Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
1. Completeness Relations | Outer Product of a Vector on Itself >>
The state vectors \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \) form an orthormal basis. Orthonormal bases have properties listed below.
For each property, write an equation that illustrates the property for both:
- The Cartesian basis vectors \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\).
- The spin state vectors \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
The elements are norm 1.
The elements are orthogonal.
- The elements form a complete set. Any vector in the space can be written as a linear combination of the two.
assignment Homework
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: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
face Lecture
30 min.
group Small Group Activity
30 min.
keyboard Computational Activity
120 min.
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 Homework
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.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.group Small Group Activity
10 min.
assignment Homework
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.)