PHY206: Atomic Spectra

1. PHY206: Atomic Spectra Lecturer: Dr Stathes Paganis Office: D29, Hicks Building Phone: 222 4352 Email: paganis@NOSPAMmail.cern.ch Text: A. C. Phillips, ‘Intro to QM’

2. Lectures 9-11: Outline • Identical Particles • Particle Exchange Symmetry and its Physical Consequences • Exchange Symmetry with Spin • Bosons and Fermions • Atomic Spectra • Atomic Quantum States • Central Field Approximation • Review Atomic Spectra

3. Identical Particles Consider a system of 2 particles with wavefunction: The (joint) probability to find particle 1 in a volume d3r1 and a particle 2 in a volume d3r2 is: But if the particles are identical, if we exchange them we should get the same probability: we cannot tell which one is 1 and which one 2! This simply means that the two wavefunctions are equal up to a c-phase: Atomic Spectra

4. Identical Particles: cannot be distinguished Obviously the exponential can only be -1 or +1 because Thus, for 2 identical particles the wavefunction must be: Antisymmetric wavefunction under the exchange of 2 identical particles Symmetric wavefunction under the exchange of 2 identical particles Atomic Spectra

5. Physical Consequences of Exchange symmetry Consider the Hamiltonian of the system of 2 non-interacting identical particles, where each particle has mass m and inside a 1D harmonic oscillator potential: As in the single particle case, each of the two particles may occupy single-particle states yn with eigen-energies: If both particle are in the same state with quantum number n: Atomic Spectra

6. Physical Consequences Single Particle State wavefunction: Here yn the eigenfuntion for particle 1 in a harmonic oscillator potential: Notice that if we exchange particles 1 and 2 we get the same eigenfunction. So, a single particle state wavefunction is always symmetric. Reverse this argument: Two identical particles with antisymmetric wavefunction cannot occupy the same single state Atomic Spectra

7. Identical particles in different states For 2 identical particles in different energy states n and n’: Symmetric Antisymmetric Now take x1=x2, so that the 2 particles are in the same position. Then: Two identical particles with antisymmetric wavefunction are never found at the same location. Atomic Spectra

8. Addition of spins A particle with spin s has a spin magnitude is expressed as a linear superposition of 2s+1 eigenvectors: As an example the electron has spin s=1/2 and can described by two eigenstates: Where the ‘up’ arrow denotes the spin-up +1/2 state and the ‘down’ the -1/2 state Now consider a system of two electrons (1) and (2) and try to construct the eigenstates of this 2-electron system Atomic Spectra

9. Addition of spins From these three pieces we can construct 4 eigenstates: SYMMETRIC ANTISYMMETRIC Atomic Spectra

10. Exchange Symmetry with spin Now we re-write the combined (spatial)x(spin) wavefunction of a particle p For a particle in a central potential like in the Hydrogen atom: We have seen that identical particles must have a definite symmetry (otherwise they are distinguishable). Now we reach the point to ask: do particles (like an electron) occasionally choose the symmetric total wavefunction, FS, and sometime the antisymmetric, FA ? How do the particles actually behave? Atomic Spectra

11. Fermions and Bosons The definite symmetry under exchange of identical particles, depends on spin: Systems of particles with integer spin (Bosons) have symmetric wavefunctions under exchange of two particles. Systems of particles with half-integer spin (Fermions) have antisymmetric wavefunctions under exchange of two particles. Since we don’t know particles that can have both Bosonic and Fermionic behaviour this means that in the real world (as we know it) identical particles do not have the freedom to choose between the two definite symmetry options! Supersymmetry: a new theory in which particles can have both Bosonic and Fermionic behaviour (like the two faces of a coin). The superpartners of known particles like electron are called super particles (selectron). The reason we haven’t seen them yet (if they exist) is because supersymmetry is badly broken so that they are heavy and after the big-bang they have already decayed to lighter particles. Atomic Spectra

12. Spin-Statistics Theorem The 2 particle fermion states must be antisymmetric and there are 2 ways to do this: Spin-statistics theorem: for identical quantum particles there are 2 ways to be indistinguishable. Being fermionic (antisymmetric states) or being bosonic (symmetric states) Atomic Spectra

13. Consequences of antisymmetry for electrons: The Pauli Exclusion Principle: Two identical fermions cannot occupy the same single particle state. A single particle state can by occupied by at most one electron. Incompressibility of Solids: When the spin part of two electrons is symmetric the spatial part is antisymmetric, which means the electrons don’t want to be close in space. Molecule formation due to covalent atomic bonding: When the spin part of two electrons is antisymmetric the spatial part is symmetric, which means the electrons prefer to be close in space. Atomic Spectra

14. Atomic Spectra An atom with atomic number Z has Z electrons in the attractive Coulomb field of the nucleus. However electrons also repel each other. In particular as we will see, electrons in outer shells feel (see) less positive nuclear charge because of screening caused by inner shell electrons. The simplest example is the Helium atom with two electrons p and q: The energy levels and eigenfunctions are found by solving the equation: We can solve numerically using the central field approximation. Atomic Spectra

15. The Central Field Approximation In the CFA: each electron in the atom moves independently in a central attractive Coulomb potential potential due to the nucleus (atomic number Z). An example is the following approximate central potential: The energy levels and eigenfunctions are found by solving the equation: However the energy levels depend not only on n, but also on l: Atomic Spectra

16. Building the atomic spectra using the Pauli exclusion principle • Pauli: no more than one electron can occupy the same state • ‘Same’ means with the same quantum numbers: n,l,ml,ms • (1s): 2 states n=1 l=0 ml=0 ms=+1/2,-1/2 • (2s): 2 states n=2 l=0 ml=0 ms=+1/2,-1/2 • (2p): 6 states n=2 l=1 ml=-1,0,1 ms=+1/2,-1/2 EXAMPLE Carbon: has 6 electrons. So, the ground state is: (1s)2(2s)2(2p)2 With energy E = 2E1s + 2E2s +2E2p Atomic Spectra

17. Build atoms ignoring electron screening We build the lowest energy states for an increasing number of electrons and we keep the spectroscopic notation from the hydrogen atom: l=0 1 2 Energy Z=18 n=3 3d Z=10 n=2 2s 2p Z=2 n=1 ↑↓ H 1 (1s) He 2 (1s)2 Li 3 (1s)2(2s) Be 4 (1s)2(2s)2 B 5 (1s)2(2s)2(2p) 1s Atomic Spectra

18. Atomic Spectra An atom with atomic number Z has Z electrons in the attractive Coulomb field of the nucleus. However electrons also repel each other. In particular as we will see, electrons in outer shells feel (see) less positive nuclear charge because of screening caused by inner shell electrons. The simplest example is the Helium atom with two electrons p and q: The energy levels and eigenfunctions are found by solving the equation: We can solve numerically using the central field approximation. Atomic Spectra

19. The Central Field Approximation In the CFA: each electron in the atom moves independently in a central attractive Coulomb potential potential due to the nucleus (atomic number Z). An example is the following approximate central potential: The energy levels and eigenfunctions are found by solving the equation: However the energy levels depend not only on n, but also on l: Atomic Spectra

20. Screening changes the energy levels Notice that extra radial peaks closer to the nucleus (example 2s) are less screened Atomic Spectra

21. Energy levels shift due to screening (E=En,l) Z=18 Z=10 Z=2 Notice that the Energy for 4f is above 4d above 4p above 4s due to screening Atomic Spectra

22. Periodic Table • Closed Shell • Noble gases – non reactive, stable • Helium Z=2 1s2 n=1 states full • Neon Z=10 1s22s22p6 n=1,2 states full • Argon Z=18 1s22s22p63s23p6 • Krypton…. • One outer electron • Alkali metals, effective screening, weak binding, easily get ions • Lithium Z=3 He + 2s • Sodium Z=11 Ne + 3s • Potassium Z=19 Ar + 4s • Rubidium Z=37 Kr + 5s • Two outer electrons • Be, Mg, Ca form 2+ ions (alkaline earth metals) • Seven outer electrons • F,Cl,Br form 1- ions to get closed shell (halogens) Atomic Spectra

23. Entangled States Now take 2 particles in different energy states n and n’: If these 2 particles were distinguishable the total wavefunction is: Entangled State Atomic Spectra

24. Building the two eigenspaces We could easily see that there are 2 separate groups of eigenstates if we used raising (or lowering operators): As an example apply this raising operator to the antisymmetric function: The lowering operator also gives 0, so this is a singlet state with total spin 0 Atomic Spectra