3.23 Electrical, Optical, and Magnetic Properties of Materials

Transcription

1 MIT OpenCouseWae Electical, Optical, and Magnetic Popeties of Mateials Fall 27 Fo infomation about citing these mateials o ou Tems of Use, visit:

2 3.23 Fall 27 Lectue 5 THE HYDROGEN ECONOMY 3.23 Electonic, Optical and Magnetic Popeties of Mateials Nicola Mazai (MIT, Fall 27)

3 Last time 1. Commuting opeatos, Heisenbeg pinciple 2. Measuements and collapse of the wavefunction 3. Angula momentum and spheical hamonics 4. Electon in a cental potential and adial solutions 3.23 Electonic, Optical and Magnetic Popeties of Mateials Nicola Mazai (MIT, Fall 27)

4 Simultaneous eigenfunctions of L 2, L z LY ˆ m,, ( θϕ) = m Y m( θϕ) z l l ˆ 2 m, 2 1, l l LY ( θ ϕ) = l( l+ ) Y m ( θ ϕ) Y ( θϕ, ) =Θ m( θ) Φ ( ϕ) m l l m

5 An electon in a cental potential 2 ˆ2 ˆ 1 d d L H = 2 Vˆ() μ d d 2μ 2 ψ ( ) = R ( ) Y ( ϑϕ, ) nlm nlm lm

6 An electon in a cental potential (III) ll ( + 1) u () = R () V () = nl nl eff μ 4 Ze πε 2 2 d + V () u () E u () 2 2 eff = nl nl nl μ d

7 The Radial Wavefunctions R R fo Coulomb V() Z R R 21 2 R R Thickness d Y R R R 31 2 R 2 31 X R 32 2 R 2 32 Figues by MIT OpenCouseWae Radial functions R nl () and adial distibution functions 2 R 2 nl () fo atomic hydogen. The unit of length is a µ = (m/ µ ) a, whee a is the fist Boh adius.

8 Solutions in a Coulomb Potential 5d 4f 5g Images emoved; please see any visualization of the 5d, 4f, and 5g hydogen obitals.

9 The Full Alphabet Soup Coutesy of David Manthey. Used with pemission.souce:

10 Good Quantum Numbes Fo an opeato that does not depend on t: d A d Ψ Aˆ Ψ ˆ ˆ ˆ = = Ψ A Ψ + Ψ A Ψ + Ψ A Ψ =... dt dt t t t 1... = AH ˆ, ˆ ih Then, if it commutes with the Hamiltonian, its expectation value does not change with time (it s a constant of motion if we ae in an eigenstate, that quantum numbe will emain constant)

11 Thee Quantum Numbes Ĥ Pincipal quantum numbe n (enegy, accidental degeneacy) E n e Z Z Z = = ( ev) = ( 1 Ry) 8πε an n n ˆL 2 Angula momentum quantum numbe l l =,1,,n-1 (a.k.a. s, p, d obitals) L m = -l,-l+1,,l-1,l ˆz Magnetic quantum numbe m

12 How do you measue angula momentum? Coupling to a (stong!) magnetic field B Image emoved due to copyight estictions. Please see any expeimental setup fo obseving the Zeeman Effect.

13 Right expeiment wong theoy (Sten-Gelach) ˆ ˆ μ B ˆ ˆ μb H H + L B= H + Lˆ zb z Image coutesy Teesa Knott. Used with Pemission. ˆ ˆ μ B ˆ ˆ ˆ B H H + ( L+ 2S) B= H + μ ( Lˆ 2ˆ z + Sz) Bz Goudsmit and Uhlenbeck

14 Spin Diac deived the elativistic extension of Schödinge s equation; fo a fee paticle he found two independent solutions fo a given enegy Thee is an opeato (spin S) that commutes with the Hamiltonian and that can only have two eigenvalues In a magnetic field, the spin combines with the angula momentum, and they couple via ˆ ˆ μb H H + ( Lˆ+ 2 Sˆ ) B

15 Spin Eigenvalues/Eigenfunctions Nom (s intege bosons, half-intege femions) ˆ 2 2 spin s s 1 S Sˆ ( ) Ψ = + Ψ Ψ =± Ψ 2 z spin spin spin Z-axis pojection (electon is a femion with s=1/2) Spin-obital: poduct of the space wavefunction and the spin wavefunction

16 Pauli Exclusion Pinciple We can t have two electons in the same quantum state Any two electons in an atom cannot have the same 4 quantum numbes n,l,m,m s

17 ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI Auf-bau 6s 6p 5d 4f LOW ENERGY HIGH ENERGY 5s 4s 3s 5p 4p 3p 4d 3d 2p 2s 1s Figue by MIT OpenCouseWae.