# Exercise 2

In this exercise you will create a superposition and work with the position operator.
Open the file "Exercise 2 - probability density.flow" including the following nodes.

• Spatial dimension: A definition of an x-axis, as well as the number of points (n = 1024, can be adjusted as needed).
• Potential: A definition of a potential.
• Hamiltonian: A definition of the Hamiltonian operator.
• Spectrum: Calculating the 2 lowest eigen energies (can be changed to several by altering the value of $N_{eigenstate}$.
• Linear Combination: Here, $c_1$ and $c_2$ are defined and the linear combination $c_1 * \psi_1 + c_2 * \psi_2$ is calculated as output $\psi$.
• Position Plot: The calculated wave function is displayed with the mean position $\langle x\rangle \pm \sigma_x$

You can in principle build these charts yourself, but we have premade these for you so that you can spend your time on understanding the physics.

1. Try changing the coefficients $c_1$ and $c_2$.
Does the norm-square change as you expect?
Look at the real-part and the imaginary part.
Investigate "Normalize output" and "Normalize coeff."
2. Generalize to the 3 lower energy eigenvalues with the variables $c_1$, $c_2$ and $c_3$.
Try to form a linear combination positioned as far to the right as possible. What can you say about the spread $\sigma_x$ now compared to the spread of the ground state.
Check if necessary, how the individual states looks at ex.$c_1 = 1$ and $c_2 = c_3 = 0$
3. You can also calculate the position $\langle x\rangle$ and the spread $\sigma_x$.
Try to find the x operator in the "Operator" menu on the left. Its input must be connected to the selected x-axis (pull a wire between the yellow points).
Try to find the mean and scatter in the "State Analysis" menu on the left (which must be connected to wires for the wave function $\psi$ and the operator $\hat{O}$).
By combining and varying $c_1, c_2$ and $c_3$ you can vary $\sigma_x$. Which combination gives the smallest/largest $\sigma_x$? Formulate why.