PHYS274: Atomic Structure III Alpha source Schrodinger’s cat. How does this work ? cyanide Review of hydrogenic wave functions Quiz/Clicker Questions Zeeman Effect (breaking of degeneracy)

Question: What is a “gedanken experiment” ? Study more in PHYS480

The hydrogen atom: Probability distributions I States of the hydrogen atom with l = 0 (zero orbital angular momentum) have spherically symmetric wave functions that depend on r but not on θ or ϕ. These are called s states. The figure (below) shows the electron probability distributions for three of these states. 3

The hydrogen atom: Probability distributions II States of the hydrogen atom with nonzero orbital angular momentum, such as p states (l = 1) and d states (l = 2), have wave functions that are not spherically symmetric. The figure (below) shows the electron probability distributions for several of these states, as well as for two spherically symmetric s states. Question: Which wvfcns are spherically symmetric ? 4

Clicker question on the hydrogen atom This illustration shows radial probability distribution functions for three hydrogen atom wave functions, plotted versus r/a (r = distance from the center of the atom and a = 0.0529 nm). It follows that A. an electron in a 4p state is always farther from the center of the atom than is an electron in a 2p state. B. an electron in a 2p state can be found at the atom’s center. C. a 3p state has three units of orbital angular momentum. D. none of the above is true. Answer: D 5

Clicker question on the hydrogen atom This illustration shows radial probability distribution functions for three hydrogen atom wave functions, plotted versus r/a (r = distance from the center of the atom and a = 0.0529 nm). It follows that A. an electron in a 4p state is always farther from the center of the atom than is an electron in a 2p state. B. an electron in a 2p state can be found at the atom’s center. C. a 3p state has three units of orbital angular momentum. D. none of the above is true. 6

Clicker question on the hydrogen atom The Bohr model and the Schrödinger equation both make predictions about the hydrogen atom. For which of the following quantities are the predictions different? A. the energy of the lowest (n = 1) energy level B. the difference in energy between the n = 2 and n = 1 energy levels C. the orbital angular momentum of the electron in the lowest (n = 1) energy level D. more than one of A., B., and C. E. none of A., B., or C.—the predictions are identical for all of these Answer: C 7

Clicker question on the hydrogen atom The Bohr model and the Schrödinger equation both make predictions about the hydrogen atom. For which of the following quantities are the predictions different? A. the energy of the lowest (n = 1) energy level B. the difference in energy between the n = 2 and n = 1 energy levels C. the orbital angular momentum of the electron in the lowest (n = 1) energy level D. more than one of A., B., and C. E. none of A., B., or C.—the predictions are identical for all of these 8

Which of the following statements are true ? Q30.1 Here l=0,1,2,….n-1 Which of the following statements are true ? Only the n = 0 state can have l = 0 Only the n = 1 state can have l = 0 Only the n = 2 state can have l = -1 Only the n = 3 state can have l = 3 None of the above D 9

Which of the following statements are true ? Q30.1 Here l=0,1,2,….n-1 Which of the following statements are true ? Only the n = 0 state can have l = 0 Only the n = 1 state can have l = 0 Only the n = 2 state can have l = -1 Only the n = 3 state can have l = 3 None of the above D 10

Q30.2 How many quantum states are there in the n=2 hydrogen level ? (neglect intrinsic spin for today) 6 2 3 4 8 Here l=0,1,2,….n-1 Here m=0,±1, ±2,…. ±l Count the states, l=0,1  l=0, m=0; l=1, m=-1,0,1 4 states. 11

Q30.2 How many quantum states are there in the n=2 hydrogen level ? (neglect intrinsic spin for today) 6 2 3 4 8 Here l=0,1,2,….n-1 Count the states: l=0,1  l=0, m=0; l=1, m=-1,0,1 4 states. Here m=0,±1, ±2,…. ±l Count the states, l=0,1  l=0, m=0; l=1, m=-1,0,1 4 states. 12

In the n=4 state, what is the total angular momenum? Q30.3 In the n=4 state, what is the total angular momenum? Count the states, l=0,1  l=0, m=0; l=1, m=-1,0,1 4 states. 13

Q30.3 In the n=4 state, what is the total angular momenum? Here l=0,1,2,….n-1 Recall: n=4  l=3 Here m=0,±1, ±2,…. ±l Count the states, l=0,1  l=0, m=0; l=1, m=-1,0,1 4 states. 14

Cover story in European Journal of Physics B Published 6 October 2016. Journal of Physics B: Atomic, Molecular and Optical Physics, Volume 49, Number 20 Charge density distribution in two overlapping Rydberg atoms, which are in different QM electronic states. 15

Review: Application of the Bohr model In an alkali “Rydberg atom” the principal quantum number may reach n=1000. Question: How big is a Rydberg atom ? 16

Wolfgang Pauli and Niels Bohr having fun Maybe Niels is thinking about an electron spinning on its axis and circling around the nucleus ? Looking at a small toy gyroscope 17

Angular Momentum Quantization and Heisenberg Question: Why does the L vector point along a cone ? Why couldn’t the L vector point in a single direction ? Or always lie in the x-y plane ? Ans: Suppose L was in the x-y plane then Lz=0 Δpz=0 by Heisenberg Δz∞  not possible for a localized state such as a hydrogen atom (contradiction). Also cannot simultaneously measure Lz and Lx or Ly 18

Magnetic moments and the Zeeman effect Electron states with nonzero orbital angular momentum (l = 1, 2, 3, …) have a magnetic dipole moment due to the electron motion. Hence these states are affected if the atom is placed in a magnetic field. The result, called the Zeeman effect, is a shift in the energy of states with nonzero ml. This is shown below. 19

The Zeeman effect and selection rules An atom in a magnetic field can make transitions between different states by emitting or absorbing a photon. A transition is allowed if l changes by 1 and ml changes by 0, 1, or –1. (This is because a photon itself carries angular momentum.) A transition is forbidden if it violates these selection rules. See lower right. Energy levels 20

The Zeeman effect and B field in sunspots The 0.4T B field in the sunspot leads to the Zeeman effect for one spectral line. Pieter Zeeman, 1902 Physics Nobel Prize 21

The anomalous Zeeman effect and electron spin For certain atoms the Zeeman effect does not follow the simple pattern that we have described (see the Figure below). This is because an electron also has an intrinsic angular momentum, called spin angular momentum. 22

The anomalous Zeeman effect and electron spin In 1928, Samuel Goudsmit and George Uhlenbeck, two graduate students in the Netherlands, proposed that the electron has a spin quantum number to explain the anomalous results on the sodium doublet line. (It is split into two lines: 589.0nm and 589.6 nm) Using semi-classical arguments they introduced the spin quantum number s=1/2, with sz=±1/2 23

Magnetic moments (from the Bohr model to QM) Go back to PHYS272 and current loops. Question: What is the potential energy of a magnetic dipole in a B field ? Let’s calculate the magnetic dipole moment in the Bohr model (assume electron moves with velocity v at a radius r around the nucleus) Question: How long does it take the electron to go around the Bohr atom ? 24

Magnetic moments (from the Bohr model to QM) Let’s calculate the magnetic dipole moment in the Bohr model (assume electron moves with velocity v at a radius r around the nucleus) This is the “Bohr magneton” 25

Magnetic moments from orbital ang. momentum Now let’s put aside the Bohr model and get the precise results from QM Here ml=0,±1,±2…±l This result explains the Zeeman effect. 26

For next time Quantum Mechanics to understand the world Read material in advance Concepts require wrestling with material