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Lecture 25
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) [9 responses]
•The Energy levels are NOT tied directly to the angular
momentum. [2 responses]
•There are several different states with the same energy in the
QM atom [4 answers addressed this but in different words]
•Angular momentum is more involved, and more subtle than in
the Bohr atom [3 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 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
http://en.wikipedia.org/wiki/Atomic_orbital
http://www.corrosionsource.com/handbook/periodic/periodic_table.gif
Alternative periodic table of Benfey
http://en.wikipedia.org/wiki/File:Elementspiral.svg
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 am not going to lie, I cannot quite figure out what this question
is asking for. (I think that this was true for many, only 3 said so).
• …However, since we cannot precisely know the other two
components of L, we must assume that L is not exactly
quantized, and that in fact only one component at any time,
L(z), is quantized. (This is an important point, but be careful).
•The magnitude of the L vector in 3 dimensions must be greater
than or equal to the magnitude of the z component. (this is what
I was after).
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 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 25
Stern-Gerlach Experiment
http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment
•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
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 25
Anomalous Zeeman Effect
From Gasioriowicz “Quantum “Physics”
It may be better to think of this as the
“Generalized” Zeeman effect
Guidelines for Term Paper Assignment
Due 23 Nov. 2009
• You are to read an article from early in the era of “Modern
Physics” and compose a concise (2 page) 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.
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.
•I think that the effects would be largest at l = 1 because the
electron is most tightly bound to the nucleus . (13 answered this
way, for some this reflects a slight misunderstanding, for others I
think it was just a silly mistake l= 0 is possible!)
• judging from figure 7.12, the largest l will give you the smallest
expected value for radial distance, which in turn should give you
the biggest effects with the nucleus. . (3 answered this way, it
reflects a good observation, but it is not correct).
•l=0 or “the lowest value of (6; this is correct, because the nucleus
is so small it is only the limit as r->0 that matters not <r>.
Lecture 26
Radial Wavefunctions
From Gasioriowicz “Quantum “Physics”
Lecture 26
Radial Wave Functions
From T&R Fig 7.12
Lecture 27
The periodic Table
Lecture 27
Many-electron Atoms
Lecture 26
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?
•in multi-electron atoms there are more energy levels for the
atoms to be in so different n's dont have the same energy
•Because now there are not only spin-orbital interactiona but
spin-spin and orbital-orbital as well.
•Electrons with higher l-values have less elliptical orbits and
thus feel smaller Coulomb force than those with lower l-values.
Lecture 26
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?
•Spin-orbit interaction. (5 answers, be careful this is in H as well)
• More/different states (3 answers; Careful, more different values
for E, but the q#’s available to each electron are the same!!)
•Interactions between the electrons/screening (5; this is correct,
and the book states this, but it is easy to lose it because of the
emphasis they put on S-O coupling, addition of angular
momentum etc. .
•5 answers were basic non-sequitors
•in multi-electron atoms there are more energy levels for the atoms to be in so different
n's dont have the same energy
•Because now there are not only spin-orbital interactiona but spin-spin and orbital-orbital
as well.
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/has10/SpecExec.cgi
Lecture 26
Radial Wave Functions
From Gasioriowicz “Quantum “Physics”
Lecture 26
Angular Momentum
From Gasioriowicz “Quantum “Physics”
Lecture 25
Combining angular momentum
Lecture 27
Energy splitting for 2 electrons
in the 4p/4d states