Dimensional Analysis of Kets

This problem is used in the following sequences
1. << Compare \& Contrast Kets \& Wavefunctions | Completeness Relations |

dirac notation dimensions probability completeness relations

Compare \& Contrast Kets \& Wavefunctions

Lecture

30 min.

Compare & Contrast Kets & Wavefunctions

Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density

Completeness Relations
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
Going from Spin States to Wavefunctions

Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)
Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations
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.
Unknowns Spin-1/2 Brief
Homework

Unknowns Spin-1/2 Brief
Quantum Fundamentals 2023 (3 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?
Completeness Relation Change of Basis
Homework

Completeness Relation Change of Basis
change of basis spin half completeness relation dirac notation
Completeness Relations
Quantum Fundamentals 2023 (3 years)
1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}
  Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
2. Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where $a$ and $b$ are known. Find coefficients $c$ and $d$ such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
3. Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the $S_y$ basis. In otherwords, find $j$ and $k$ such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
Hydrogen Probabilities in Matrix Notation

Small Group Activity

30 min.

Hydrogen Probabilities in Matrix Notation
Central Forces 2023 (2 years)
$|\pm\rangle$ Forms an Orthonormal Basis

Small Group Activity

30 min.

$|\pm\rangle$ Forms an Orthonormal Basis
Quantum Fundamentals 2023 (3 years)
Cartesian Basis $S_z$ basis completeness normalization orthogonality basis

Completeness Relations
Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
Outer Product of a Vector on Itself

Small Group Activity

30 min.

Outer Product of a Vector on Itself
Quantum Fundamentals 2023 (2 years)
Projection Operators Outer Products Matrices

Completeness Relations
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
Changing Spin Bases with a Completeness Relation

Small Group Activity

10 min.

Changing Spin Bases with a Completeness Relation
Quantum Fundamentals 2023 (3 years)
Completeness Relations Quantum States

Completeness Relations
Students work in small groups to use completeness relations to change the basis of quantum states.
Quantum Particle in a 2-D Box
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.
Frequency

Homework

Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2023 (3 years) Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable $M$ is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where $c$ is real and positive. Let the initial state of the system be $\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle $, where $\left|{m_1}\right\rangle $ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? This frequency is the Bohr frequency.

1. $\left\langle {\Psi}\middle|{\Psi}\right\rangle =1$ Identify and discuss the dimensions of $\left|{\Psi}\right\rangle $.
2. For a spin $\frac{1}{2}$ system, $\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1$. Identify and discuss the dimensions of $\left|{+}\right\rangle $ and $\left|{-}\right\rangle $.
3. In the position basis $\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1$. Identify and discuss the dimesions of $\left|{x}\right\rangle $.