One-dimensional quantum systems

Roberto Di Remigio, Luca Frediani

In this assignment we require you to look at some one-dimensional quantum systems and leverage the knowledge of Python acquired during the seminars. Many of the tasks can also be achieved with pen-and-paper derivations, but we encourage to use Python as much as possible. Remember that:

1. We have looked at similar tasks together already
2. numpy, scipy, matplotlib and sympy are very rich modules, with extensive online documentation that you are encouraged to search through
3. You are encouraged to work through this assignment as a group
4. You can ask us for help if you get stuck on a specific task
5. The deadline for submission is in two weeks

Exercise 1. Hydrogen-like atoms

The eigenfunctions for the hydrogen-like atoms can be expressed as the product of a radial and angular part: \begin{equation} \psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{l}^{m}(\theta, \phi) \end{equation} These eigenfunctions are expressed in spherical polar coordinates. $r$ is the distance between the electron and the nucleus, $\theta$ is the polar angle and $\phi$ the azimuthal angle.

1. Given a value of $n>0$, what are the possible valid values of $l$?
2. Given a value of $n>0$ and a valid value of $l$, how many nodes does $R_{nl}(r)$ have?

The general form of the radial function is: \begin{equation} R_{nl} (r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \frac{2 Z r}{n a_{\mu}} \right ) \end{equation} where:

• $L_{n-l-1}^{2l+1}$ are the generalized Laguerre polynomials.
• $a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}} = \frac{\hbar c}{\alpha\mu c^2} ={{m_{\mathrm{e}}}\over{\mu}} a_0$, $\alpha$ is the fine structure constant. Here, $\mu$ is the reduced mass of the nucleus-electron system, that is, $\mu = {{m_{\mathrm{N}} m_{\mathrm{e}}}\over{m_{\mathrm{N}}+m_{\mathrm{e}}}}$

1. Choose $Z=1$ and plot the radial wavefunctions for $n=1, 2, 3$ and for all corresponding, valid values of $l$. Hint: scipy has a module for special functions Check there if you can find a function for the generalized Laguerre polynomials
2. Plot the corresponding radial probability distributions. Note that this is given by $4\pi r^2|R_{nl}(r)|^2$, since we are in spherical polar coordinates!
3. Now repeat the same plots for different nuclei, i.e. change the atomic number $Z$ and the mass of the nucleus entering the definition of the reduced mass.

Exercise 2. Spherical harmonics

Now that we have looked at the radial part of hydrogen-like atoms wavefunctions, we can have a closer look at their angular part. The angular part is expressed in terms of spherical harmonics: \begin{equation} Y_l^m(\theta, \phi) = (-1)^{(m+|m|)/2}\sqrt{\frac{(2l+1)(l-|m|)!}{4\pi(l+|m|)!}}P_l^{|m|}(\cos\theta)\mathrm{e}^{\mathrm{i}m\phi} \end{equation} where $P_l^{|m|}(\cos\theta)$ are associated Legendre polynomials.

1. Given a value of $l$, what is the degeneracy of the angular part? That is, how many $m$ values are valid?
2. The spherical harmonics are, in general, complex functions. For $l$ assigned, is there a way to get the spherical harmonics for that $l$ value to be real functions? Hint: it might be easier to consider the explicit form of the $l=0$, $l=1$ and $l=2$ spherical harmonics to start reasoning about this question.
3. Plot the module, $|Y_l^m(\theta, \phi)|$, the real part, $\Re(Y_l^m(\theta, \phi))$ and the imaginary part, $\Re(Y_l^m(\theta, \phi))$ of the spherical harmonics for $l=1, 2, 3$ and all corresponding, valid values of $m$, following this code example for $l=0, m=0$. As you can see, scipy already has the definition of the spherical harmonics

Exercise 3. Finite potential wells

In the lectures and seminars we have considered potential wells with infinite barries, i.e. cases where the quantum particle is confined inside a finite space and cannot escape. More interesting systems are those where the particle is confined within a well, with finite barriers:

The potential is thus: \begin{equation} V(x) = \begin{cases} 0 \quad\quad \text{if} \,\, -L/2\leq x \leq L/2 \\ V_0 \quad\quad \text{otherwise} \end{cases} \end{equation}

1. Using finite differences, solve the quantum mechanical problem for the finite potential well. You should prepare a table with the first 5 eigenvalues, plot the first 5 eigenvectors and corresponding probability distributions. Hint: Avoid full diagonalization of $\mathbf{H}$ This can be done by using sparse matrix operations as provided by scipy. You can then ask for the $k$ lowest eigenvectors and eigenvalues. The general function to use is scipy.sparse.linalg.eigs. For symmetric matrices it's preferable to use scipy.sparse.linalg.eigsh Read the documentation to understand how to use these functions

2. Can this model be used as a crude approximation to the harmonic oscillator potential? Why? If yes, how would you compare the results from the two models? Hint: Think about the turning points of the harmonic oscillator

3. What happens when the energy is higher that $V_0$?

Bonus

Another interesting potential is the semi-infinite potential well:

which is a crude model for the bond energy curve of a diatomic molecule. Use the finite difference method to compute eigenvalues and eigenvectors for this potential. Plot some of the eigenvectors and the corresponding probability distributions.