Lecture 25 The Hydrogen Atom revisited

Lecture 25The 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 25Spherical Polar Coordinates

http://en.citizendium.org/wiki/Spherical_polar_coordinates

http://en.wikipedia.org/wiki/Spherical_coordinate_system

•r defines the sphere• defines the cone•defines the plane and• the intersection of the three is the point of interest

Lecture 25Spherical Harmonics

http://www.physics.umd.edu/courses/Phys402/AnlageSpring09/spherical_harmonics.gif

See also the hydrogen atom viewer at: http://www.falstad.com/qmatom/

Lecture 24Spherical Harmonics

http://en.wikipedia.org/wiki/Atomic_orbital

http://www.corrosionsource.com/handbook/periodic/periodic_table.gif

http://en.wikipedia.org/wiki/File:Elementspiral.svg

Alternative periodic table of Benfey

Lecture 25Angular Momentum

There is another uncertainty relation among the components of angular momentum (Lx 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 25Zeeman 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” ml=+1,0,-1. The “anomalous” effect is what shows up if the electron spin plays a role, not just orbital angular momentum.

Lecture 25Zeeman Effect

http://faculty.gvsu.edu/majumdak/public_html/OnlineMaterials/ModPhys/QM/QM3D/zeeman_fig1.gif

Lecture 26Dipole 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 25Stern-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 26Stern-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 25Anomalous Zeeman Effect

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

Guidelines for Term Paper AssignmentDue 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 26Radial 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 26Radial Wavefunctions

From Gasioriowicz “Quantum “Physics”


From T&R Fig 7.12

Lecture 27The periodic Table

Lecture 27Many-electron Atoms

Lecture 26Multi-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 26Multi-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.•Electrons with higher l-values have less elliptical orbits and thus feel smaller Coulomb force than those with lower l-values.

Lecture 27Hydrogen 3d, 4s and 4p

4s3d

4p

We can get some insight into the relative Energies of these three orbitals fromthe website:

http://keisan.casio.com/has10/SpecExec.cgi



Lecture 26Angular Momentum


Lecture 25Combining angular momentum

Lecture 27Energy splitting for 2 electrons

in the 4p/4d states

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Lecture 25 The Hydrogen Atom revisited