Exercises on differential equation¶

The pendulum equation¶

We consider the equation

$\theta^{\prime\prime} = -\sin{\theta}$

Write a function that returns an array containing the position and the angular velocity of the pendulum for $N$ instants $t_i$ between 0 and $T$ .

The initial position is $\theta=0$ . Plot the phase space trajectory for different values of the initial velocity. Angle will be represented between -π and π.

Solving the Schrödinger equation using the finite element method¶

Let us consider the Schrödinger equation with hbar = m = 1

$\frac{d\psi}{dt} =-i\left( -\frac12 \frac{d^2\psi}{dx^2} + V(x)\psi\right)$

The potential is $V(x) = \frac12 k x^2$ with $\kappa=.5$ .

To solve the equation, we will truncate the x-axis to values between $x_\mathrm{min}$ and $x_\mathrm{max}$ . We will also discretize the x-axis with small steps ( $\Delta x$ ).