Activity: Time Evolution Refresher (Mini-Lecture)

Central Forces 2023 (3 years)
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.

Begin with Schrödinger's Equation: \begin{equation} i\hbar\frac{d}{dt}\left|{\Psi(t)}\right\rangle =\hat{H}\left|{\Psi(t)}\right\rangle \end{equation} If we know how the Hamiltonian acts on a state, we can use Schrodinger's equation to find out how it evolves in time.

Let's do this for the case of the particle on a ring (c.f. McIntyre 3.1 for a spin 1/2 system). Write out the arbitrary state as a superposition of energy eigenstates, putting the time dependence we are trying to find in the expansion coefficient. \begin{equation} i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle =\hat{H}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle \end{equation} We are writing our state in this basis because we know what the Hamiltonian does to these kets \(\left|{m}\right\rangle \) via the energy eigenvalue equation. \begin{equation} \hat{H}\left|{m}\right\rangle =E_m\left|{m}\right\rangle \end{equation} Using this result, we can calculate: \begin{align} i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle &=\hat{H}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty} c_m(t) \hat{H}\left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty}\left(\frac{\hbar^2}{2I}\, m^2\right)\, c_m(t) \left|{m}\right\rangle \label{energy} \end{align} The energies \(E_m\) are the energies for the physical system we care about, i.e. a particle on a ring. If we had a different system, a different Hamiltonian, the structure of this equation would be similar but the values of energy in Eqn. (\ref{energy}) would be different.

Because we are solving for the time-dependent coefficients, let's get rid of the sum using an inner product with an arbitratry state \(\left\langle {k}\right|\) by using the orthonormality condition. \begin{align} \left\langle {k}\right|i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle &=\left\langle {k}\right|\sum_{m=-\infty}^{\infty} E_m c_m(t) \left|{m}\right\rangle \\ i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left\langle {k}\middle|{m}\right\rangle &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \left\langle {k}\middle|{m}\right\rangle \\ i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \delta_{km} &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \delta_{km}\\ i\hbar\frac{d}{dt} c_k(t) &= E_k c_k(t)\\ \end{align} We now have a first-order ODE for the unknown time-dependent coefficients. \begin{equation} \frac{d}{dt}c_k(t)=-\frac{iE_k}{\hbar}c_k(t) \end{equation} Because this ODE is linear with constant coefficients, the solution is an exponential. \begin{equation} c_k(t)=Ae^{-\frac{iE_k}{\hbar}t} \end{equation} The constant factor \(A\) is what we have been calling the time-independent version of \(c_k\) up to this point; it is the probability amplitude of the eigenstate it accompanies at \(t=0\). You can plug in \(t=0\) to help verify this.

The solution is: \begin{align} \left|{\Psi(t)}\right\rangle &=\sum_{m=-\infty}^{\infty} e^{-\frac{iE_m}{\hbar}t}c_m \left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty} e^{-\frac{i\hbar}{2I}\, m^2t}c_m \left|{m}\right\rangle \end{align} Here are some important features of the solution:

  • We can only put time evolution phases on states that are written in the energy basis;
  • The time dependent phase is pure imaginary; it does not effect the normalization of the state;
  • Each eigenstate gets its own time-dependent phase that includes the energy of that eigenstate.

It would be valuable for you to go through this derivation and see which parts depend on the particular system we are using (the quantum ring) and which parts are generic to any solution of Schrödinger's Equation.

Lecture Notes: Time Evolution Refresher

Ask students to write on a small whiteboard something they know/remember about Schrödinger's Equation. Use their responses to write the general form: \begin{equation} i\hbar\frac{d}{dt}\left|{\Psi(t)}\right\rangle =\hat{H}\left|{\Psi(t)}\right\rangle \end{equation} If we know how the Hamiltonian acts on a state, we can use Schrodinger's equation to find out how it evolves in time.

Let's do this for the case of the particle on a ring (c.f. McIntyre 3.1 for a spin 1/2 system). Write out the arbitrary state as a superposition of energy eigenstates, putting the time dependence we are trying to find in the expansion coefficient. \begin{equation} i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle =\hat{H}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle \end{equation} Remind the students that we are writing our state in this basis because we know what the Hamiltonian does to these kets \(\left|{m}\right\rangle \). This could be a good place to ask students to write the energy eigenvalue equation. \begin{equation} \hat{H}\left|{m}\right\rangle =E_m\left|{m}\right\rangle \end{equation} Use their responses to get calculate: \begin{align} i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle &=\sum_{m=-\infty}^{\infty}\hat{H} c_m(t) \left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \left|{m}\right\rangle \\ &=\sum_{m=-\infty}^{\infty}\left(\frac{\hbar^2}{2I}\, m^2\right)\, c_m(t) \left|{m}\right\rangle \end{align} Empasize that the energies \(E_m\) are the energies for a particle on a ring. If we had a different system, a different Hamiltonian, the structure of this equation would be similar but the values of energy would be different.

Because we are solving for the time-dependent coefficients, let's try to get rid of the sum using an inner product with an arbitratry state \(\left\langle {k}\right|\) so we can use the orthonormality condition. \begin{align} \left\langle {k}\right|i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left|{m}\right\rangle &=\left\langle {k}\right|\sum_{m=-\infty}^{\infty} E_m c_m(t) \left|{m}\right\rangle \\ i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \left\langle {k}\middle|{m}\right\rangle &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \left\langle {k}\middle|{m}\right\rangle \\ i\hbar\frac{d}{dt}\sum_{m=-\infty}^{\infty} c_m(t) \delta_{km} &=\sum_{m=-\infty}^{\infty} E_m c_m(t) \delta_{km}\\ i\hbar\frac{d}{dt} c_k(t) &= E_k c_k(t)\\ \end{align} We now have a first-order, linear ODE with constant coefficients. \begin{equation} \frac{d}{dt}c_k(t)=-\frac{iE_k}{\hbar}c_k(t) \end{equation} See if the students recall what the solution to this equation is. There should be someone who recognizes the solution to be an exponential. \begin{equation} c_k(t)=Ae^{-\frac{iE_k}{\hbar}t} \end{equation} Explain that the constant factor \(A\) is what they have been calling the time independent version of \(c_k\) up to this point, it is the probability amplitude of the eigenstate it accompanies at \(t=0\). You can have them plug in \(t=0\) to help verify this. \begin{equation} \left|{\Psi(t)}\right\rangle =\sum_m e^{-\frac{iE_m}{\hbar}t}c_m \left|{m}\right\rangle \end{equation} Point out important features of the solution:

  • We can only put time evolution phases on states that are written in the energy basis;
  • The time dependent phase is pure imaginary; it does not effect the normalization of the state;
  • Each eigenstate gets its own time-dependent phase that includes the energy of that eigenstate.

  • assignment Visualization of Wave Functions on a Ring

    assignment Homework

    Visualization of Wave Functions on a Ring
    Central Forces 2023 (3 years) 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.)
    1. 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.
    2. 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.
    3. 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.
    4. 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 Normalization of Quantum States

    assignment Homework

    Normalization of Quantum States
    Central Forces 2023 (3 years) 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}
  • group Superposition States for a Particle on a Ring

    group Small Group Activity

    30 min.

    Superposition States for a Particle on a Ring

    central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

    Quantum Ring Sequence

    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.
  • keyboard Position operator

    keyboard Computational Activity

    120 min.

    Position operator
    Computational Physics Lab II 2022

    quantum mechanics operator matrix element particle in a box eigenfunction

    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.
  • assignment Radiation in an empty box

    assignment Homework

    Radiation in an empty box
    Thermal physics Radiation Free energy Thermal and Statistical Physics 2020

    As discussed in class, we can consider a black body as a large box with a small hole in it. If we treat the large box a metal cube with side length \(L\) and metal walls, the frequency of each normal mode will be given by: \begin{align} \omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2} \end{align} where each of \(n_x\), \(n_y\), and \(n_z\) will have positive integer values. This simply comes from the fact that a half wavelength must fit in the box. There is an additional quantum number for polarization, which has two possible values, but does not affect the frequency. Note that in this problem I'm using different boundary conditions from what I use in class. It is worth learning to work with either set of quantum numbers. Each normal mode is a harmonic oscillator, with energy eigenstates \(E_n = n\hbar\omega\) where we will not include the zero-point energy \(\frac12\hbar\omega\), since that energy cannot be extracted from the box. (See the Casimir effect for an example where the zero point energy of photon modes does have an effect.)

    Note
    This is a slight approximation, as the boundary conditions for light are a bit more complicated. However, for large \(n\) values this gives the correct result.

    1. Show that the free energy is given by \begin{align} F &= 8\pi \frac{V(kT)^4}{h^3c^3} \int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi \\ &= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3} \\ &= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3} \end{align} provided the box is big enough that \(\frac{\hbar c}{LkT}\ll 1\). Note that you may end up with a slightly different dimensionless integral that numerically evaluates to the same result, which would be fine. I also do not expect you to solve this definite integral analytically, a numerical confirmation is fine. However, you must manipulate your integral until it is dimensionless and has all the dimensionful quantities removed from it!

    2. Show that the entropy of this box full of photons at temperature \(T\) is \begin{align} S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3 \\ &= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3 \end{align}

    3. Show that the internal energy of this box full of photons at temperature \(T\) is \begin{align} \frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3} \\ &= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3} \end{align}

  • assignment Quantum Particle in a 2-D Box

    assignment Homework

    Quantum Particle in a 2-D Box
    Central Forces 2023 (3 years) 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.)
    1. 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.
    2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
    3. 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*}

    4. 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.
  • group Working with Representations on the Ring

    group Small Group Activity

    30 min.

    Working with Representations on the Ring
    Central Forces 2023 (3 years)
  • 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 \(kT\).

  • assignment Sum Shift

    assignment Homework

    Sum Shift
    Central Forces 2023 (3 years)

    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:

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

  • computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

    computer Mathematica Activity

    30 min.

    Visualization of Quantum Probabilities for a Particle Confined to a Ring
    Central Forces 2023 (3 years)

    central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

    Quantum Ring Sequence

    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.

Author Information
Corinne Manogue
Keywords
schrodinger equation time dependence stationary states
Learning Outcomes