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.)
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.
This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive.
\[\left|D\right\rangle\doteq
\begin{pmatrix}
7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\
\end{pmatrix}\\
\left|E\right\rangle\doteq
\begin{pmatrix}
i\\ 4\\
\end{pmatrix}\\
\left|F\right\rangle\doteq
\begin{pmatrix}
2+2i\\ 3-4i\\
\end{pmatrix}
\]
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
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.
Students set up a spherical coordinate system on a balloon, draw a spherical harmonic, and use the balloon as a prop to describe the main features of their spherical harmonic to the class.
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.
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \). (Since \(\left|{\psi_3}\right\rangle \) has already been covered in class, please only do \(\left|{\psi_4}\right\rangle \) )
Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
Find the entropy of a set of \(N\) oscillators of frequency
\(\omega\) as a function of the total quantum number \(n\). Use the
multiplicity function: \begin{equation}
g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}
\end{equation} and assume that \(N\gg 1\). This means you can
make the Sitrling approximation that
\(\log N! \approx N\log N - N\). It also means that
\(N-1 \approx N\).
Let \(U\) denote the total energy \(n\hbar\omega\) of the
oscillators. Express the entropy as \(S(U,N)\). Show that the total
energy at temperature \(T\) is \begin{equation}
U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}
\end{equation} This is the Planck result found the hard
way. We will get to the easy way soon, and you will never again need
to work with a multiplicity function like this.
A beam of spin-\(\frac{1}{2}\) particles is prepared in the initial state \[ \left\vert \psi\right\rangle = \sqrt{\frac{2}{5}}\; |+\rangle_x - \sqrt{\frac{3}{5}}\; |-\rangle_x \](Note: this state is written in the \(S_x\) basis!)
What are the possible results of a measurement of \(S_x\), with what probabilities?
Repeat part a for measurements of \(S_z\).
Suppose you start with a particle in the state given above, measure \(S_x\), and happen to get \(+\hbar /2\). You then take that same particle and measure \(S_z\). What are the possible results and with what probability would you measure each possible result?
Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
Click and drag a new connection from the analyzer to empty space to create a new element. A new analyzer is one of the options.
Measure \(S_z\) twice in succession.
What is the probability that a particle leaving the first analyzer with \(S_z=\frac{+\hbar}{2}\) will be measured by the second analyzer to have \(S_z=\frac{-\hbar}{2}\)?
Try all four possible combinations of input/outputs for the second analyzer.
What have you learned from these experiments?
Try All Combinations of Sequential Measurements
In the table, enter the probability of a particle exiting the 2nd analyzer with the spin indicated in row if the particle enters the 2nd analyzer with the spin indicated in each column.
You can rotate the Stern-Gerlach analyzers to any direction you want (using spherical coordinates).
Choose an arbitrary direction (not along one of the coordinate axes) for the 1st analyzer and measure the spin along the coordinate directions for the 2nd analyzer.
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
Students become acquainted with the Spins Simulations of Stern-Gerlach Experiments and record measurement probabilities of spin components for a spin-1/2 system. Students start developing intuitions for the results of quantum measurements for this system.
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
Show that if a linear combination of ring energy eigenstates is normalized, then
the coefficients must satisfy
\begin{equation}
\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1
\end{equation}
Describe the energy eigenstates for the ring system algebraically and graphically.
List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
Use an expansion in energy eigenstates to find the time dependence of a given state.
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?
Consider a two-state quantum system with a Hamiltonian
\begin{equation}
\hat{H}\doteq
\begin{pmatrix}
E_1&0\\ 0&E_2
\end{pmatrix}
\end{equation}
Another physical observable \(M\) is described by the operator
\begin{equation}
\hat{M}\doteq
\begin{pmatrix}
0&c\\ c&0
\end{pmatrix}
\end{equation}
where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle
=\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the expectation value of \(M\) as a function of time? What is the frequency of oscillation of the expectation value of \(M\)?
For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).
Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
For \(\hat{Q}\doteq
\begin{pmatrix}
2 & i \\ -i & -2
\end{pmatrix}
\), calculate
\(\langle1|\hat{Q}|2\rangle\), \(\langle2|\hat{Q}|1\rangle\),
\(\langle a|\hat{Q}| b \rangle\) and \(\langle b|\hat{Q}|a \rangle\).
What kind of mathematical object is \(|a\rangle\langle b|\)? What is the result if you multiply a ket (for example, \(| a\rangle\) or \(|1\rangle\)) by this expression? What if you multiply this expression by a bra?
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).
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 \(\sigma_z\).
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.
For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length
\(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D
box, then you just multiply together the eigenfunctions for a 1-D box in each
direction. (This is what the separation of variables procedure tells you to do.)
Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L\)
in the \(x\)-direction and length \(W\) in the \(y\)-direction.
Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e.
\begin{equation}
\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)
\end{equation}
Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing
energy.
You may find it easier to work in bra/ket notation:
\begin{align*}
\left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\
&=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle
\end{align*}
Find a formula for the \(c_{nm}\)s in part (c). Find the formula first in bra ket notation and then rewrite it in wave function notation.
Consider one particle
confined to a cube of side \(L\); the concentration in effect is
\(n=L^{-3}\). Find the kinetic energy of the particle when in the
ground state. There will be a value of the concentration for which
this zero-point quantum kinetic energy is equal to the temperature
\(kT\). (At this concentration the occupancy of the lowest orbital is
of the order of unity; the lowest orbital always has a higher
occupancy than any other orbital.) Show that the concentration \(n_0\)
thus defined is equal to the quantum concentration \(n_Q\) defined by
(63): \begin{equation}
n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}
\end{equation} within a factor of the order of unity.
The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
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.
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).