Math Review

Note

Quantum Mechanics is a difficult subject. Often students focus on the math, thinking this is the hardest part. However, the hardest part is mastering the concepts first and then do the math, which then seems pretty obvious, almost magically – Vincent Meunier, Rensselaer Polytechnic Institute

Note

The unitary circle and complex number, a crash introduction, can be found here.

Practice and Review Packets

A general Review Packet can be downloaded here: pdf
Complex numbers can be reviewed with this packet here: pdf
Linear algebra (eigen-value problems, etc) can be reviewed here: pdf

Screencast of live session

You can watch the screencast from 01/29/2021 (recitation 1) here.

Test your knowledge

$a$ , $b$ and $\epsilon$ are real numbers and $\epsilon \ll 1$ . Evaluate $e^{i\epsilon a} b e^{-i\epsilon a}$ to first order in $\epsilon$ . (hint: start by Taylor expanding $f(a) = e^{i\epsilon a}$ to first order)
$A$ and $B$ are $n\times n$ matrices and $\epsilon\ll 1$ is a real number. Evaluate $e^{i\epsilon A} B e^{-i\epsilon A}$ to first order in $\epsilon$ . (hint: your answer should be written in terms of the commutator $[A,B]$ )
Consider the matrix $A = \mqty(0&-i\\i&0)$ . Is $A$ Hermitian? Explain your answer.
Consider the matrix $A = \mqty(0&-i\\i&0)$ . Find the eigenvalues and normalized eigenvectors of $A$ .
Consider the matrix $A = \mqty(0&-i\\i&0)$ . Construct a unitary matrix $U$ whose columns are the normalized eigenvectors of $A$ , and show by explicit matrix multiplication that $U$ is unitary.
Consider the matrix $A = \mqty(0&-i\\i&0)$ . Show by explicit matrix multiplication that some product involving $A$ and $U$ produces a diagonal matrix. (hint: the diagonal elements should be the eigenvalues from question 4.)