*assignment*Visualization of Wave Functions on a Ring*assignment*Homework##### Visualization of Wave Functions on a Ring

Central Forces Spring 2021 Using either this*Geogebra*applet or this*Mathematica*notebook, explore the wave functions on a ring. (Note: The*Geogebra*applet may be a little easier to use and understand and is accessible if you don't have access to*Mathematica*, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)- Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
- Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
- Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
- Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

*assignment*Theta Parameters*assignment*Homework##### Theta Parameters

AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022The function \(\theta(x)\) (the

*Heaviside*or*unit step function*) is a defined as: \begin{equation} \theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases} \end{equation} This function is discontinuous at \(x=0\) and is generally taken to have a value of \(\theta(0)=1/2\).Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}

*assignment*Quantum Particle in a 2-D Box*assignment*Homework##### Quantum Particle in a 2-D Box

Central Forces Spring 2021(2 points each)

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 \(L_x\) in the \(x\)-direction and length \(L_y\) 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 (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

*assignment*Circle Vector, Version 2*assignment*Homework##### Circle Vector, Version 2

AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

- Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
Make a sketch of the graph \begin{equation} \vert \vec{r} - \vec{a} \vert = 2 \end{equation}

for each of the following values of \(\vec a\): \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

- Derive a more familiar equation equivalent to \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation} for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
- Write a brief description of the geometric meaning of the equation \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation}

*face*Time Evolution Refresher (Mini-Lecture)*face*Lecture30 min.

##### Time Evolution Refresher (Mini-Lecture)

Central Forces Spring 2021 The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.*computer*Approximating Functions with Power Series*computer*Computer Simulation30 min.

##### Approximating Functions with Power Series

Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021 AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Central Forces Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022Taylor series power series approximation

Students use a prepared*Mathematica*notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.*assignment*Gibbs entropy is extensive*assignment*Homework##### Gibbs entropy is extensive

Gibbs entropy ProbabilityConsider two

*noninteracting*systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.- Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
- Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

##### Note

In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.*assignment*Series Notation 1*assignment*Homework##### Series Notation 1

AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022Write out the first four nonzero terms in the series:

\[\sum\limits_{n=0}^\infty \frac{1}{n!}\]

- \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
- \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

*assignment*Series Notation 2*assignment*Homework##### Series Notation 2

AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

\[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]

- \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]

*group*Representations of the Infinite Square Well*group*Small Group Activity120 min.

##### Representations of the Infinite Square Well

Quantum Fundamentals Winter 2021-
Central Forces Spring 2021
In each of the following sums, shift the index \(n\rightarrow n+2\). Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

- \begin{equation} \sum_{n=0}^3 n \end{equation}
- \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
- \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}