lecture21to22

lecture21to22 - 5.61 Fall 2007 Lecture#21 page 1 HYDROGEN...

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Unformatted text preview: 5.61 Fall 2007 Lecture #21 page 1 HYDROGEN ATOM Schrodinger equation in 3D spherical polar coordinates: − !2 ⎡ 1 ∂ ⎛ 2 ∂ ⎞ 1 ∂⎛ ∂⎞ 1 ∂2 ⎤ r +2 sin θ ⎟ + 2 2 ⎢ ⎥ ψ r ,θ ,φ + U r ,θ ,φ ψ r ,θ ,φ = Eψ r ,θ ,φ 2 µ ⎣ r 2 ∂r ⎜ ∂r ⎟ r sin θ ∂θ ⎜ ∂θ ⎠ r sin θ ∂φ 2 ⎦ ⎝ ⎠ ⎝ ( with Coulomb potential U (r ) = ) ( )( ) − Ze2 4πε 0 r Rewrite as ⎡ 2 ∂⎛ 2 ∂⎞ ⎤ 2 ˆ2 ⎡ ⎤ ⎢−! ⎜ r ∂r ⎟ + 2 µr ⎣U r − E ⎦ ⎥ ψ r ,θ ,φ + L ψ r ,θ ,φ = 0 ∂r ⎝ ⎠ ⎣ ⎦ () ( ) function of r only ˆH r is separable ⇒ ( ) function of θ,φ only ψ is separable Angular momentum: solutions are spherical harmonic wavefunctions ( ) () ( ) (θ ,φ ) = ! l ( l + 1)Y (θ ,φ ) ψ r ,θ ,φ = R r Yl m θ ,φ with ˆ L2Yl m 2 m l l = 0,1,2,... Radial equation for the H atom: () ( ) 2 ⎤ !2 d ⎛ 2 dR r ⎞ ⎡ ! l l + 1 − +U r − E⎥ R r = 0 ⎟+⎢ ⎜r dr ⎠ ⎢ 2 µr 2 2 µr 2 dr ⎝ ⎥ ⎣ ⎦ () () () Solutions R r are the H atom radial wavefunctions Simplest case: l = 0 yields solution ⎛Z⎞ R r = 2⎜ ⎟ ⎝ a0 ⎠ () 32 e − Zr a0 Y ml exponential decay away from nucleus ( ) 5.61 Fall 2007 Lecture #21 page 2 with E = − Z 2 e2 8πε 0 a0 lowest energy eigenvalue B ohr radius a0 ≡ ε 0 h2 πµe2 General case: solutions are products of (exponential) x (polynomial) Energy eigenvalues: − Z 2 e2 − Z 2 µ e4 = 222 8ε 0 h n 8πε 0 a0 n2 E= n = 1,2,3,... Radial eigenfunctions: Rnl 12 l +3 2 ⎡ ⎤ ⎛ ⎞ − Z r na0 2 l + 1 2 Zr ⎢ n − l − 1 ! ⎥ ⎛ 2Z ⎞ r = − r le Ln + l ⎜ ⎜ na ⎟ ⎟ 3⎥ ⎢ ⎝ na0 ⎠ 2 n ⎡ n + l !⎤ ⎥ ⎝ 0 ⎠ ⎢ ⎦⎦ ⎣⎣ ( () ) ) ( ( ) where L2 l++l1 2 Zr na0 are the associated Laguerre functions, the first few of which are: n n=1 l=0 L1 = −1 1 n=2 l=0 ⎛ ⎞ L12 = −2!⎜ 2 − Zr ⎟ a0 ⎠ ⎝ l =1 L3 = −3! 3 n=3 l=0 ⎛ ⎞ 22 L1 = −3!⎜ 3 − 2 Zr + 2 Z r 2⎟ 3 a0 9 a0 ⎠ ⎝ l =1 ⎛ ⎞ L3 = −4!⎜ 4 − 2 Zr 4 3a0 ⎟ ⎝ ⎠ l=2 L5 = −5! 5 Normalization: Spherical harmonics ∫ 2π Radial wavefunctions ∫ ∞ 0 0 ()() π dφ ∫ dθ sin θ Yl m* θ ,φ Yl m θ ,φ = 1 0 () () * dr r 2 Rnl r Rnl r = 1 5.61 Fall 2007 Lecture #21 page 3 TOTAL HYDROGEN ATOM WAVEFUNCTIONS ( ) () ( ) ψ nlm r ,θ ,φ = Rnl r Yl m θ ,φ principle quantum number angular momentum quantum number magnetic quantum number ENERGY depends on n: n = 1,2,3,... l = 0,1,2,..., n − 1 m = 0, ±1, ±2,..., ± l E = − Z 2 e2 8πε 0 a0 n2 ( ORBITAL ANGULAR MOMENTUM depends on l: ) L = ! l l +1 ANGULAR MOMENTUM Z-COMPONENT depends on m: Lz = m! Total H atom wavefunctions are normalized and orthogonal: ∫ 2π 0 π ∞ 0 0 ( ) ( ) * dφ ∫ sin θ dθ ∫ r 2 dr ψ nlm r ,θ ,φ ψ n ' l ' m ' r ,θ ,φ = δ nn 'δ ll 'δ mm ' () ( ) since components Rnl r Yl m θ ,φ are normalized and orthogonal. Lowest few total H atom wavefunctions, for n = 1 and 2 (with σ = Zr a0 ): 5.61 Fall 2007 n=1 n=2 Lecture #21 l=0 l=0 m=0 m=0 page 4 3/ 2 ψ 100 1 ⎛Z⎞ = ⎜⎟ π ⎝ a0 ⎠ 3/ 2 ψ 200 ⎛Z⎞ = ⎜⎟ 32π ⎝ a0 ⎠ 3/ 2 ψ 210 ⎛Z⎞ = ⎜⎟ 32π ⎝ a0 ⎠ e− σ = ψ 1s 1 1 (2 − σ ) e −σ / 2 = ψ 2s σ e− σ / 2 cosθ = ψ 2 p l =1 m=0 l =1 ⎛Z⎞ −σ / 2 m = ±1 ψ 21±1 = sin θ e± iφ ⎜ a ⎟ σ e 64π ⎝ 0 ⎠ z 3/ 2 1 or the alternate linear combinations ψ2p ψ2p ⎛Z⎞ = ⎜⎟ 32π ⎝ a0 ⎠ 1 x ⎛Z⎞ = ⎜⎟ 32π ⎝ a0 ⎠ 1 y 3/ 2 σ e− σ / 2 sin θ cos φ = 3/ 2 σ e− σ / 2 sin θ sin φ = 1 2 1 2i (ψ 21+ 1 + ψ 21−1 ) (ψ 21+ 1 − ψ 21−1 ) The value of l is denoted by a letter: l = 0,1,2,3... s,p,d,f orbitals The value of m is denoted by a letter for l = 1 : m = 0, ± 1 linear combinations pz , px ,p y orbitals HYDROGEN ATOM ENERGIES Potential energy of two electrons separated by the Bohr radius: U = e2 4πε 0 a0 _ one “atomic unit” (a.u.) of energy. H atom energies: E = − Z 2 e2 8πε 0 a0 n2 = − Z 2 2 n2 a.u. n 1 2 3 4 5 ! En (a.u.) -1/2 -1/8 -1/18 -1/32 -1/50 ! ∞ 0 5.61 Fall 2007 Lecture #21 page 5 H atom energies &transitions n E E/hc (a.u.) (cm-1) 4 3 0 0 -1/32 -6,855 -1/18-12,187 2 -1/8 1 -1/2 -109,680 H atom emission spectra Lyman series Balmer Paschen λ(A) 12 ν(10 Hz) DEGENERACIES OF H ATOM ENERGY LEVELS As n increases, the degeneracy of the level increases. What is the degeneracy gn of each level as a function of n? Does this help understand the periodic table? 5.61 Fall 2007 Lecture #21 page 6 SHAPES AND SYMMETRIES OF THE ORBITALS S ORBITALS () 3 ψ 1s = π a0 −1 2 e − r / a0 ( )( ) 3 ψ 2 s = 32π a0 2 − r a0 e l=0 − r / 2 a0 spherically symmetric n - l -1 = 0 radial nodes l=0 angular nodes n -1 = 0 n - l - 1 = 1 l = 0 n - 1 = 1 total nodes ( Electron probability density given by ψ r ,θ ,φ ) 2 Probability that a 1s electron lies between r and r + dr of the nucleus: ∫ 2π 0 P ORBITALS: ( π )( () ) * 3 dφ ∫ dθ sin θ ψ 1s r ,θ ,φ ψ 1s r ,θ ,φ r 2 dr = 4π π a0 0 1l= −1 e −2 r / a0 2 r dr wavefunctions Not spherically symmetric: depend on θ ,φ m = 0 case: ψ 2 p independent of φ ⇒ z ( 3 ψ 210 = ψ 2 p = 32π a0 z ) (r a ) e −1/ 2 0 − r / 2 a0 cosθ symmetric about z axis radial nodes n - l -1 = 0 angular nodes l = 1 total nodes () n - 1 = 1 (note difference from 2s: Rnl r depends on l as well as n) xy nodal plane – zero amplitude at nucleus m = ±1 case: Linear combinations give Equivalent probability distributions ( ) (r a ) e = ( 32π a ) ( r a ) e 3 ψ 2 p = 32π a0 x ψ2p y 3 0 −1/ 2 0 −1/ 2 0 − r / 2 a0 sin θ cos φ − r / 2 a0 sin θ sin φ 5.61 Fall 2007 Lecture #21 page 7 H atom radial probability densities 0.6 0.5 () 0.4 2 r 2 Rnl r a0 220nlrRra () 0.3 0.6 0.6 0.2 0.5 0.5 2s 0.1 0.4 0.4 1s 0.3 0 0.6 0.3 0.5 0.2 5 10 15 20 25 30 10 30 15 20 25 30 0.2 0.4 0.1 0 2p 0.1 0.6 0 0.3 0 5 10 15 0 20 0 25 5 0.5 0.2 0.4 0.1 0.6 3s 0.3 0 0.5 0 5 10 15 20 25 30 20 25 30 20 25 30 0.2 0.4 0.1 3p 0.3 0 0 5 10 15 0.2 3d 0.1 0 0 5 10 15 s = r/a0 5.61 Fall 2007 Lecture #21 page 8 MAGNETIC FIELD EFFECTS Electron orbital angular momentum (circulating charge) μ=− magnetic moment e L 2 me Magnetic field B applied along z axis interacts with μ: Potential energy U = −μ•Β = −μ zΒz = eBz Lz 2 me Include in potential part of Hamiltonian operator: eB ˆ ˆ ˆ H = H 0 + z Lz 2 me ˆ ˆ H atom wavefunctions are eigenfunctions of both H 0 and Lz operators ˆ eigenfunctions of new H operator. Energy eigenvalues are the sums eB Z 2 e2 E= + zm 2 2 me 8 0 a0 n Energy depends on magnetic quantum number m when a magnetic field is applied. 2p orbitals: m = -1,0,+1 states have different energies Splitting proportional to applied field B z. 5.61 Fall 2007 Lecture #21 page 9 Applied magnetic field No magnetic field 2p m = +1 m=0 m = -1 Energy m=0 1s 2p → s Emission spectra 1 One line Three lines ˆ Complex functions ψ 21−1 and ψ 21+1 are eigenfunctions of Lz with eigenvalues ± m! . ˆ ˆ ψ 2 p and ψ 2 p are eigenfunctions of H 0 but not of Lz ⇒ no longer energy x y eigenfunctions once magnetic field is applied. ...
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

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