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Students re-represent a state given in Dirac notation in matrix notation

Students are asked to:

• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.

Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

• Eigenvalues and eigenstates
• Measurements of energy and angular momentum for hydrogen atom
• Quantum probabilities
• Superposition of states
• New degeneracies

Expectation Values and Uncertainty

You have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the \(z\)-direction.

Consider one of the different initial spin states described below:

A spin 1/2 particle described by:

1. \(\left|{+}\right\rangle \)
2. \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)

3. \(\left|{+}\right\rangle _x\)

   A spin 1 particle described by:

4. \(\left|{0}\right\rangle \)
5. \(\left|{-1}\right\rangle _x\)
6. \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)

• List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.

  
  

• Plot a histogram of the probabilities.

  
  

• Find the expectation value of the z-component of spin.

  
  

• Find the uncertainty of the z-component of spin.

Introduction

I like to break this activity into two parts:

(1) Calculating expectation values and relating them to the associated distributions of the probabilities of results, and

(2) Calculating the quantum uncertainty of the state and relating the uncertainty to distributions of the probabilities of results.

Therefore, I have my students do the first part of the activity before I introduce quantum uncertainty.

I introduce the activity by reminding students about two ways of calculating the expectation value. Given a quantum state \(\left|{\psi}\right\rangle \), for a measurement of an observable represented by an operator \(\hat{A}\) with eigenstates \(\left|{a_i}\right\rangle \) and eigenvalues \(a_i\): \begin{align*} \langle \hat{A} \rangle &= \sum_{i} a_i\mathcal{P}(a_i) \\ &= \left\langle {\psi}\right|\hat{A}\left|{\psi}\right\rangle \end{align*}

After the students calculate expectation values and we have a whole class discussion about 1 of the examples, then I do a lecture introducing the quantity of quantum uncertainty (relating it to the standard deviation of the distribution of probabilities by spin component value) and deriving the simplified equation:

\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}\]

Student Conversations

1. One could have each group report out, or the instructor could discuss a few key examples.

   For expectation value, I like to talk about Case 2: \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \), where the probabilities of the two outcomes are not equal to show how the weighting plays out. Also, the expectation value is not a possible measurement value, and I like to talk about that. “Expectation” value is a misleading name for this quantity - it characterizes the distribution and is not necessarily a result of an individual measurement.

   I also like to discuss an example like Case 5: \(\left|{1}\right\rangle _x\) where the distribution is symmetric around \(0\hbar\).

2. I think it's important to encourage students to calculate expectation values both ways (with probabilities and as a bracket with matrix notation) while the teaching team is available to help them.

3. For quantum uncertainty, I like to talk about an example like Case 3: \(\left|{+}\right\rangle _x\) where all the individual measurements are the same ”distance” away from the expectation value as a sensemaking exercise to connect to a conceptual interpretation of physics.

   I also like to discuss an example like Case 5: \(\left|{-1}\right\rangle _x\), where the fact that we're taking an rms average is apparent: half the measurements are \(\hbar\) away from the expectation value and the other half are \(0\hbar\) away, but the uncertainty is \(\hbar/\sqrt{2}\).

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

1-D Particle-in-a-Box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \end{align} Eigenstates: \begin{align} \left|{n}\right\rangle &\doteq\sqrt{\frac{2}{L}}\, \sin\frac{n\pi x}{L}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align}

Eigenvalue Equations: \begin{align} \hat{H}\left|{n}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu L^2}\, n^2 \left|{n}\right\rangle \\ \end{align}

Particle-on-a-Ring

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I}\frac{\partial^2}{\partial \phi^2} \end{align} Eigenstates: \begin{align} \left|{m}\right\rangle &\doteq\frac{1}{\sqrt{2\pi r_0}}\, e^{im\phi}\\ m&=\left\{\dots -2, -1, 0, 1, 2, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{m}\right\rangle &=\frac{\hbar^2}{2I}\, m^2 \left|{m}\right\rangle \\ \hat{L}^2\left|{m}\right\rangle &=\hbar^2\, m^2 \left|{m}\right\rangle \\ \hat{L}_z\left|{m}\right\rangle &=\hbar\, m \left|{m}\right\rangle \end{align}

2-D Particle-in-a-Box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \end{align} Eigenstates: \begin{align} \left|{mn}\right\rangle &\doteq\sqrt{\frac{2}{L_x}}\sqrt{\frac{2}{L_y}}\, \sin\frac{m\pi x}{L_x}\sin\frac{n\pi y}{L_y}\\ m&=\left\{1, 2, 3, \dots\right\}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{mn}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu}\, \left(\frac{m^2}{L_x^2}+\frac{n^2}{L_y^2}\right) \left|{mn}\right\rangle \\ \end{align}

Particle-on-a-Sphere

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I} \left[\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \left(\sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] \end{align} Eigenstates: \begin{align} \left|{\ell m}\right\rangle &\doteq Y_{\ell}^m(\theta, \phi)\\ &=(-1)^{\frac{m+|m|}{2}}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \ell&=\left\{0, 1, 2, \dots\right\}\\ m&=\left\{-\ell, \dots , 0, \dots,\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{\ell m}\right\rangle &=\frac{\hbar^2}{2I}\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}^2\left|{\ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}_z\left|{\ell m}\right\rangle &=\hbar\, m \left|{\ell m}\right\rangle \end{align}

Hydrogen Atom

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2\mu} \nabla^2 - \frac{ke^2}{r} \\ &\doteq -\frac{\hbar^2}{2\mu r^2} \left[\frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \Big) + \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \Big(\sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] - \frac{ke^2}{r} \end{align} Eigenstates: \begin{align} \left|{n\ell m}\right\rangle &\doteq R_{n\ell}(r)\, Y_{\ell}^m(\theta, \phi)\\ &=-\sqrt{\left(\frac{2Z}{na_0}\right)^3 \frac{(n-\ell-1)!}{2n[(n+\ell)!]^3}} \left(\frac{2\rho}{n}\right)^{\ell}\, e^{-\frac{\rho}{n}}\, L_{n+\ell}^{2\ell+1}{\scriptstyle{\left(\frac{2\rho}{n}\right)}} (-1)^{\frac{m+|m|}{2}} \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \rho&=\frac{Zr}{a_0}\\ n&=\left\{1, 2, 3,\dots\right\}\\ \ell&=\left\{0, 1, 2, \dots, n-1\right\}\\ m&=\left\{-\ell, \dots , 0, \dots,\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{n\ell m}\right\rangle &=-\frac{1}{2}\left(\frac{Ze^2}{4\pi\epsilon_0}\right)^2 \frac{\mu}{\hbar^2}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ &=-13.6 \text{eV}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ \hat{L}^2\left|{n \ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{n \ell m}\right\rangle \\ \hat{L}_z\left|{n \ell m}\right\rangle &=\hbar\, m \left|{n \ell m}\right\rangle \end{align}

For \(\ell=1\), the operators that measure the three components of angular momentum in matrix notation are given by: \begin{align} L_x&=\frac{\hbar}{\sqrt{2}}\left( \begin{matrix} 0&1&0\\ 1&0&1\\ 0&1&0 \end{matrix} \right)\\ L_y&=\frac{\hbar}{\sqrt{2}}\left( \begin{matrix} 0&-i&0\\ i&0&-i\\ 0&i&0 \end{matrix} \right)\\ L_z&=\;\;\;\hbar\left( \begin{matrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{matrix} \right) \end{align}

Show that:

1. Find the commutator of \(L_x\) and \(L_y\).
2. Find the matrix representation of \(L^2=L_x^2+L_y^2+L_z^2\).
3. Find the matrix representations of the raising and lowering operators \(L_{\pm}=L_x\pm iL_y\). (Notice that \(L_{\pm}\) are NOT Hermitian and therefore cannot represent observables. They are used as a tool to build one quantum state from another.)
4. Show that \([L_z, L_{\pm}]=\lambda L_{\pm}\). Find \(\lambda\). Interpret this expression as an eigenvalue equation. What is the operator?
5. Let \(L_{+}\) act on the following three states given in matrix representation. \begin{equation} \left|{1,1}\right\rangle =\left( \begin{matrix} 1\\0\\0 \end{matrix} \right)\qquad \left|{1,0}\right\rangle =\left( \begin{matrix} 0\\1\\0 \end{matrix} \right)\qquad \left|{1,-1}\right\rangle =\left( \begin{matrix} 0\\0\\1 \end{matrix} \right) \end{equation} Why is \(L_{+}\) called a “raising operator”?

Instructor's Guide

Introduction

This activity is meant to lay the foundation of what raising and lowering oporators are and how they can be used. This material will become very important for students' study of symmetry matrices in PH427 and the Quantum Harmonic Oscillator in the Quantum Capstone.

Student Conversations

At this stage, students will not have seen commutators or done much matrix multiplication in a while, so students may progress lower here than you'd expect. It will be important for the teaching team to be on the look out for groups that are confused at the beginning since some will forget that a commutator can have the form \([A,B]=AB-BA\), which is necessary to progress.

Making sure the teaching team has a good handle on the results of each calculation so they can help trouble shoot errors made during matrix multiplication which are hard to catch in the act and usually can most easilty be inferred from an erronous result (which the students themselves won't usually recognize).

Wrap-up

It is a good idea to reinforce the patterns seen in orbital angular momentum to their experiences with spin angular momentum, such as that cross product-like relationship between commutators of cartesian directed angular momenta. Then it becomes easy to contrast those patterns with that of the raising and lower operators and emphasize that these are not observables which correspond to measures of angular momentum but a different object entirely.

While their importance should be emphasized for study of periodic systems and the quantum harmonic oscilator, it should also be mentioned these operators will not be a major focus of this course or our study of the Hydrogen atom as we head into the home stretch of the course. This content is largely a very important detour.

Representations of the Infinite Square Well

Consider three particles of mass \(m\) which are each in an infinite square well potential at \(0<x<L\).

The energy eigenstates of the infinite square well are:

\[ \left\langle {x}\middle|{E_n}\right\rangle =\phi_n(x) = \sqrt{\frac{2}{L}}\sin{\left(\frac{n \pi x}{L}\right)}\]

with energy eigenvalues \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\)

The particles are initially in the states, respectively: \begin{eqnarray*} |\psi_a(0)\rangle &=& A \Big[\left|{E_1}\right\rangle + 2i \left|{E_4}\right\rangle - 3\left|{E_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[\sqrt{\frac{2}{L}}\sin{\left(\frac{\pi x}{L}\right)} + i \sqrt{\frac{8}{L}}\sin{\left(\frac{4\pi x}{L}\right)} - \sqrt{\frac{18}{L}}\sin{\left(\frac{10\pi x}{L}\right)} \right]\\[6pt] \psi_c(x,0) &=& C x(x-L) \end{eqnarray*}

For each particle:

1. Determine the value of the normalization constant.
2. At \(t=0\), what is the probability of measuring the energy of the particle to be \(\frac{8\pi^2\hbar^2}{mL^2}\)?
3. Find the state of the particle at a later time \(t\).
4. What is the probability of measuring the energy of the particle to be the same value \(\frac{8\pi^2\hbar^2}{mL^2}\) at a later time \(t\)?
5. What is the probability of finding the particle to be in the left half of the well?

Student Conversations

1. Help students recognize that particle \(a\) and particle \(b\) are in the same state.
2. For normalization, emphasize that you must calculate the square of the norm of the state BEFORE you integrate.
3. The energy value given is simplified - students need to recognize that this energy corresponds to \(n=4\).
4. Time evolving particle \(c\) is brutal for the students. Reassure students that they have to leave it as a sum. Setting up the integral is the point here. For time expediancy, encourage students to leave the integral to be evaluated later.
5. For Hamiltonian's that don't don't depend on time, the probabilities of measuring energies are time independent.
6. Emphasize to students that you can't calculate the probability of finding a particle in a region in Dirac notation.

1. Set-Up a Sequential Measurement

   1. Add an analyzer to the experiment by:

      1. Break the links between the analyzer and the counters by clicking on the boxes with up and down arrow labels on the analyzer.
      2. 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.

   2. 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 \(S_z=\frac{-\hbar}{2}\)?

   3. Set the first analyzer to measure \(S_z\) and the second analyzer to measure \(S_x\).

      

      What have you learned from these experiments?

   
   

2. 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.

   

   
   

3. 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 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.

\(\boxed{\begin{array}{lll} \ell& m & \quad\quad\quad\; Y_\ell^m(\theta,\phi) \\[.35cm] \hline \\[.03cm] 0 & 0 & \quad\;\; Y_0^0=\sqrt{\frac{1}{4\pi}} \\[.35cm] 1 & 0 & \quad\;\; Y_1^0=\sqrt{\frac{3}{4\pi}}\cos\theta \\[.35cm] & \pm1 & \quad Y_1^{\pm1}=\mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi} \\[.35cm] 2 & 0 & \quad\;\;Y_2^0=\sqrt{\frac{5}{16\pi}}\left(3\cos^2\theta-1 \right) \\[.35cm] & \pm1 & \quad Y_2^{\pm1}=\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos \theta e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_2^{\pm2}=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm2i\phi} \\[.35cm] 3 & 0 & \quad\;\;Y_3^0=\sqrt{\frac{7}{16\pi}}\left(5\cos^3\theta-3 \cos\theta\right) \\[.35cm] & \pm1 & \quad Y_3^{\pm1}=\mp\sqrt{\frac{21}{64\pi}}\sin\theta \left(5\cos^2\theta-1\right)e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_3^{\pm2}=\sqrt{\frac{105}{32\pi}} \sin^2\theta\cos\theta e^{\pm2i\phi} \\[.35cm] & \pm3 & \quad Y_3^{\pm3}=\mp\sqrt{\frac{35}{64\pi}}\sin^3\theta e^{\pm3i\phi} \\[.001cm] \end{array}}\)

Spherical harmonics are continuous functions on the surface of a sphere.

The \(\ell\) and \(m\) values tell us how the function oscillates across the surface.

Spherical harmonics are complex valued functions.

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.

Students do calculations for time evolution for spin-1.

In a Stern-Gerlach experiment, the arrival of an atom at a measurement counter is a random process. I would like to use the results of the experiments to answer the question:

What is the probability \(\mathcal{P}\) that an atom will arrive at the top counter?

In the case where all the atoms arrive at the top counter, the probability is 1. However, what if I send 10 atoms through the analyzer and detect 3 atoms in the top counter? How confidently can I conclude that the probability is 0.3? What if I repeat my experiment and send 10 more atoms though the analyzer but detect 4 atoms in the top counter? I probably want to revise my estimate. If I do a bunch of sets of experiments, I will get a distribution of probabilities. Therefore, I'm going to need statistical tools to answer my questions:

1. What is the best estimate of the probability, given the experimental data?
2. How confident am I of that estimate?

To find the best estimate of the probability, I'm going to do a bunch of sets of experiments and take the mean. The mean probability will be my best estimate of the probability.

To determine how confident I am in the estimate, I'm going to consider the shape of the distribution. (For random processes like the Stern-Gerlach experiment - or coin flipping experiments, where there are 2 possible outcomes for each experiment - the underlying distribution is a binomial distribution.) To get a distribution, I can't do just one Stern-Gerlach experiment, or even a one set of Stern-Gerlach experiments - I have to do a bunch of sets of Stern-Gerlach experiments.

Some Definitions

\(\mathcal{P}\) is the “true value” of probability of ending up in the top counter for the physical system (measuring \(S_z = \hbar/2\)). (This probability is the number that I'm trying to experimentally estimate.)

In 1 Stern-Gerlach experiment, as single atom passes through the analyzer and is detected at a counter.

I'm going to do a bunch of experiments and organize them into \(N\) sets. Each individual set \(n\) will include \(M\) particles being sent into an analyzer and counted in a counter.

For example, I can click the "10k" button and send 10,000 particles through the analyzer (i.e., 10,000 experiments). I can record the number of particles in the top counter and then repeat so that I end up with 5 sets of experiments.

\(M\) = the number of Stern-Gerlach experiments in each set. This is the number of particles I send through the analyzer in 1 set. I'll assume that each set has the same number of experiments.

\(x_n\) = the (integer) number of atoms in the top counter after \(M\) Stern-Gerlach experiments

\(\mathcal{P}_n\) is the probability I determine for 1 set of \(M\) Stern-Gerlach experiments.

\(N\) = the number of sets of Stern-Gerlach experiments (note: \(n\) is an index that indicates a single set of experiments)

\(\bar{\mathcal{P}}\) is the mean probability determined from \(N\) sets of \(M\) experiments. This will be my estimation of the true probability.

Best Estimate of the Probability: the Mean

The probability I determine for a set of experiments (like in the table above) is:

\[\mathcal{P}_n = \frac{x_n}{M}\]

The mean of these probabilities is:

\[\bar{\mathcal{P}}= \frac{1}{N}\sum_{n=1}^N \mathcal{P}_n\]

If I want to, I can also write the mean probability in terms of the number of atoms counted: \begin{align*} \bar{\mathcal{P}} &= \frac{1}{N}\sum_{n=1}^N \mathcal{P}_n \\ &= \frac{1}{N}\sum_{n=1}^N \frac{x_n}{M}\\ &= \frac{1}{NM}\sum_{n=1}^N x_n\\ &= \frac{1}{M}\bar{x} \end{align*}

Experimental Uncertainty - the Standard Error

In this section I'm going to argue that the standard error is a sensible thing to report as the experimental uncertainty (and for making statistical inferences). In order to understand the standard error, I'm first going to talk about the variance and the standard deviation.

The Variance

In order to quantify how spread out the distribution is, conceptually I'm tempted to find the average of the difference between each probability \(\mathcal{P}\) and the mean of the distribution. The problem with this approach is that this average should be zero - the average is at the center of all the observations!

\begin{align*} \frac{1}{N}\sum_{n=1}^N (\bar{\mathcal{P}}-\mathcal{P}_n) &= \frac{1}{N}\left(\sum_{n=1}^N \bar{\mathcal{P}}\right)-\left(\frac{1}{N}\sum_{n=1}^N\mathcal{P}_n\right)\\[12pt] &= \frac{1}{N}N\bar{\mathcal{P}}-\bar{\mathcal{P}} \\[12pt] &=\bar{\mathcal{P}}-\bar{\mathcal{P}}\\[6pt] &= 0 \end{align*}

One way to get around this is to square all the differences first. The variance is the squared difference between the probability for one set of SG experiments and the mean probability:

\[var = \frac{1}{N}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2\]

All contributions to the variance are positive, so the variance is greater than zero (though a zero variance is still technically possible if the distribution is one number). The larger the variance, the more spread out the distribution.

The Standard Deviation

The standard deviation is the square root of the variance. \begin{align*} SD &= \sqrt{var}\\ &= \sqrt{\frac{1}{N}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2}\\ &\rightarrow \sqrt{\frac{1}{N-1}\sum_{n=1}^{N} (\bar{\mathcal{P}} - \mathcal{P}_n )^2} \quad \mbox{ for small N} \end{align*}

(For N < 30ish, there are theoretical arguments about how the standard deviation of the sample underestimates the true standard deviation of the system, so the prefactor is front is made a smidge larger.)

• The standard deviation does not decrease with more sets of experiments. The standard deviation does not vary with \(N\). The standard deviation comes from taking an average (you add up N things and then divide by \(N\)). As \(N\) increases, the standard deviation does not change with the number of experiments (it might fluctuate a little because of the random nature of additional experiments, especially if the total number of experiments is small, but if you plot \(SD\) vs. \(N\) (e.g., the number of particles in the top counter), the best fit line should have a near-zero slope). Therefore, the standard deviation is a characteristic of the system.

  

• The standard deviation is a characteristic of the combined physical and measurement system, including information about the distribution of the physical system and sources of random uncertainty during the measurement process.

• Binomial vs “normal” distribution For large numbers of experiments (\(M\)), a binomial distribution is very close to a normal (or Gaussian) distribution. For a normal distribution, 68% of measurements will lie within 1 standard deviation from the mean.

  

  

• The standard deviation is a special kind of average, an rms average. The \(rms\) stands for “root mean square” and describes the order of operations in the calculation (first you square, then you average, then you take a square root). So, the rms average allows me to get a sense of how far away individual probabilities \(\mathcal{P}_n\) are from \(\bar{\mathcal{P}}\) without running into the problem with doing a regular average, as described above.

• Subtle difference between the distributions of number of atoms and probability. The standard deviation does depend on the number of measurements in each set (which conceptually makes sense to me because the standard deviation is a characteristic of the combined physical and measurement system). For binomial distributions, the standard deviation for the distribution of the number of atoms is

  \[SD_{x_n} = \sqrt{M\mathcal{P}(1-\mathcal{P})}\]

  where \(\mathcal{P}\) is the true probability I'm trying to measure. This equation for standard deviation is not general; it is only true for binomial distributions, where each experiment is a coin flip, atom through a Stern-Gerlach analyzer, etc. This equation tells me a system with a characteristic probability \(\mathcal{P}\), the standard deviation will be twice as large if each set includes 100 experiments than if each set includes 25 experiments. (The mean will also be bigger because here I'm counting particles.)

  

  In contrast, the standard deviation of the distribution of the probabilities is different by a factor of \(M\)

  \begin{align*} SD_{\mathcal{P}} &= \sqrt{\frac{1}{N} \sum_{n=1}^N (\bar{\mathcal{P}_n}-\mathcal{P}_n)^2} \\[12pt] &= \sqrt{\frac{1}{N} \sum_{n=1}^N \left(\frac{\bar{x_n}}{M}-\frac{x_n}{M}\right)^2}\\[12pt] &= \sqrt{\frac{1}{M^2N} \sum_{n=1}^N (\bar{\mathcal{x}_n}-\mathcal{x}_n)^2} \\[12pt] &= \frac{1}{M}\sqrt{\frac{1}{N} \sum_{n=1}^N (\bar{x_n}-x_n)^2} \\[12pt] &= \frac{1}{M} s_{x_n} \\[12pt] &= \frac{1}{M}\sqrt{M\mathcal{P}(1-\mathcal{P})}\\[12pt] &= \sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{M}} \end{align*}

  This tells me that the distribution of probabilities will get narrower as the number of experiments in each set gets larger.

  

The Standard Error

The standard error (a.k.a. the standard deviation of the mean) \(\sigma\) is an a measure of how well I know the mean. In this lab, I'm estimating the true value of the probability by doing many (N) sets of Stern-Gerlach experiments and finding the mean of these sets. Now imagine that I repeat this whole process many times (N times) so that I get many means. Each mean is a better estimate of the true value of the probability than any individual probabily I measure, and the distribution of these means is much narrower than the distribution of the probabilities that I determined from each set of Stern-Gerlach experiments. If I compute the standard deviation of the distribution of means, it turns out that:

\[StErr_{\mathcal{P}} = \frac{SD_{\mathcal{P}}}{\sqrt N}\]

(see Taylor, pp. 147-148 for a nice derivation) If I only find one mean (\(\bar{\mathcal{P}}\) from my original \(N\) sets of Stern-Gerlach experiments), I can be confident that there is a 68% chance that my mean lies is within 1 standard error from the true value of the probability.

In the case of Stern-Gerlach experiments (which following a binomial distribution):

\begin{align*} StErr_{\mathcal{P}} &= \frac{\sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{M}}}{\sqrt{N}} \\ &= \sqrt{\frac{\mathcal{P}(1-\mathcal{P})}{MN}} \end{align*}

• The standard error of the probability varies with the total number of experiments. Notice that \(MN\) is the total number of Stern-Gerlach experiments that I run (\(M\) experiments in each set for \(N\) sets). The standard error is inversely proportionally to the square root of the total number of Stern-Gerlach experiments. It doesn't matter how I group them. If I do 10,000 Stern-Gerlach experiments, the standard error is the same as if I do 10 sets of 100 experiments, 20 sets of 50 experiments, or 10,000 sets of 1 experiment. It's hard to tell from looking at the plot alone that the standard error is the same:

  

• The standard error as a measure of uncertainty The standard error tells me about how well my mean probability estimates the true value of the probability. Conceptually, it makes sense that the more experiments I do, the more confidence I should have in my estimate.

Reporting Uncertainty

To answer the question of how confident I am in my estimates, I could choose to report the uncertainty as the standard deviation or the standard error. These two options have different meanings (for this discussion, I'm going to assume that \(M\) is large and we have an approximately normal distribution):

\(\bar{\mathcal{P}} \pm SD_{\mathcal{P}}\)
Meaning: If I do one more set of SG experiments, there is a 68% chance that the probability I measure will fall in this range.

\(\bar{\mathcal{P}} \pm StErr_{\mathcal{P}}\)
Meaning: If I repeat the entire exercise, doing \(N\) sets of \(M\) SG experiments, there is a 68% change that the average probability I determine will fall in this range.

In this case, the standard error of the mean is closer to the thing I mean by my confidence in my estimate.

Comparing Values

If I wanted to compare my estimate of the probability to either (1) someone else's measurement or (2) a theoretically expected answer, both of which I'll call \(\mathcal{P}_{exp}\), I might describe the difference between values in terms of the number of standard errors.

\[ t = \frac{|\bar{\mathcal{P}} - \mathcal{P}_{exp}|}{StErr}\]

A smaller \(t\) corresponds to a higher likelihood that the two values come from the same normal distribution. The boundary between acceptable and unacceptable differences is a matter of opinion, to be decided by the experimenter (and the reader). For normal distributions, many scientists consider differences of:

\(t<2\) to be acceptable (“the discrepancy is insignificant”) and

\(t>2\) to be unacceptable (“the discrepancy between values is significant.”).

Differences that are \(t\approx 2\) (1.9-2.6) are generally considered inconclusive.

For a normal distribution, there is a 95% likelihood that the true values lies with 2 standard errors of mean, meaning \(t<2\).

For Your Information:

Inferential statistical tests can be used to formally compare values, for example:

• One Sample T-Test: A one sample t-test allows us to test whether a sample mean (of a normally distributed variable) significantly differs from a hypothesized value.
• Independent Samples T-Test: An independent samples t-test is used when you want to compare the means of a normally distributed dependent variable for two independent groups.
• Binomial Test: A one sample binomial test can be used to determine whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value. (Remember that for large values of \(M\), a binomial distribution approximates a normal distribution, so the first two tests might be applicable.)

For each statistical test, a set of assumptions need to be met in order for the test to give reliable, meaningful results. For example, a one sample t-test assumes that the data are normally distributed.

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.

What I call the Quantum State Postulate (McIntyre's Postulate #1, p. 27) says:

“The state of a quantum mechanical system, including all the information you can known about it, is represented mathematically by a normalized ket.”

These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.

• Practice time evolution
• To calculate quantum probabilities in Dirac and Wavefunction notation
• To visualize how wave wavefunctions depend on time after time evolution
• To notice that not all probabilities of operators will depend on time

Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.

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 a PhET simulation to explore the time evolution of a particle in an infinite square well potential.