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.

- 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." - 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 $c_2 = c_3 = 0$ - 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.