Lecture #21: Hydrogen Atom II

Transcription

1 561 Fall, 217 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) = R nl (r)y m l (θ,φ) Y l m ( θ,φ) is universal what is the difference between H atom and rigid rotor? R nl (r) is unique for each atom Worthwhile to develop intuitive interpretation: semi-clssical Experimental evidence for an unexpected internal degree of freedom: electron spin [ ] = [ L z, f (r)] =? different coordinates: φ, θ vs r Why does L 2, f (r) Today: 1 Patterns in spectrum of H atom 2 Semi-classical interpretation of R nl (r) 3 Rydberg states 4 Get ready for many-electron atoms

2 561 Fall, 217 Lecture #21 Page 2 H Atom Separation of variables in TISE gives ψ nlml separation of r from θ,φ gives radial TISE ( r,θ,φ) = R nl (r) " $$ # Y m l ( θ,φ) " $$$$ # unique universal all central force problems Effective potential! 2 d 2µr 2 dr r d 2 dr + V l(r) E R nl(r) = V l (r) = + "2 l( l +1) 2µr 2 centrifugal barrier Ze 2 ( 4πε )r Coulomb attraction µ = m e m ion m e + m ion E nlml = hcrz 2 n =1,2,3, degeneracy: l=,1, n 1 m l = l 1 l+1,, l total degeneracy: g n, g l s p d g l n = 1 g n = 1 n = 2 g n = = 4 n = 3 g n = = 9 g n = Number of nodes: radial angular total for nl state n l 1 l n 1

3 561 Fall, 217 Lecture #21 Page 3 We want to understand R nl (r) for H atom and then extend the cartoon to all many-e atoms and even to molecules M + + e at n = 3 Series converges according to E n+1 E n 2R 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 E n, nʹ hc = R 1 n 1ʹ 2 n ʹ > n Each value of n is associated with a convergent series in nʹ Easy to recognize E n, nʹ hc = E n hc + RZ 2 nʹ 2 each series converges at nʹ Ionization Energy, thus each series reveals the value of E n Structure

4 561 Fall, 217 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 Ionization energy: I n = hcrz n = hcrz I n 1/2 Measure I n empirically determine n (or n*, the effective principal quantum number, for non-h systems) I n n all n,l dependent properties! McQuarrie, page 333: expectation values of r k k r k nlm l 2 1 a 2 n l( l +1) 1/ 3 1 Z 2 2 a 1+ 1 l( l +1) 1 Z 2 (basis for intuition about orbital size) n Z a n Z 2 ( l +1/ 2) a 2 n 3 Z 3 a 3 n 3 l l ( l +1 )

5 561 Fall, 217 Lecture #21 Page 5 a is the Bohr radius a = 529Å Note that r k scales as (a /Z) k Most importantly, for all k -2, r k 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 χ nl (r) n 3/2 Generalization from the 1e atom or ion to many-e atoms and molecules E n H + a point hc = R H n integer Na + spherical charge surrounding + 11 charge on the nucleus E nl hc = R Na * n l n * l = n δ l δ l is quantum defect C O a not-round charge distribution shielding C +6 and O +8 nuclei E nlλ hc = R CO * n lλ * n lλ = n δ lλ λ is projection of! l on bond axis Semiclassical theory for R nl (r) * almost everything scales as the amplitude of the innermost lobe of χ nl (r) * inner lobe (first node) lined up for all n of given l We need to go to the R nl (r) = 1 r χ nl(r) picture The 3-D spherical polar volume element is r 2 sinθ dr dθ dφ R nl r k R n l = r k r 2 R nl R n l dr = r k χ nl (r)χ n l (r)dr χ(r) gets rid of all of the extra factors of r in the radial Schrödinger equation and in all integrals involving R nl (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 = boundary condition R ns() χ ns () = (surprise!)

6 561 Fall, 217 Lecture #21 Page 6 Why is R ns ()? The centrifugal barrier term is absent V l= (r = ) = The radial wavefunction is pulled in toward the nucleus There is no barrier that prevents R ns () from being non-zero But this is a forgivable singularity lim r 2 R * ns (r)r ns (r) r because of the r 2 in the volume element! There is one extremely important consequence of R ns () 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 R ns ()! Hyperfine is very important in NMR Set up a framework for semi-classical analysis V l (r) = Ze2 1 4πε r + "2 l(l +1) 2µr 2 attractive repulsive

7 561 Fall, 217 Lecture #21 Page 7 6 IE V`= (r) - r I n? purely attractive

8 561 Fall, 217 Lecture #21 Page 8 IE 6 V`=1 (r) - repulsive at small r Turning points: where E nl = V l (r ± ) E nl = hcr Solve for r ± (n,l) r ± (n,l) = a 1± 1 l(l +1) 1/2 Note that r +(n,1) = a r + (n,) = a (2) 1/2 < r + (n,) a little bit surprising

9 561 Fall, 217 Lecture #21 Page 9 Classical mechanical radial momentum 1/2 p nl (r) = 2µ ( E nl V l (r)) λ nl (r) = h p nl (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 E n τ n = Node to next-node probability h E n+1/2 E n 1/2 = h ( ) n+3 hcr 2 n 3 We will find: δt node to node t turning point to turning point ( ) 2 ( ) µ δt node to node = λ r center of lobe p r center of lobe t turning point to turning point = τ n /2 Because inner wall of V l (r) is nearly vertical for n! > 6, innermost radial nodes line up at nearly the same value of r (for n > 6) Because p n>6 r +δr turning point ( ) p n=6 ( r +δr), λ is approximately independent of n near inner Thus we get an n-dependent amplitude of the corresponding inner lobes that scales as n 3/2! (from τ n n +3 )

10 561 Fall, 217 Lecture #21 Page 1 If the system we are interested in is not H, we consider shielding of the nucleus by core electrons, Z-shielding = Z eff eff Z l goes from Z (charge on the bare nucleus) at r = to +1 at r radius of core The degree of penetration of Rydberg e into core region is l-dependent n n l * < n n s * n p * < n d * < n f * n because high-l cannot penetrate inside the core So the inner lobe scales as n l * Stationary Phase Expectation values and matrix elements of r k χ nl (r) r k χ n l dr two rapidly oscillating functions Integral accumulates only in the region where the two functions are oscillating at the same spatial frequency I nl, n l = χ nl (r) r k χ n l (r)dr accumulates as r

11 561 Fall, 217 Lecture #21 Page 11 I nl,n l r stationary 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 n l 3/2 3/2 n l 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!

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