Completeness Relations

1. Activity: $|\pm\rangle$ Forms an Orthonormal Basis

Introduces the idea of a complete basis and completeness as one of the characteristics of a basis.

Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.

2. Activity: Outer Product of a Vector on Itself

Introduces the idea of a completeness relation as a sum of outer products.

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).

3. Activity: Changing Spin Bases with a Completeness Relation

Students practice using a completeness relation to change between spin component bases for a spin-1/2 system

Students work in small groups to use completeness relations to change the basis of quantum states.

4. Homework Problem: Completeness Relation Change of Basis

Students practice at home using completeness relations to change basis from a polar coordinates basis to a cartesian coordinates basis, as well as from an z-component of quantum spin basis to an y-component of quantum spin basis.

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: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
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: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
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 \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]

5. Homework Problem: Matrix Elements and Completeness Relations

Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

What if I want to calculate the matrix elements using a different basis??

The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: $\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y$

In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

where $I$ is the identity operator: $I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}$. This effectively rewrite the $\left|{+}\right\rangle $ in the $\left|{\pm}\right\rangle _y$ basis.

Find the top row matrix elements of the operator $\hat{S}_y$ in the $S_z$ basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

6. Activity: Going from Spin States to Wavefunctions

Instructor introduces the position basis and the completeness relation for the position basis.

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.

7. Activity: Compare \& Contrast Kets \& Wavefunctions

The instructor discusses how to translate from dirac notation to wavefunction notation through the use of the completeness relation for the position basis.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

8. Homework Problem: Dimensional Analysis of Kets

Students use a completeness relation to determine the dimensions of the position basis kets.

$\left\langle {\Psi}\middle|{\Psi}\right\rangle =1$ Identify and discuss the dimensions of $\left|{\Psi}\right\rangle $.
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 $.
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 $.