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.
1. << Using Arms to Represent Time Dependence in Spin 1/2 Systems | Arms Sequence for Complex Numbers and Quantum States |
2. << Matrix Elements and Completeness Relations | Completeness Relations | Compare \& Contrast Kets \& Wavefunctions >>
This activity is part of the Arms Sequence for Complex Numbers and Quantum States.
Students should be familiar with spin-1/2 and spin-1 systems, particularly represently quantum states with Dirac notation, histograms, matrices, and arms.
Set-Up
Instructor writes a quantum state in Dirac notation on the board. “How would you represent this state using arms?”
The instructor then asks the pair of volunteers (volunteers should write their answer on the board):
Instructor asks.“What would we have to do in order to represent a spin-1 state with arms?” Answer - add a student.
Instructor repeats exercise for a spin-3 system (7 basis kets)
Instructor asks the students to consider a new situation: “Now imagine that I have a particle and the position of the particle in an observable I'm interested in.”
“How could we present a quantum state in the position basis with Arms?” Ask as many students as volunteer to join the Arms representation. Still not enough - we'd need an infinite number of people! And an infinite number of people in between each person.
What would the histogram look like? Answer: A function of x. This is the wavefunction.
What would a quantum state written in a 1-D position basis look like? Answer: An infinitely long column with each entry corresponding a position eigenstate. The set of all the entries is the wavefunction.
in Dirac notation, label with the eigenvalue:
\[\hat{X}\left|{x}\right\rangle =x\left|{x}\right\rangle \]
Complete - can write down a completion relation. The sum for a discrete basis, like spin, become an integral for a continuous basis:
\[ \sum_{s} \left|{s}\right\rangle \left\langle {s}\right| = 1 \]
\[ \int \left|{x}\right\rangle \left\langle {x}\right| dx = 1 \]
Talk about the dimensions of \(\left|{x}\right\rangle \rightarrow\) 1/\(\sqrt{\mbox{length}}\)
Write a state in the position basis by applying a completeness relation:
\begin{align*} \left|{\psi}\right\rangle &= \left(\int \left|{x}\right\rangle \left\langle {x}\right| dx \right) \left|{psi}\right\rangle \\ &= \int \left|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int \left\langle {x}\middle|{\psi}\right\rangle \left|{x}\right\rangle dx \\ &= \int \psi(x) \left|{x}\right\rangle dx \\ \end{align*}
Notice: \(\left\langle {x}\middle|{\psi}\right\rangle \) is the wavefunction. Because of the integral, we can't just take the norm squared of the wavefunction at a particular x to get the probability of the particle being at a particular location. The wavefunction doesn't have the right dimensions! Instead, the wavefunction is a “probability amplitude density”. If you square the wavefunction, you get a probability density. You have to integrate the norm square of the wavefunction over a length to get a probability.
In fact, asking the question “What is the probability of the particle being at a particular point?” is not a question we can answer with the formalism of quantum mechanics. We can only calculate the probability of a particle being in an infinitesimal region:
\[\mathcal{P}(\mbox{between }x_0 \mbox{and }x_0+dx) = |\left\langle {x}\middle|{\psi}\right\rangle |^2 dx\]
The multiplication by \(dx\) makes the quantity dimensionless.
To calculate the probability of the particle being in a finite region:
\[\mathcal{P}(\mbox{between }x=a \mbox{and }x=b) = \int |\left\langle {x}\middle|{\psi}\right\rangle |^2 dx\]
accessibility_new Kinesthetic
10 min.
Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation
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.accessibility_new Kinesthetic
10 min.
accessibility_new Kinesthetic
10 min.
quantum states complex numbers arms Bloch sphere relative phase overall phase
Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).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.accessibility_new Kinesthetic
10 min.
keyboard Computational Activity
120 min.
inner product wave function quantum mechanics particle in a box
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.format_list_numbered Sequence
accessibility_new Kinesthetic
10 min.
arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.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.accessibility_new Kinesthetic
30 min.