Students find a wavefunction that corresponds to a Gaussian probability density.
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The Gaussian \begin{equation} {\cal P}(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation} is normalized so that the area under the curve is equal to one. If this Gaussian represents the probability density for a free quantum mechanical particle, what is a possible wavefunction?
group Small Group Activity
10 min.
group Small Group Activity
60 min.
Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation
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.keyboard Computational Activity
120 min.
probability density particle in a box wave function quantum mechanics
Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.face Lecture
30 min.
Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.assignment Homework
For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
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.assignment Homework
keyboard Computational Activity
120 min.
group Small Group Activity
30 min.