5.61 F17 Lecture 21: Hydrogen Atom II

5.61 Fall, 2017 Lecture #21 Page 1 Lecture #21: Hydrogen Atom II Last time: TISE For H atom: final exactly solved problem Ĥ in spherical polar coordinates Separation: ψ nℓmℓ ( r,θ , φ ) = Rnℓ (r)Yℓm (θ , φ ) Yℓm (θ , φ ) is universal — what is the difference between H atom and rigid rotor? Rnℓ(r) is unique for each atom. Worthwhile to develop intuitive interpretation: semi-clssical Experimental evidence for an unexpected internal degree of freedom: electron spin Why does [ L2 , f (r)] = [ Lz , f (r)] = 0 ? different coordinates: φ, θ vs. r Today: 1. Patterns in spectrum of H atom 2. Semi-classical interpretation of Rnℓ(r) 3. Rydberg states 4. Get ready for many-electron atoms. revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 2 H Atom Separation of variables in TISE gives ψ nℓmℓ ( r,θ , φ ) = Rnℓ (r)Yℓm (θ , φ ) "$$# "$$$$# unique universal all central force problems separation of r from θ,φ gives radial TISE ⎡ !2 d 2 d ⎤ ⎢ 2 µr 2 dr r dr + Vℓ (r) − E ⎥ Rnℓ (r) = 0 ⎣ ⎦ Effective potential " 2 ℓ ( ℓ + 1) Ze2 Vℓ (r) = + − 2 µr 2 ( 4πε 0 ) r centrifugal barrier µ= Enℓmℓ = − degeneracy: Coulomb attraction me mion me + mion hcℜZ 2 n2 n = 1,2, 3,… ℓ=0,1,…n–1 mℓ = –ℓ1 – ℓ+1, …, ℓ total degeneracy: s p gℓ 1 3 gn, gℓ d 5 n = 1 gn = 1 n = 2 gn = 1 + 3 = 4 n = 3 gn = 1 + 3 + 5 = 9 gn = n2 Number of nodes: radial n–ℓ–1 angular ℓ total for nℓ state n–1 revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 3 We want to understand Rnℓ(r) for H atom and then extend the cartoon to all many-e– atoms and even to molecules M+ + e– at ∞ Series converges according to 2R En+1 − En ≈ 3 n n=3 n=2 n=1 Absorption spectrum is simple and easy to assign because all population starts in n = 1 (1s) and all ℓ for the same value of n are degenerate (concealing the ∆ℓ = ±1 selection rule). In a discharge, see many-line emission spectrum. Many overlapping convergent series. ⎡1 ∆ En,nʹ 1 ⎤ = ℜ⎢ 2 − 2 ⎥ ⎣ n nʹ ⎦ hc nʹ > n Each value of n is associated with a convergent series in nʹ. Easy to recognize ∆ En,nʹ −En ℜZ 2 = + 2 hc hc nʹ each series converges at nʹ → ∞ Ionization Energy, thus each series reveals the value of En. “Structure” revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 4 Most electronic properties scale as the expectation value of an integer power of r. Each such expectation value or matrix element scales as a specific power of n and ℓ. It is possible to empirically measure n (both for H atom and any other atom or molecule) by using the Rydberg Formula relationship between the ionization energy out of the n-orbital. In = Ionization energy: hcRZ n2 1/2 ⎡ hcRZ ⎤ n=⎢ ⎥ ⎣ In ⎦ Measure In → empirically determine n (or n*, the effective principal quantum number, for non-H systems) In → n → all n,ℓ dependent properties! McQuarrie, page 333: expectation values of rk k rk 2 a02 n 4 ⎧ 3 ⎡ ℓ ( ℓ + 1) − 1 / 3 ⎤ ⎫ ⎨1+ 1− ⎥⎦ ⎬⎭ Z 2 ⎩ 2 ⎢⎣ n2 1 a0 n 2 ⎧ 1 ⎡ ℓ ( ℓ + 1) ⎤ ⎫ ⎨1+ 1− ⎬ Z ⎩ 2 ⎢⎣ n 2 ⎥⎦ ⎭ (basis for intuition about orbital size) 0 n0 –1 Z an n 2 –2 Z2 (ℓ + 1 / 2) a02 n 3 –3 nℓmℓ Z3 1⎞ ⎛ a03n 3ℓ ℓ + ( ℓ + 1) ⎝ 2⎠ revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 5 a0 is the Bohr radius a0 = 0.529Å. Note that 〈rk〉 scales as (a0/Z)k. Most importantly, for all k ≤ -2, 〈rk〉 ∝ n–3. This reflects the amplitude in the innermost lobe of the radial wavefunction. I will show you that the amplitude in this first lobe of χnℓ(r) ∝ n–3/2. Generalization from the 1e– atom or ion to many-e– atoms and molecules. H+ Na + C a point spherical charge surrounding + 11 charge on the nucleus Enℓ ℜ = − Na hc nℓ* En ℜ = − 2H hc n n integer O a not-round charge distribution shielding C+6 and O+8 nuclei Enℓλ ℜ = − *CO hc nℓλ nℓ* = n − δ ℓ δℓ is “quantum defect” nℓ*λ = n − δ ℓλ ! λ is projection of ℓ on bond axis Semiclassical theory for Rnℓ(r) * almost everything scales as the amplitude of the innermost lobe of χnℓ(r) * inner lobe (first node) lined up for all n of given ℓ. 1 We need to go to the Rnℓ (r) = χ nℓ (r) picture. r The 3-D spherical polar volume element is r2 sinθ dr dθ dφ. ∞ Rnℓ r Rn′ℓ = ∫ r k r 2 Rnℓ Rn′ℓ dr k 0 ∞ = ∫ r k χ nℓ (r) χ n′ℓ (r)dr 0 χ(r) gets rid of all of the extra factors of r in the radial Schrödinger equation and in all integrals involving Rnℓ(r). The radial TISE and all integrals now look like the ordinary 1–D V(x) problems we have studied. One more special point: The r = 0 boundary condition. Rns (0) ≠ 0 (surprise!) χ ns (0) = 0 revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 6 Why is Rns(0) ≠ 0? The centrifugal barrier term is absent. Vℓ=0(r = 0) = –∞ The radial wavefunction is pulled in toward the nucleus. There is no barrier that prevents Rns(0) from being non-zero. But this is a forgivable singularity. lim r 2 Rns* (r)Rns (r) → 0 r→0 because of the r2 in the volume element! There is one extremely important consequence of Rns(0) ≠ 0. The electron actually comes into contact with the nucleus. If there is a non-zero nuclear spin, I, there is “Fermi Contact” hyperfine interaction. This gives rise to a very small splitting in the energy levels whenever there is a half-filled orbital with ns character. Hyperfine measures Rns(0)! Hyperfine is very important in NMR. Set up a framework for semi-classical analysis Ze2 1 " 2 ℓ(ℓ + 1) Vℓ (r) = − + 2 4πε 0 r 2 µr attractive repulsive revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 7 6 IE - 0 V`=0(r) In ................... ............................................... ............................. . . . . . . . . . . . . . . . . . . . . . . ................... ................. ............... . . . . . . . . . . . . . ............ ........... .......... . . . . . . . . . . ......... ........ ....... . . . . . . ....... ...... ..... . . . .... .... .... . . . . .... ... ... . . .. ... .. ... . .. .. ... .. .. .... .. .. .... .. .. .... . .. .. ... . r purely attractive ? revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ............................................ ............................................ ... ...................... . . . . . . ... . . . . . . . . . . . . ... ............... ... ............. ............ ... . . . . . . . . . . ... .......... ... ......... ... ......... . . . . . . . . ... ....... ... ........ ... ........ . . . . ... . . . ... ....... ....... ... ....... ... . . . . . ... .... ... ....... ... .... . . . ... .... ... .... .... ... . . . . .... . ........ .................. ...... ... 6 ... IE V`=1(r) Page 8 0 - repulsive at small r Turning points: where Enℓ = Vℓ (r± ) E =6− ? nℓ 5 (x) 5 (x) hcℜ n2 6 E Solve for r±(n,ℓ) ⎛ ⎡ ℓ(ℓ + 1) ⎤1/2 ⎞ r± (n,ℓ) = a0 n ⎜ 1 ± ⎢1− ⎟ ⎝ ⎣ n 2 ⎥⎦ ⎠ 2 0 E5 = 1/2 ⎡ ⎛ 2⎞ ⎤ r (n,1) = a0 n ⎢1+ ⎜1− 2 ⎟ ⎥ < r+ (n,0) Note that + ⎣ ⎝ n ⎠ ⎦ h2 (25) 8ma2 2 a little bit surprising 2 r+ (n,0) = a0 n (2) 0 x a revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 9 Classical mechanical radial momentum pnℓ (r) = ⎡⎣ 2 µ( Enℓ − Vℓ (r)) ⎤⎦ 1/2 λnℓ (r) = h pnℓ (r) hard inner wall soft outer wall We want to know where are the nodes and what is the probability for the e– between each pair of nodes. Classical oscillation period at En τn = h h = ∝ n+3 3 En+1/2 − En−1/2 hcℜ ( 2 n ) Node to next-node probability δ t node to node ∆ t turning point to turning point δ t node to node λ ( rcenter of lobe ) 2 = p ( rcenter of lobe ) µ ∆tturning point to turning point = τn/2 We will find: Because inner wall of Vℓ(r) is nearly vertical for n > 6 , innermost radial nodes line up at ! nearly the same value of r (for n > 6) Because pn>6 ( r− + δ r ) ≈ pn=6 ( r− + δ r ) , λ is approximately independent of n near inner turning point. Thus we get an n-dependent amplitude of the corresponding inner lobes that scales as n–3/2! (from τ ∝ n +3 n ) revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 10 If the system we are interested in is not H, we consider shielding of the nucleus by core electrons, Z-shielding = Zeff.. Z ℓeff goes from Z (charge on the bare nucleus) at r = 0 to +1 at r ≈ radius of core. The degree of penetration of Rydberg e– into core region is ℓ-dependent. n → nℓ* < n ns* ≪ n*p < nd* < n*f ≈ n because high-ℓ cannot penetrate inside the core. So the inner lobe scales as nℓ* . Stationary Phase Expectation values and matrix elements of rk ∫ ∞ 0 χ nℓ (r)r χ n′ℓ′ dr k two rapidly oscillating functions Integral accumulates only in the region where the two functions are oscillating at the same spatial frequency. r′ I nℓ,n′ℓ′ = ∫ χ nℓ (r)r k χ n′ℓ′ (r)dr 0 accumulates as revised 11/3/17 8:52 AM 5.61 Fall, 2017 Lecture #21 Page 11 Inℓ,n′ℓ′ 0 rstationary phase r′ For most nℓ ≠ nʹℓʹ integrals (as in transition moment from 1s to np) the only stationary phase region is in the innermost lobe. I nℓ,n′ℓ′ ≈ nℓ−3/2 nℓ′′−3/2 This is a framework for training your intuition and then making intuition-based predictions about properties of many-electron systems for atoms and molecules! Rydberg States! revised 11/3/17 8:52 AM