Quantum Harmonic Oscillator

Schrödinger's Equation

Weird things happening in QM

• What does a state mean?

• Why can't we encounter the effects of QM in our scales? (A = Decoherence)

Quantum Harmonic Oscillators (QHO)

1. Ubiquitous

2. Theorists LOVE them they're really elegant

3. Interesting Features

   • Eigenstates are coherent states

Once you describe the Hamiltonian of QHO you get that (for $\hbar = 1, \omega$ given) $H\vert\varphi> = \omega(n+\frac{1}{2})\vert\varphi>$

We can perturb the QHO to obtain the hamiltonian (family) that depends on $\lambda$:

Note that if a classical particle were to be moving in the now quantic potential it would not feet much of a difference if ir were near $x = 0$ since the potential looks close enough to quartic there. That means that for low energies the classical motion will be harmonic in nature. For high energy, other things happen (remove).

Numerical Matrix Diagonalization

• Focus on numerical methods to diagonalize matrices

• You can do it instantaneously on numpy, scipy, etc but the point is undestanding why that works

Jacobi Algorithm for Real Symmetric Matrices

1. At each step find the largest off diagonal element, say $S_{pq}$ (or cycle thru each off-diagonal index $p,q$ one at a time)

2. Transform it to an orthogonal matrix $J$ s.t. that element and its symmetric version are both 0

3. Check whether the relative error $\delta <\epsilon$ where $\epsilon$ is the desired error. If not, repeat the previous steps

Complex Matrices

For a complex number $U = H_r + iH_{im}$ we get that its matrix representation is:

One can check that if $O = \begin{pmatrix} O_r & -O_{in} \\ O_{in} & O_r \end{pmatrix}$ is orthogonal (i.e $OO^T = \mathbb{I},\ O_rO_{im}^T-O_{im}O_r^T = 0$) then $U$ is unitary (i.e $UU^T = \mathbb{I}$).