Lecture 12 REVIEW

Hydrogen atom

Potential energy, from Coulomb's law

spherical harmonics

Potential is spherically symmetric. Therefore, solutions must have form

n is principal quantum numberl is orbital angular momentum quantum numberm is magnetic quantum number

Energy levels of the hydrogen atom

Note: energy depends only on principalquantum number n in non-relativistic approximation.

There are letters associated with values of orbital angular momentum. The first few are:

For example, state withn=1 l=0 is referred to as 1s,

n=2 l=1 is referred to as 2p, and so on.n=2 l=0 is referred to as 2s,

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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.

Angular momentum: summary

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

and one state with spin s = 0, m = 0:

This is called a

triplet configuration.

This is called asinglet configuration.

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Relativistic corrections of energy terms

Proper treatment: solve Dirac equation. However, the corrections are small so we can use ournon-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:

Putting all three effects together, we get the following expression for the energy:

Note that this expression does not depend on ! Then, 2s1/2 and 2p1/2 levels of hydrogen still have the same energy.

fine structure

"Fine-structure" refers to splitting of levels with the same but different j. For example,

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Lamb shiftIn 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 zero-point 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)

-1.8×10-4eV

3.6×10-5eV1s hyperfine splitting5.8×10-6eV (1420.5 MHz)

Fine-structuresplitting4.5×10-5eV 2s1/2-2p1/2

splitting due to Lamb shift4.4×10-6eV

2s hyperfine splitting

7.3×10-7eV (177.6 MHz)

Final summary: H energy level structure

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Neutral Helium (Z=2), two electrons

Hamiltonian

Two hydrogenic Hamiltonians (with Z=2), one for electron 1 andone 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 constructtwo-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 acceptthe 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 beantisymmetric.

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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 1S0

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 ↓

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.

1s shell (2 electrons)

2s subshell (2 electrons)

2p subshell (6 electrons)

Maximum number of electrons in a subshell

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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 , 7Li

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

Simplified energy-level diagram for Na

Singly-charged ions of alkaline-earth metal atomsNote: first excited level becomes metastable (long-lived).

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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 aperturbation 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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