3.012 Fund of Mat Sci: Bonding Lecture 5/6. Comic strip removed for copyright reasons.

Transcription

1 3.12 Fund of Mat Sci: Bonding Lectue 5/6 THE HYDROGEN ATOM Comic stip emoved fo copyight easons.

2 Last Time Metal sufaces and STM Diac notation Opeatos, commutatos, some postulates

3 Homewok fo Mon Oct 3 Study: 18.4, 18.5, 2.1 to 2.5. Read befoe 3.14 stats next week: 22.6 (XPS and Auge)

4 Second Postulate Fo evey physical obsevable thee is a coesponding Hemitian opeato

5 Hemitian Opeatos 1. The eigenvalues of a Hemitian opeato ae eal 2. Two eigenfunctions coesponding to diffeent eigenvalues ae othogonal 3. The set of eigenfunctions of a Hemitian opeato is complete 4. Commuting Hemitian opeatos have a set of common eigenfunctions

6 The set of eigenfunctions of a Hemitian opeato is complete Figue by MIT OCW.

7 Thid Postulate In any single measuement of a physical quantity that coesponds to the opeato A, the only values that will be measued ae the eigenvalues of that opeato.

8 Position and pobability Gaph of the pobability density fo positions of a paticle in a one-dimensional had box emoved fo copyight easons. Gaphs of the pobability density fo positions of a paticle in a one-dimensional had box accoding to classical mechanics emoved fo copyight easons. See Motime, R. G. Physical Chemisty. 2nd ed. San Diego, CA: Elsevie, 2, p. 554, figue See Motime, R. G. Physical Chemisty. 2nd ed. San Diego, CA: Elsevie, 2, p. 555, figue 15.3.

9 Quantum double-slit Souce: Wikipedia

10 Quantum double-slit Image of the double-slit expeiment emoved fo copyight easons. See the simulation at "Samples fom Visual Quantum Mechanics": "Double-slit Expeiment." Above: Thomas Young's sketch of two-slit diffaction of light. Naow slits at A and B act as souces, and waves intefeing in vaious phases ae shown at C, D, E, and F. Souce: Wikipedia

11 Fouth Postulate If a seies of measuements is made of the dynamical vaiable A on an ensemble descibed by Ψ, the aveage Ψ Aˆ Ψ ( expectation ) value is A = Ψ Ψ

12 Deteministic vs. stochastic Classical, macoscopic objects: we have welldefined values fo all dynamical vaiables at evey instant (position, momentum, kinetic enegy ) Quantum objects: we have well-defined pobabilities of measuing a cetain value fo a dynamical vaiable, when a lage numbe of identical, independent, identically pepaed physical systems ae subject to a measuement.

13 Spheical Coodinates z P = x = sin θ cos ϕ θ y y = sin θ sin ϕ φ x z = cos θ Figue by MIT OCW.

14 3-d Integation Diagam of an infinitesimal volume element in spheical pola coodinates emoved fo copyight easons. See Motime, R. G. Physical Chemisty. 2nd ed. San Diego, CA: Elsevie, 2, p. 16, figue B.4.

15 Angula Momentum Classical Quantum L = p

16 Commutation Relation Lˆ Lˆ Lˆ Lˆ 2 = x y z ˆ2 ˆ ˆ2 ˆ ˆ2 ˆ x y z L, L = L, L = L, L = Lˆ, Lˆ = ihlˆ x y z

17 Angula Momentum in Spheical Coodinates Lˆ Lˆ z = ih ϕ = h sinθ sinθ θ θ sin θ ϕ 2 2

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

19 Spheical Hamonics in Real Fom Figue by MIT OCW.

20 An electon in a cental potential (I) h 2m 2 ˆ 2 2 = + V( ) needs to be in spheical coodinates H e * 2 2 ˆ h H = sin ϑ V( ) me sinϑ ϑ ϑ sin ϑ ϕ 2 ˆ2 ˆ h 1 2 L H = V() me h

21 An electon in a cental potential (II) 1 ˆ ˆ h d d L H = + + V() 2m d d 2m e e ψ ( ) = R( ) Y( ϑϕ, )

22 An electon in a cental potential (III) 2 2 h 1 d 2 d h l( l+ 1) V() R () () nl = Enl Rnl 2me d d 2me

23 What is the V() potential? 2 1 V centipetal () 1 18 v() (J) 1 1 (m) V eff () V Coulomb () -2 Figue by MIT OCW.

24 The Radial Wavefunctions R 1 R fo Coulomb V() R R R R 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. Figue by MIT OCW.

25 The Radial Density 2 R 1 2 R Z R R R 21 2 R 2 21 Thickness d y X R R R 31 2 R 2 31 Figue by MIT OCW R 32 2 R 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. Figue by MIT OCW.

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

27 Emission and absoption lines Coutesy of the Depatment of Physics and Astonomy at the Univesity of Tennessee. Used with pemission.

28 Balme lines in a hot sta Coutesy of the Depatment of Physics and Astonomy at the Univesity of Tennessee. Used with pemission.

29 XPS in Condensed Matte Diagam of Moon composition as seen in X-ays, emoved fo copyight easons.

30 The Gand Table

31 Solutions in the cental Coulomb Potential: the Alphabet Soup Table of obitals emoved fo copyight easons. See "n and l vesus m" at

32 Obital levels in multi-electon atoms s 3s 4p 3p 4d 3d 4f s 3s 4p 3p 4d 3d 4f Obital Enegy (kj / mol) s 2p s 2p s Hydogen s Multielecton Atoms Figue by MIT OCW.

33 Sceening

34 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 OCW.