Activity: Representations of the Infinite Square Well

Quantum Fundamentals 2022 (2 years)
  • group Small Group Activity schedule 120 min. build Tabletop Whiteboard with markers, Computers with Maple, Voltmeter, Coordinate Axes, A handout for each student description Student handout (PDF)

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:

\[ E_n(x) = \sqrt{\frac{2}{L}}\sin{\left(\frac{n \pi x}{L}\right)}\]

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

The particles are initially in the states, respectively: \begin{eqnarray*} |\psi_a(0)\rangle &=& A \Big[ 2i \left|{E_4}\right\rangle - 3\left|{E_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[ 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 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 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 first 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 the normal square happens 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 evolve with 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.
  • assignment Wavefunctions

    assignment Homework

    Wavefunctions
    Quantum Fundamentals 2022 (2 years)

    Consider the following wave functions (over all space - not the infinite square well!):

    \(\psi_a(x) = A e^{-x^2/3}\)

    \(\psi_b(x) = B \frac{1}{x^2+2} \)

    \(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)

    In each case:

    1. normalize the wave function,
    2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
    3. find the probability that the particle is measured to be in the range \(0<x<1\).

  • assignment Quantum concentration

    assignment Homework

    Quantum concentration
    bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 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.
  • assignment One-dimensional gas

    assignment Homework

    One-dimensional gas
    Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of \(N\) particles, each of mass \(M\), confined to a one-dimensional line of length \(L\). The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature \(T\). You may assume that the temperature is high enough that \(k_B T\) is much greater than the ground state energy of one particle.
  • assignment Spin Fermi Estimate

    assignment Homework

    Spin Fermi Estimate
    Quantum Fundamentals 2022 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:
    1. 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.
    2. 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.
  • group Time Evolution of a Spin-1/2 System

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    Time Evolution of a Spin-1/2 System
    Quantum Fundamentals 2022 (2 years)

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    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.
  • assignment Ideal gas in two dimensions

    assignment Homework

    Ideal gas in two dimensions
    Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
    1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

    2. Find an expression for the energy \(U\) of the gas.

    3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

  • face Fermi and Bose gases

    face Lecture

    120 min.

    Fermi and Bose gases
    Thermal and Statistical Physics 2020

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  • group Quantum Expectation Values

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    30 min.

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    Quantum Fundamentals 2022 (2 years)
  • keyboard Sinusoidal basis set

    keyboard Computational Activity

    120 min.

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    Computational Physics Lab II 2022 (2 years)

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  • assignment Unknowns Spin-1/2 Brief

    assignment Homework

    Unknowns Spin-1/2 Brief
    Quantum Fundamentals 2022 (2 years) 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 \(\left|{\psi_4}\right\rangle \).
    1. 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 \(\left|{-}\right\rangle \).
    2. 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.
    3. 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.
    4. 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?

Learning Outcomes