Homework 6A

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

The energy of a single electron around a proton is

\[ E_n = -\dfrac {m_e e^4}{8\epsilon_0^2 h^2 n^2}\]

with \(n=1,2,3 ...\infty\). The corresponding eigenstates can be expressed as the product of a angular and radial functions

\[ |\psi(r,\theta,\phi) \rangle = | R(r) \rangle |Y_{\ell}^{m_l}(\theta,\phi) \rangle \]

which have three quantum numbers to describe an unique wavefunction, \(n\), \(l\) and \(m_l\) each with unique range permissible. The fact that energy depends only on \(n\) indicates that this system has degenerate wavefunction. List all the eigenstates (explicitly pointing out the appropriate quantum numbers) that have the following energies:

1. \( E_n = -\dfrac {m_e e^4}{32\epsilon_0^2 h^2}\)
2. \( E_n = -\dfrac {m_e e^4}{72\epsilon_0^2 h^2}\)
3. \( E_n = -\dfrac {m_e e^4}{128\epsilon_0^2 h^2}\)

Q2

How many radial nodes (i.e., nodes in \(R(r)\)) are there in the following hydrogen orbitals?

1. \(| \psi_{1s} \rangle\)
2. \(| \psi_{2s} \rangle\)
3. \(| \psi_{3s} \rangle\)
4. \(| \psi_{2p_x} \rangle\)
5. \(| \psi_{2p_z} \rangle\)
6. \(| \psi_{3d_{z^2}} \rangle\)
7. \(| \psi_{100} \rangle\) (the subscript refers to the \(n\),\(l\),\(m_l\) triad of quantum numbers)

Q3

Explain why the radial distribution function \(r^2R(r)^*R(r)\) should be used to discussed the probability of finding an electron from a nucleus rather than the square of the radial component of the wavefunction \(R^2\). Draw a plot of both the radial distribution function and the square of the radial component of the wavefunction.

Q4

Show that the Hamiltonian operator \(\hat{H}=(p^2/2m)\) of a rigid rotor commutes with all three components of \(\vec{L}\). Thus \(H\), \(L^2\), and \(L_z\) are mutually compatible observables.

Q5

Derive the following commutation relationship (one of three in 3-D space) using the representation as differential operators

\[ [\hat{L}_z,\hat{L}_x] =i \hbar \hat{L}_y\]

What does this equation and the other two commutation relationships mean (in your words)?

Q6

What is \( \langle I \rangle \), the expectation value of the moment of inertia, of the hydrogen atom for the 1s, 2s, and 2p\(_z\) states.

Q7

Construct a Grotrian diagram (Figure 2 in Lecture 18) that diagrams all possible electronic transition allowed for all hydrogen atom eigenstates with the \(n=1\), \(n=2\) and \(n=3\) quantum numbers. Use double end arrows to represent the absorption and emission transitions.

Hint: there are different ways to diagram this out, but for credit, the relevant quantum numbers must be clearly indicated.

Q8

Consider a hydrogen atom. Is an electron in the 1s orbital (on average) further, closer, or the same distance from the nucleus in the 2s orbital? Confirm this intuition quantum mechanically by calculating the expectation value.

Q9

What is the least probable radius of a 2p electron in a hydrogen atom? What is the most probable radius?

Q10

What is the orbital angular momentum of an electron in the following orbitals

1. 1s
2. 2s
3. 2p
4. 3d
5. 5f

How many angular and radial nodes exist for the wavefunctions described by the above states?

Q11

Given:

\[ \hat{L}_z = -i \hbar \frac{\partial}{\partial \varphi} \]

\[\hat{L^2} = - \hbar^2\left( \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2} {\partial\varphi^2}\right) \]

Show that \( Y_0^0 \) and \( Y_1^{-1} \) are eigenfunctions of \( \hat{L^2} \) and\( \hat{L}_z \).