Lecture 24 The Hydrogen Atom revisited Major differences between the “QM” hydrogen atom and Bohr’s model (my list): The electrons do not travel in orbits, but in well defined states (orbitals) that have particular shapes (probability distributions for the electrons, or linear combinations thereof) [8 responses, although expressed in about 8 different ways] New quantum numbers introduced ( l and m l ) [4 responses] The Energy levels are NOT tied directly to the angular momentum. [DVB] There are several different states with the same energy in the QM atom [DVB] Other 6 responses NOTE: the energy levels are (nominally) the same, until we account for subtle effects that lift degeneracy.

Lecture 25 Spherical Polar Coordinates r defines the sphere  defines the cone  defines the plane and the intersection of the three is the point of interest

Lecture 25 Spherical Polar Coordinates Radial Equation Theta Equation (  equation just gives exp{im l  })

Lecture 25 Spherical Harmonics

See also the hydrogen atom viewer at:

Lecture 24 Spherical Harmonics E n does not depend on l or m l (for a single-electron atom), but only on n. We have the following conditions on the three Q.N’s: l <n ; - l <=m l <= l

Lecture 25 Angular Momentum There is another uncertainty relation among the components of angular momentum (  L x  L y >0.5 hbar |L| z, which says that you cannot know precisely more than one component of the angular momentum. Comment on the connection between this result and the relation between |L z | and (|L| 2 ) 1/2. I don't see the relationship. My best guess is that the uncertainty relationship has something to do with why L^2 = l(l+1) instead of l^2. … [several had trouble understanding the Q, but several zeroed in on this point, ‘though none cut to the chase better. Good Guess!]

Lecture 25 Zeeman effect The “Normal Zeeman effect is just what you’d expect on the basis of quantizing only orbital angular momentum (all state-splittings are of the same size, and we have the “selection rule”  m l =+1,0,-1. The “anomalous” effect is what shows up if the electron spin plays a role, not just orbital angular momentum.

Lecture 25 Zeeman Effect

Lecture 25 Anomalous Zeeman Effect From Gasioriowicz “Quantum “Physics” It may be better to think of this as the “Generalized” Zeeman effect

Lecture 26 Dipole in non-uniform field Fig. 7.7 A uniform field exerts only a torque on a dipole, but a non-uniform field can exert a force

Lecture 26 Stern-Gerlach Experiment Figures from J. W. Rohlf “Modern Physics from  to Z o ” The Stern-Gerlach experiment looked for direct evidence of quantization of angular momentum projection by looking at the deflections of silver atoms in a strong magnetic field gradient. They saw the atoms deflected into bands (as expected), rather than the smooth blob expected classically; surprisingly, they saw all atoms deflected up or down (none went through undeflected as expected for the m l =0 state). ONLY TWO PROJECTIONS APPEARED TO BE ALLOWED!

Lecture 26 Stern-Gerlach Experiment This is a computer simulation that can give you a bit of insight into the way quantum mechanical angular momenta behave.

Lecture 26 Radial Wave Functions There are some phenomena in atomic physics that depend on the direct interactions between the electrons and the nucleus. By looking at figure 7.12, identify the value(s) of l (the angular momentum quantum number) for which you’d expect these effects to be largest. l = 0 or l “the smallest it could be” (15 answered one of these ways.) l=2 or the largest it could be. (4 answered this way).

Lecture 27 Spherical Polar Coordinates Radial Equation Theta Equation (  equation just gives exp{im l  }) In the limit of very small r, you can show that the radial equation has solutions that behave like R n l ( r) ~ r l. It can also be shown that this solution has n- l -1 radial nodes (n-l “bumps” in the radial distribution). These have important consequences for the structure of the periodic table and how electrons interact with nuclei.

Lecture 27 Radial Wavefunctions From Gasioriowicz “Quantum “Physics”

Lecture 27 Radial Wave Functions

Guidelines for Term Paper Assignment Due 22 Nov You are to read an article from early in the era of “Modern Physics” and compose a concise (no more than 2 pages) summary of its contents. The summary should provide some of the context of the work (what was known, or believed going into the work, and what influence this work had on future development) as well as a summary of the key points in experimental design or interpretation that made the work successful. You will find a collection of suitable papers in electronic form on the syllabus page of the website (under the link “Historical Articles for Term Paper”). If you have another article that you would like to summarize instead of one of these, that is allowed, but if you want to use this path, please check with me about the suitability of the article you have in mind (and have a copy for me to look at) before you get started. A subfolder contains an example historical paper (Anderson’s discovery of the positron) with an example summary (from me). Anderson’s paper is not eligible for you to use in your summary!

Lecture 27 The periodic Table

Alternative periodic table of Benfey

Lecture 27 Many-electron Atoms

Lecture 27 Multi-electron Atoms In the hydrogen atom, all states with a given value of the principal quantum number (n) have equal energies (they are “degenerate”). What is the primary reason that this is no longer the case for multi-electron atoms? Because it depends on l and m, not just on n [2 responses] Pauli Exclusion principle keeps two electrons from occupying the same state: [6 responses] Coulomb interactions among the electrons: Screening [6 responses] Other [6 responses] No answer: 26

Lecture 27 Hydrogen 3d, 4s and 4p 4s 3d 4p We can get some insight into the relative Energies of these three orbitals from the website:

Lecture 27 Hydrogen 2p, 3s, 3p 3s 2p 3p These are some of the orbitals providing Shielding for the 3d and 4s,p orbitals.

Lecture 27 Combining angular momentum

Lecture 27 Energy splitting for 2 electrons in the 4p/4d states