Show that \(\sigma_x \sigma_y = i\sigma_z\) and \(\sigma_y \sigma_x =
-i\sigma_z\). (Note: These identities also hold under a cyclic
permutation of \(\left\{x,y,z\right\}\), e.g. \(x\rightarrow y\),
\(y\rightarrow z\), and \(z\rightarrow x\)).
The commutator of two matrices \(A\) and \(B\) is defined by \(\left[A,
B\right]\buildrel \rm def \over = AB-BA\). Show that \(\left[\sigma_x,
\sigma_y\right] = 2i\sigma_z\). (Note: This identity also holds
under a cyclic permutation of \(\left\{x,y,z\right\}\), e.g.
\(x\rightarrow y\), \(y\rightarrow z\), and \(z\rightarrow x\)).
The anti-commutator of two matrices \(A\) and \(B\) is defined by
\(\left\{A, B\right\}\buildrel \rm def \over = AB+BA\). Show that
\(\left\{\sigma_x, \sigma_y\right\}
= 0\). (Note: This identity also
holds under a cyclic permutation of \(\left\{x,y,z\right\}\), e.g.
\(x\rightarrow y\), \(y\rightarrow z\), and \(z\rightarrow x\)).
Explain what each of the matrices “does” geometrically when thought
of as a linear transformation acting on a vector.
The commutator of two matrices \(A\) and \(B\) is defined by \(\left[A,
B\right]\buildrel \rm def \over = AB-BA\). Find the following
commutators: \(\left[A,B\right]\), \(\left[A,C\right]\),
\(\left[B,C\right]\).
Two matrices are said to commute, if their commutator is zero.
Thought of as linear transformations, two matrices commute if it
doesn't matter in which order the transformations act. For all pairs
of the matrices \(A\), \(B\), and \(C\), show geometrically that the order
of the transformations doesn't matter when the matrices commute and
does matter when they don't commute.
Spin Matrix
S0 4962S
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.