In this unit, you will explore the most common partial differential equations that arise in physics contexts. You will learn the separation of variables procedure to solve these equations.
Motivating Questions
How are partial differential equations (PDEs) different from ordinary differential equations (ODEs)?
What new kinds of physics can we learn from solving partial differential equations?
What can we learn about physics and geometry from the separation of variables procedure?
This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.
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.
On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)
For each of the following complex numbers, determine the complex conjugate, square, and
norm. Then, plot and clearly label each \(z\), \(z^*\), and \(|z|\) on an Argand diagram.
\(z_1=4i-3\)
\(z_2=5e^{-i\pi/3}\)
\(z_3=-8\)
In a few full sentences, explain the geometric meaning of the complex
conjugate and norm.
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.
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 \(\sigma_z\).
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).
This activity gives links to some external resources (2 simulations and 1 video) that allow students to explore circle trigonometry. There are no prompts and nothing specific to turn in.
Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)
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?
In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
vector differentialrectangular coordinatesmath Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s)Found in: Integration Sequence sequence(s)
In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
