Download Lecture 12: Review.

Lecture 12 REVIEW
Hydrogen atom
Potential energy, from Coulomb's law
Potential is spherically symmetric. Therefore, solutions must have form
n is principal quantum number
l is orbital angular momentum quantum number
m is magnetic quantum number
spherical harmonics
Energy levels of the hydrogen atom
Note: energy depends only on principal
quantum number n in non-relativistic approximation.
There are letters associated with values of orbital angular momentum. The first few are:
For example, state with
n=1 l=0 is referred to as 1s,
n=2 l=0 is referred to as 2s,
n=2 l=1 is referred to as 2p, and so on.
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Angular momentum: summary
For a given value of l, there are 2l+1 values of m:
Generally, half-integer values of the angular momentum are also allowed but not for orbital angular moment.
Spin
, therefore
and there are two eigenstates
We will call them spin up
and spin down
.
Addition of angular momenta
Addition of two spin 1/2
Three states
with spin s = 1, m = 1, 0, -1
This is called a
triplet configuration.
and one state with spin s = 0, m = 0:
This is called a
singlet configuration.
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Relativistic corrections of energy terms
Proper treatment: solve Dirac equation. However, the corrections are small so we can use our
non-relativistic solution and apply first-order perturbation theory.
Relativistic effects:
(1) The relativistic dependence of electron mass on its velocity → small decrease in kinetic energy.
(2) Darwin term - smears the effective potential felt by the electron → change in the potential energy.
(3) Spin-orbit term: the interaction of the magnetic moment of the electron [due to electron spin] with the
effective magnetic field the electrons see due to orbital motion around the nucleus.
Energy shift due to spin-orbit interaction:
"Fine-structure" refers to splitting of levels with the same
but different j. For example,
fine structure
Putting all three effects together, we get the following expression for the energy:
Note that this expression does not depend on
have the same energy.
! Then, 2s1/2 and 2p1/2 levels of hydrogen still
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Lamb shift
In quantum electrodynamics, so-called "radiative corrections" to
the Dirac theory are obtained by taking into account
interactions of electrons with the quantized electromagnetic
field.
In QED, a quantized radiation field in the lowest-energy state of
NOT the one with ZERO electromagnetic fields, but there exist
zero-point oscillations. Then, there are non-zero
electromagnetic fields that are present even in the absence of
any external radiation due to the fields associated with zeropoint energy. Such fields are referred to as "vacuum"
fluctuations since they are present even in the "vacuum", i.e.
absence of observable photons.
There are two dominant QED contributions that cause the Lamb shift: self-energy and
vacuum polarization.
The corresponding Feynman diagram for self-energy is
The Feynman diagram for vacuum polarization QED term is
Hyperfine structure
All levels of hydrogen, including
fine structure components, are
split into two more components.
Additional splitting of the atomic energy levels appear because of the interaction of the nuclear
moments with the electromagnetic fields of the electrons. The level splitting caused by this interaction
is even smaller than the fine structure, so it is called hyperfine structure.
We consider magnetic-dipole hyperfine interaction, i.e. the interaction of the nuclear magnetic moment
with the magnetic field produced by the electrons at the nucleus, and the Hamiltonian is given by:
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We now define total angular momentum (nucleus + electrons)
Example: hydrogen ground state J=1/2, I=1/2 (proton) F=0,1, so two levels (see picture on previous page)
Final summary: H energy level structure
Fine-structure
splitting
4.5×10-5eV
2s hyperfine splitting
7.3×10-7eV (177.6 MHz)
2s1/2-2p1/2
splitting due to Lamb shift
4.4×10-6eV
-1.8×10-4eV
3.6×10-5eV
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1s hyperfine splitting
5.8×10-6eV (1420.5 MHz)
Neutral Helium (Z=2), two electrons
Hamiltonian
Two hydrogenic Hamiltonians (with Z=2), one for electron 1 and
one for electron 2.
Term that describes repulsion of two electrons.
The two electrons of the He atom are identical particles. Let's review how to treat this.
Identical particles: bosons and fermions
There are two possible ways to deal with indistinguishable particles, i.e. to construct
two-particle wave function that is non committal to which particle is in which state:
Therefore, quantum mechanics allows for two kinds of identical particles: bosons (for the "+"
sign) and fermions (for the "-" sign). In our non-relativistic quantum mechanics we accept
the following statement as an axiom:
All particles with integer spin are bosons, all particles with half integer spin are fermions.
From the above, two identical fermions can not occupy the same state:
It is called Pauli exclusion principle.
Electrons have spin
and; therefore, are fermions. Total atomic wave function has to be
antisymmetric.
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He ground state is 1s2 with L=0 and S=0.
He excited states have configurations 1snl and can have either S=0 (singlet states)
or S=1 (triplet states).
How do we label the atomic states in a general case?
The atomic state is described by its electronic configuration (1s2, for example) and a "term" symbol
that describes total S, L, and J of all electrons. The term symbol is always written as follows:
Numbers are used for S and J but letters S, P, D, F, etc.
are used for L.
Helium ground state 1s2
1S
0
Helium excited states 1snl 1LJ or 3LJ
Building-up principle of the electron shell for larger atoms.
Pauli principle does not allow for two atomic electrons with the same quantum numbers,
therefore each next electron will have to have at least one quantum number [n, ℓ, mℓ, ms ]
different from all the other ones. We will use ↑ for ms=1/2 and ↓ for ms=-1/2.
List of distinct sets of quantum number combinations in order of increasing n and ℓ:
n
ℓ
mℓ
ms
1s
1
0
0
↑
1s
1
0
0
↓
2s
2
0
0
↑
2s
2
0
0
↓
2p
2
1
-1
↑
2p
2
1
0
↑
2p
2
1
1
↑
2p
2
1
-1
↓
2p
2
1
0
↓
2p
2
1
1
↓
1s shell (2 electrons)
Maximum number of
electrons in a subshell
2s subshell (2 electrons)
2p subshell (6 electrons)
Shells with the same n but different l (2s, 2p) may be referred to as either shells or subshells.
There are 2 electrons in n=1 shell and 8 electrons in n=2 shell.
How do these shells get filled in a periodic table, i.e. what are electronic configurations
and terms for ground states of all elements in the periodic table?
Rule 1. The Pauli principle is obeyed.
Rule 2. The total energy of all electrons is minimum for atomic ground state.
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Hund's rules:
Hund's first rule: for every atomic ground state, the total electron spin has maximum value
tolerated by the Pauli principle.
Hund's second rule: for a given spin, the term with the largest value of the total orbital angular
momentum quantum number L, consistent with overall antisymmetrization, has the lowest energy.
Hund's third rule: if an outermost subshell is half-filled or less, the level with the lowest value of the
total angular momentum J=|L-S| has the lowest in energy. If the outermost subshell is more than
half-filled, the level with the highest value of J=L+S has the lowest energy.
Alkali-metal atoms
Z
Configuration
Stable isotopes
Lithium
Li
3
1s2 2s
6Li
Sodium
Na
11
1s22s22p6 3s
23Na
Potassium K
19
1s22s22p63s23p6 4s
39K, 40K, 41K
Rubidium
Rb
37
1s22s22p63s23p63d104s24p6 5s
85Rb, 87Rb
Cesium
Cs
55
1s22s22p63s23p63d104s24p64d105s25p6 6s
133Cs
Francium
Fr
87
1s22s22p63s23p63d104s24p64d104f145s25p65d106s26p6 7s none
, 7Li
Singly-charged ions of alkaline-earth metal atoms
Note: first excited level becomes metastable (long-lived).
Simplified energy-level diagram for Na
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The central-field approximation
The Hamiltonian for N electrons in the Ze Coulomb potential is
Kinetic and potential
energy for each electron in
the Coulomb field
Electrostatic repulsion
between two electrons.
Sum is taken for j>i to
avoid double counting
Large part of the repulsion between the electrons can be treated as a central potential S(r)
since the closed sub-shells within the core have a spherical charge distribution. Therefore, the
interaction between the different shells and the valence electron are also spherically
symmetric. Then, the total potential energy depends only on the radial coordinate:
The Hamiltonian is
When the potential is in this form, the N-electron Schrödinger equation
separates into N one-electron equations
Note that proper antisymmetric wave function has the form of the Slater determinant.
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Self-consistent solutions
How do we determine the central-field potential
?
To get the potential we need to know the wave functions, but to get the wave functions
we need to know the potential … going in a circle …
Procedure (devised by Hartree)
(a) make initial guess for the potential
(b) calculate the electron wave function in this potential
(c) use resulting wave functions to calculate new central-field potential
(d) repeat until the changes in the wave functions and potential get smaller
and converge to self-consistent solution, i.e. solving the radial Schrödinger
equation for that central potential gives back the same wave functions within
the numerical accuracy.
Interaction of atoms with external magnetic field. Zeeman effect.
In 1896 P. Zeeman observed that the spectral lines of atoms were split in the presence of the
external magnetic field.
The atom's magnetic moment has orbital and spin contributions:
The interaction of the atom with an external field is described by
The expectation value of such Hamiltonian can be calculate in the basis |LSJMJ> if
EZE<<Es-o<<Ere. In this case, the interaction of atom with the magnetic field can be treated as a
perturbation to the fine-structure levels.
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Zeeman effect and hyperfine structure
The total atomic magnetic moment is the sum of the electronic and nuclear moments:
but we can neglect the nuclear contribution for most cases since
Then, the interaction Hamiltonian is
While this interaction does not depend on the nuclear spin, the expectation values depends on
the hyperfine structure. We consider the weak-field and strong-field cases.
Zeeman effect of a weak field (
)
In the weak-field regime, the interaction with the external field is weaker than
so it
can be treated as a perturbation to the hyperfine structure. In this regime, F and MF are good
quantum numbers, but not MI and MJ. Taking the projection of the magnetic moments along F
gives
where
The corresponding Zeeman energy shift is
Zeeman effect of a strong field (
)
If the interaction with the external field is greater than
, F is not a good quantum
number. The effect of the hyperfine interaction can be calculated as a perturbation on the
eigenstates,
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Notes on the exam preparation & exam taking:
1. Make sure that you know, understand and can use all formulas and concepts from this
lecture and previous lecture notes.
2. Make sure that you can solve on your own and without looking into any notes any problem
done in class in Lectures or from homeworks (if integrals are complicated, use Maple,
Matematica, etc.)
3. During exam, look through all the problems first. Start with the one you know best and the
one that is shortest to write a solution for.
4. Make sure that you read the problem very carefully and understand what is being asked. If
you are unsure, ask the exam proctor.
5. If you are out of time and you have not finished, write an outline of what you would do to
finish the problem if you had time.
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