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 ml) [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
http://en.wikipedia.org/wiki/Spherical_coordinate_system
http://en.citizendium.org/wiki/Spherical_polar_coordinates
•r defines the sphere
•q defines the cone
•f defines the plane and
• the intersection of the
three is the point of
interest
Lecture 25
Spherical Polar Coordinates
Theta Equation (f equation just gives exp{imlf})
Radial Equation
Lecture 25
Spherical Harmonics
Lecture 25
Spherical Harmonics
See also the hydrogen atom
viewer at:
http://www.falstad.com/qmatom/
http://www.physics.umd.edu/courses/Phys402/AnlageSpring09/spherical_harmonics.gif
Lecture 24
Spherical Harmonics
En does not depend on l or ml (for a single-electron atom), but only on n.
We have the following conditions on the three Q.N’s: l<n ; -l<=ml<=l
http://en.wikipedia.org/wiki/Atomic_orbital
Lecture 25
Angular Momentum
There is another uncertainty relation among the
components of angular momentum (DLxD Ly>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
|Lz| 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
http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/zeeman.html#c4
•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” Dml=+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
http://faculty.gvsu.edu/majumdak/public_html/OnlineMaterials/ModPhys/QM/QM3D/zeeman_fig1.gif
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 a to Zo”
•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
ml=0 state). ONLY TWO
PROJECTIONS APPEARED TO
BE ALLOWED!
Lecture 26
Stern-Gerlach Experiment
http://phet.colorado.edu/simulations/sims.php?sim=SternGerlach_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
Theta Equation (f equation just gives exp{imlf})
Radial Equation
In the limit of very small r, you can show that the radial equation has solutions that
behave like Rnl(r) ~ rl. 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. 2010
• 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!
http://www.corrosionsource.com/handbook/periodic/periodic_table.gif
Lecture 27
The periodic Table
Alternative periodic table of Benfey
http://en.wikipedia.org/wiki/File:Elementspiral.svg
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
3d
4s
4p
We can get some insight into the relative
Energies of these three orbitals from
the website:
http://keisan.casio.com/
Lecture 27
Hydrogen 2p, 3s, 3p
2p
3s
3p
These are some of the orbitals providing
Shielding for the 3d and 4s,p orbitals.
http://keisan.casio.com/
Lecture 27
Combining angular momentum
Lecture 27
Energy splitting for 2 electrons
in the 4p/4d states