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Homework for Eigenvalues and Eigenvectors
- (EigenPractice) Lots and lots of practice finding eigenvalues and eigenvectors.
Find the eigenvectors and eigenvalues of all of the matrices from the Linear Transformations small group activity from Tuesday's class. Make up some $2\times 2$ and $3\times 3$ matrices of your own if you need more practice.
(The solutions to this problem need additional examples from the newest version of the activity.)
- (Eigenrotation) Straightfoward practice finding eigenvalues and eigenvectors for the particular case of a generic rotation matrix around the $z$-axis. Warning: the eigenvectors and eigenvalues in this case are complex numbers.
The orthogonal matrix $$R_z(\theta)=\pmatrix{\cos\theta&-\sin\theta&0\cr
\sin\theta&\cos\theta&0\cr 0&0&1\cr}$$corresponds to a rotation around the $z$-axis by the angle $\theta$.
Find the eigenvalues of this matrix.
Find the normalized eigenvectors of this matrix.
Describe how the eigenvectors do or do not correspond to the vectors which are held constant or “only stretched” by this transformation.
- (SpinMatrix)This problem is a prerequisite for the next one. Students find this problem very strange. It requires them to take the formal dot product of a vector with another vector whose components are matrices. The result is the spin operator for a generic spin $\frac{1}{2}$ system, with spin up in the $\hat n$-direction. This can be a useful problem if the students are going to be covering the content of the Quantum Measurement and Spin Course.
Consider the arbitrary Pauli matrix $\sigma_n=\hat n\cdot\Vec \sigma$ where $\hat n$ is the unit vector pointing in an arbitrary direction.
By drawing pictures, convince yourself that the arbitrary unit vector $\hat n$ can be written as:
$$\hat n=\sin\theta\cos\phi\, \hat\imath +\sin\theta\sin\phi\,\hat\jmath +\cos\theta\,\hat k$$
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
$$\Vec\sigma=\sigma_x\, \hat\imath + \sigma_y\, \hat\jmath +\sigma_z\, \hat k$$
and $\hat n$ is given in part (a) above.
- (EigenSpinChallenge)This problem requires the previous problem as a prerequisite. It is long and messy. It requires the students to use trigonometric identities and to persist through a messy calculation. In this problem, students find the eigenvalues and eigenvectors for the generic spin $\frac{1}{2}$ matrix in the $\hat n$-direction. Therefore, this can be a useful problem if the students are going to be covering the content of the Quantum Measurement and Spin Course. This problem needs to be updated so the phase conventions agree with Spins conventions, that the first component should be real.
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:
$$\pmatrix{\cos{\theta\over 2}e^{-i\phi/2}\cr \noalign{\smallskip} \sin{\theta\over 2}e^{i\phi/2}\cr} \qquad\qquad\qquad \pmatrix{-\sin{\theta\over 2}e^{-i\phi/2}\cr \noalign{\smallskip} \cos{\theta\over 2}e^{i\phi/2}\cr}$$
It is not sufficient to show that this answer is correct by pluging 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 $\sigma_z$.
