Page 1 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.
Page 2 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.
Page 3 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! Then, 2s 1/2 and 2p 1/2 levels of hydrogen still have the same energy.
Page 4 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:
Page 5 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 10-5 ev 2s1/2-2p1/2 splitting due to Lamb shift 4.4 10-6 ev 2s hyperfine splitting 7.3 10-7 ev (177.6 MHz) -1.8 10-4 ev 3.6 10-5 ev 1s hyperfine splitting 5.8 10-6 ev (1420.5 MHz)
Page 6 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.
Page 7 He ground state is 1s 2 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 (1s 2, 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 1s 2 1 S 0 Helium excited states 1snl 1 L J or 3 L J 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, m l, m s ] different from all the other ones. We will use for m s =1/2 and for m s =-1/2. List of distinct sets of quantum number combinations in order of increasing n and l: n l m l m s 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) 2s subshell (2 electrons) 2p subshell (6 electrons) Maximum number of electrons in a subshell 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.
Page 8 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 1s 2 2s 6 Li, 7 Li Sodium Na 11 1s 2 2s 2 2p 6 3s 23 Na Potassium K 19 1s 2 2s 2 2p 6 3s 2 3p 6 4s 39 K, 40 K, 41 K Rubidium Rb 37 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 5s 85 Rb, 87 Rb Cesium Cs 55 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 6s 133 Cs Francium Fr 87 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 4f 14 5s 2 5p 6 5d 10 6s 2 6p 6 7s none Simplified energy-level diagram for Na Singly-charged ions of alkaline-earth metal atoms Note: first excited level becomes metastable (long-lived).
Page 9 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.
Page 10 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 LSJM J > if E ZE <<E s-o <<E re. In this case, the interaction of atom with the magnetic field can be treated as a perturbation to the fine-structure levels.
Page 11 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 M F are good quantum numbers, but not M I and M J. 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,
Page 12 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.