Quantum Theory of Angular Momentum and Atomic Structure

Transcription

1 Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0

2 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1

3 Motivation...the questions Concepts in Materials Science I Material Music Patterns in Periodic Table Rotational spectra of molecules VBS/MRC Angular Momentum 2

4 Model of Hydrogen Atom FIx the nucleus at origin Concepts in Materials Science I Hamiltonian H = P x 2 +Py 2 +Pz 2 2m + V (r), r = x 2 + y 2 + z 2, V (r) = 4πɛ 1 e 2 o r Can we estimate ground state energy? Yes, we can! Boundstate...electron found within l of nucleus Kinetic energy (uncertainty principle) Potential energy 4πɛ 1 e 2 o l E g (l) = 2ml 2 1 e 2 2 4πɛ o l Ground state energy estimate 1 2 me 4 (4πɛ 0 ) ml 2 = -13.5eV! VBS/MRC Angular Momentum 3

5 Spherical Polar Coordinates Alternate way of describing points in space Concepts in Materials Science I (x,y,z) θ r ϕ x = r sin θ cos φ y = r sin θ sin φ z = r cos θ Suited for spherically symmetric problems VBS/MRC Angular Momentum 4

6 A different look a the Hamiltonian Classical kinetic energy H = p2 r 2m + 2mr L2 + V (r) 2 L 2 - magnitude square of the angular momentum In Quantum Mechanics L 2 is an operator In fact, L, the angular momentum vector is an operator What is the position representation of L? In Cartesian coordinates, L x = Y P z ZP y etc... See more details later VBS/MRC Angular Momentum 5

7 Hamiltonian in Polar Representation If ψ, is an energy eigenket, the wavefunction r, θ, φ ψ = ψ(r, θ, φ) satisfies 2 2m 1 2mr 2 ( 1 2 ) (rψ) r r 2 + ( 1 2 sin θ θ sin θ θ ) sin 2 θ φ }{{ 2 ψ + V (r)ψ = } L 2! Can be thought of as ( P 2 r 2m + 1 ) 2mr 2 L2 + V (r) ψ = Eψ VBS/MRC Angular Momentum 6

8 Hamiltonian in Polar Representation Try the ansatz, ψ(r, θ, φ) = R(r)Y (θ, φ) If Y is eigenfunction of L 2, then L 2 Y = l(l + 1) 2 Y (eigenvalues written anticipating results) The radial part then satisf ies the equation ( P r 2 2m + 2 l(l + 1) 2mr 2 + V (r))r = ER What are the allowed values of l, and E? VBS/MRC Angular Momentum 7

9 Angular Momentum What are the eigenstates of L 2? Concepts in Materials Science I L 2 commutes with the Hamiltonian, [L 2, H] = 0...Rotational kinetic energy is conserved since no one is applying any torque! What about L? Is it conserved?...should be! Lets see...with a bit of painful algebra ( L x = i sin φ + cot θ cos φ θ φ L y = i L z = i φ ( cos φ θ VBS/MRC Angular Momentum 8 ) cot θ sin φ φ )

10 Angular Momentum Concepts in Materials Science I It can now be shown that [L x, H] = 0 and [L x, L 2 ] = 0, similarly for y and z So angular momentum will be conserved! But there is more...very importantly [L x, L y ] = i L z, [L y, L z ] = i L x, [L z, L x ] = i L y This means that all components of angular momenta cannot be determined simultaneously without uncertainty! Since [L z, L 2 ] = 0, we can choose eigenstates of L 2 and L z simultaneously...and this is what we will do VBS/MRC Angular Momentum 9

11 Angular Momentum Concepts in Materials Science I It turns out that the eigenstates are given by Yl m (θ, φ) such that L 2 Yl m (θ, φ) = 2 l(l + 1)Yl m (θ, φ), L zyl m m (θ, φ) = m Yl (θ, l can take only non-negative integer values (0,1,2..etc) For a given value of l, m can take values between l and l...and thus 2l + 1 states Angular energy state given by l is therefore 2l + 1 fold degenerate VBS/MRC Angular Momentum 10

12 Angular Momentum Eigenstates The function Yl m (θ, φ) are called spherical harmonics They satisfy Yl m m (θ, φ)yl (θ, φ)dω = δ ll δ mm (no surprise there!) Related to Legendre polynomials (look up somewhere, Pauling-Wilson, for example) VBS/MRC Angular Momentum 11

13 Angular Momentum Eigenstates Some examples (how do you interpret this?) l = 0 l = 1 Y 0 0 = 1 π Y 0 1 = 3 4π ±1 3 cos θ, Y 1 = 8π sin θe±iφ l = 2 Y 0 2 = π (3 cos2 θ 1) Y 2 ±1 = 8π sin θ cos θe±iφ VBS/MRC Angular Momentum 12 Y ±2 = 15 sin 2 θe ±i2φ

14 Understand Angular Momentum If you put a particle in the state given by l, m, you will have a definite value of L 2 and L z...measurement of these quantities in this state will produce no uncertainty What about L x and L y? It can be shown that L x = l, m L x l, m = 0! (same of L y ) Thus L 2 x = L 2 x and L 2 y = L 2 y Clearly L 2 x + L 2 y = L 2 L 2 z = 2 ( l(l + 1) m 2) VBS/MRC Angular Momentum 13

15 Angular Momentum - Main Ideas If you let a quantum particle live on a unit sphere, rotational energy (L 2 ) states are given by l(l + 1) 2 (l is a non-negative integer) and put the particle in an l state, you can specify but one component of angular momentum precisely...the other two cannot be specif ied; also, the component can be specified only as m where m is an integer from l to l Think of the same situation in a classical context...and feel how very different quantum mechanics is! Also, make sure that you understand how you get back all classical results from quantum mechanics (hint: go to large values of l) VBS/MRC Angular Momentum 14

16 Back to Hydrogen Atom Radial Equation ( P 2 r 2m + 2 l(l+1) 2mr 2 Concepts in Materials Science I ) + V (r) R = ER Allowed values of E are E n = E o n 2, E o = 13.5eV, n = 1, 2,... For each value of n, l takes values between 0 and n 1...tells us how energy is shared between radial and rotational degrees (contrast classical picture)! For a given n and l, the radial wavefunction is R l n(r) = ( 2 na o ) 3 (n l 1)! 2n[(n + l)!] 3 e r nao ( r na o ) l L 2l+1 n+1 ( r na o ) a o Bohr radius, L 2l+1 n+1 Associated Laguerre polys. VBS/MRC Angular Momentum 15

17 Radial Wave Functions Concepts in Materials Science I (Beiser) VBS/MRC Angular Momentum 16

18 Radial Wave Functions Probabilities (Beiser) VBS/MRC Angular Momentum 17

19 Radial Wave Functions - Key Points R l n has n (l + 1) nodes! Roughly, this means that when n is large and l is small, there is more energy in the radial degree of freedom At what radius rn,l max is it most likely to find the particle? Turns out that, for a given n, l = 0 is the outer most and l = n 1 are the inner most! Recall, f shells being called as deep shells! Most chemistry is due to this! For example, this is why transition metals are very happy to part with their s-electrons! VBS/MRC Angular Momentum 18

20 Complete Wavefunctions The full wave functions for H-atom are r, θ, φ n, l, m = Rn(r)Y l l m (θ, φ) Concepts in Materials Science I We are more familiar with s, p, d, f orbitals, how are they related to the full wave functions? Let us look at some specific cases VBS/MRC Angular Momentum 19

21 Complete Wavefunctions Concepts in Materials Science I VBS/MRC Angular Momentum 20

22 Orbitals! Key idea: Any linear combination of degenerate energy states is also an energy state Useful to create orthogonal states with symmetries that ref lect the crystalline environment s-orbitals: 1s = 1, 0, 0 p-orbitals: 2p z = 2, 1, 0, 2p x = 2,1,1 + 2,1, 1 2 2p y = 2,1,1 2,1, 1 2i and d-orbitals: 3d 3z2 r 2 = 3, 2, 0, 3d xz = 3,2,1 + 3,2, 1 2, 3d yz = 3,2,1 3,2, 1 2i 3d xy = 3,2,2 3,2, 2 2i 3,2,2 + 3,2, 2, 3d x2 y2 = 2, Can understand things like crystal f ield splitting from VBS/MRC Angular Momentum 21

23 Structure of Multi-Electron Atoms Need to take care of the following things Spin! Pauli s Principle Coulomb interactions (+ spin Hund s Rule) Spin-orbit Coupling Even relativistic effects, sometimes! Angular momentum states no longer degenerate (Aufbau principle) Gives rise to the material music VBS/MRC Angular Momentum 22