Quantum Mechanics and Atomic Physics Lecture 21: Pauli Exclusion Principle and MultiMulti--electronelectron atoms http://www. physics. rutgers. edu/ugrad/361 Prof. Sean Oh Last time : Fine structure constant Electron mass : mc2 : ~0.5 MeV Bohifdhr energies: of order α2mc2 : ~10 eV Fine structure: of order α4mc2 : ~10--44 eV Lamb shift: of order α5mc2 : ~10--66 eV 4 2 Hyperfine splitting: of order (m/mp)α mc ::: ~10--66 eV Multi-Electron Atoms Atoms with 2 or more electrons have a new feature: Electrons are indistinguishable! ΨA(1) There is no way to tell them apart! Any measurable quantity (probability , Ψ (2) expectation value, etc.) must not B depend on which electron is labeled 1, 2, etc. S.E. for Multi-electron atoms Let’s consider two electrons in Helium with coordinates: The total Hamiltonian operator for this system is 1 2 So the Schrodinger equation is: S.E. for Multi-electron atoms The total potential Vtot has 3 contributions: 1. V between electron 1 and the nucleus 2. V between electron 2 and the nucleus 3. V between electron 1 and electron 2 For now, let’s consider only #1 and #2 So, Note that the potential function is the same for both electrons S.E. for Multi-electron atoms We get the usual separation of variables Each Ψ will depend on quantum numbers n, ll , mll , ms So, A and B stand for the particular sets of quantum numbers So, let’s call ΨA(1) eigenfunction for electron #1 and has the quantum numbers symbolized by A . S.E. for Multi-electron atoms So, total eigenfunction solution is: And with this separability assumption, the S. E. becomes: This equation suggests that we write the total energy E as: S.E. for Multi-electron atoms So, we can separate this neatly into two independent expressions: ((yThis can be extended to any number of non-non- interacting particles!) Exchange electrons So, we said that the total eigenfunction is: If we exchange electrons 1 and 2 we get: But the first equation gives: And the second (exchanged) equation gives: But these two probabilities are different! This is not acceptable! This means that the expressions for Ψtot above are not valid solutions! Exchange electrons Since: We need this to be satisfied: So the total eigenfunction: + sign: Symmetric eigenfunction ΨSymm --sisiggyn: Anitsymmetric eig enfunction ΨAnti Symmetric and Antisymmetric ΨSymm and ΨAiAnti are degenerate! Same energy They exhibit “exchange degeneracy” They have t he r ig ht propert ies: Note 1/√2 is for normalization, assuming that Ψtot is normalized: Pauli Exclusion Principle Principle was formulated by Wolfgang Pauli in 1925. “Weak form ”: In an atom, no two electrons can be in the same quantum state, i .e . the same set of quantum numbers: n, ll , mll , ms Pauli Exclusion Principle Suppose electrons 1 and 2 are in the same quantum state A . Then: So, ΨSymm permits 2 electrons in the same state. So, ΨSymm violates the Pauli exclusion principle ΨAnti obeys the P auli excl usi on pri nci pl e. Pauli Exclusion Principle “Strong” form of Pauli Exclusion Principle: A multimulti--electronelectron system must have an antisymmetric total eifigenfunct ion. “Strong” because it also incorporates indistinguishability. All particles of half-integer spin (1/2, 3/2, …) have antisymmetric total eigenfunctions and are called “Fermions”, obeying FermiFermi--DiracDirac statistics Electrons, protons, neutrons All particles of integer spin (0, 1, 2, …) have symmetric total eigenfunctions, and are called “Bosons”, obeying BoseBose-- Einstein statistics . Photons, alpha, W and Z particles Required for Bosons Ψtot ∝ΨA (1)ΨB (2)± ΨA (2)ΨB (1) Required for Fermions Helium Example 4 Normal Helium ( 2 He) Even number of spin 1/2 constituents 2 protons , 2 neutrons and 2 electrons Is a Boson 3 Helium-3(3 ( 2 He) 2 protons, 1 neutrons, 2 electrons Is a FrFerm ion Generally, Even number sum of protons+neutrons+electrons Î Boson Odd number sum Î Fermion Total Fermion Eigenfunction So,,,g for Fermion, total eigenfunction must be antisymmetric Can write: So, Ψ(space) and Ψ(spin) must have opposite symmetry idfin order for ΨAnti to bibe antisymmetr ic We had used A and B as abbreviations for particular sets of n , ll , mll , ms Now let’s use a and b as abbreviations for particular sets of n, ll , mll i.e. just the space part Space and Spin Eigenfunctions The space wavefunctions are analogous to RΘΦ for Hydrogen For spins, there is no spatial wavefunction, so Symbolically, Spin Eigenfunctions Example: But this does not have definite symmetry: So, the antisymmetric χ, corresponding to ΨSymm(space) (singlet state): The symmetric χ, corresponding to ΨAnti(space); there are three ways to do it (triplet state): ( +1/2,+1/2 ) ( -1/2,-1/2 ) Spin Angular Momentum In a two electron atom (Helium) the spin angulrmmlar momen tmtum o fthtf the two el ectr ons coupl e to give the total spin: But what is s’? Spin Angular Momentum If s’=0, ms’ = 0 only, and this is the singlet state, which is antisymmetric. We have opposite spins, and S’=0 If s’=1, ms’ = -1,-1, 0, +1, and this is why we get the triplet state which is symmetric. We hlllihave parallel spins Parallel Spins For parallel spins: ss1’=1 triplet state in Helium (called OrthoheliumOrthohelium)) χ symmetric Ψ(space) antisymmetricantisymmetric.. Suppose electrons get close, so a=b (same spatial quantum numbers): Low probability for electrons to have similar coordinates ParallelParallel--spinspin electron repel each other, over and above the Coulomb repulsion. This “exchange” force mainly reflects the exclusion principle Antiparallel Spins For antiparallel spins: s’0’=0 Singlet state in Helium (called Parahelium)Parahelium) χ antisymmetric Ψ(space) symmetric Suppose electrons get close, so a=b (same spatial quantum numbers): This is large, so antiparall el --spspiltin electrons attract each other via the “exchange” force. Example Construct total ( spatial+spin) ground state electron configuration of a Helium atom using the spatial hydrogenic wavefunction notation of Example Construct total ( spatial+spin) first excited state configuration of a Helium atom using the spatial hydrogenic wavefunction If you consider Coulomb repulsion between electrons, will the first excited state be in a singlet state ((paraheliumparahelium)) or in a triplet state (orthohelium)? Example If we put 5 electrons (fermions!) in an infinite square well, what is the ground state energy? Recall the energies for an infinite square well are: n2π 2 2 π 2 2 E = h = n2E , where E = h n 2mL2 1 1 2mL2 Since electrons are fermions, 2 electrons will populate n=1, 2 in n=2,,, and 1 in n=3. So, the ground state energy of this system is: E = 2 ⋅(E1 )+ 2 ⋅(4E1) +1⋅(9E1 ) =19E1 What if we put 5 bosons in the well? All 5 bosons can go into n=1! So ground state energy is E=5E1 Example For two electrons in an infinite potential well, construct the total wavefunction of the ground state. FliifiiilllFor two electrons in an infinite potential well, construct total wavefunctions of the first excited state, wihith an d w ihithout cons iderat ion o f t he electronelectron--electronelectron Coulomb potential Summary/Announcements Next time: MultiMulti--electronselectrons HW 12 will be posted on Nov. 30th (Wednesday). There will be a quiz next class (30th) .