5.73 F2018 Lecture 28: Hydrogen Radial Wavefunctions

5.73 Lecture #28 28 - 1 Hydrogen Radial Wavefunctions The Hydrogen atom is special because it has electronic states and properties that scale with the principal, n, and orbital angular momentum, !"quantum numbers, in a simple and global way. This is “structure” that is more than a collection of unrelated facts. The H atom serves as our model for “electronic structure” of manyelectron atoms, molecules, and possibly solids. By showing how E, ⟨" ! ⟩ (size and shapes), ⟨$%| "|' "ℓ"⟩ (general matrix elements) scale with the n and ℓ quantum numbers, it tells us the kind of behavior to look for and how to understand it in more complex systems. * * as a perturbation on the Hydrogen atom (quantum defects) as a hint of relationships that are useful for extrapolation, assignment, and for recognizing when something behaves differently from naive expectations. TODAY 0. A warning. 1. Simplified Radial Equation – a trick to make it look exactly like the 1D problems we have already solved. 2. Boundary conditions as r → 0 and r → ∞# 3. qualitative features of Rnℓ(r) 4. n-scaling of 〈rσ〉 , where � is an integer 5. mathematical form of Rnℓ(r) 6. regular and irregular Coulomb functions Warning: This first lecture on the Hydrogen atom is not what you expect. Most textbooks treat the H atom as an exactly solved problem. The solution is based on standard techniques for solving differential equations and the solutions are expressed in terms of named and tabulated special function. The exact and analytic solution for the H atom is presented as a unique, beautiful gem to be admired and stored in a safe place, out of view and out of use. What I present here is based on techniques and approximations discussed in previous lectures. My goal is to arm you with methods to answer questions that apply to many-electron atoms, molecules, and to quantum number “scaling-rules” that provide a framework for describing intra-species and inter-quantum-state patterns of observable properties. These quantum number scaling ideas are mostly based on the stationary phase approximation (in what local region of space does an integral accumulate to its final value, what does the WKB approximation tell you about wavefunction amplitudes and node-spacing in this special region) and perturbation theory. These are techniques and ideas that you will use throughout your career to build insights that will amaze your colleagues. 1 5.73 Lecture #28 28 - 2 For every central force problem ⎡ p̂2 ℓ̂ 2 ⎤ r Ĥ = ⎢ + ⎥ + V (r) 2 2µ 2µr ⎢⎣ ⎦⎥ ( ) *We know that Ĥ, ℓ̂ 2 , and ℓ̂ z commute, so spherical harmonics, Yℓm θ,φ , are eigenfunctions of Ĥ with eigenvalues " 2ℓ(ℓ +1). ( ) ( ) *ψ r,θ,φ = R(r)Yℓm θ,φ *trial form for separation of variables in ψ ⎞ m ⎛ p2 ℓ2 r Ĥψ = ⎜ + + V r ⎟ Yℓ θ,φ R r = Eψ 2 ⎝ 2µ 2µr ⎠ Operate on the Yℓm Ĥψ = Yℓm () ( ) () (θ,φ) angular wavefunction and move it through to left: ⎛ ⎞ ⎜ p2 " 2ℓ ℓ +1 ⎟ ⎟ R r = Eψ θ,φ ⎜ r + + V r 2µr 2 ⎜ 2µ #% % %$%%% & ⎟⎟ ⎜⎝ Vℓ (r ) ⎠ ( ) ( ) () () So we can take Yℓm (θ,φ) out of the Schrödinger Equation and we are left with a 1-D radial equation where the only remaining hint of the angular part is the ℓ-dependence of Vℓ(r), the effective potential energy function. Since the differential equation depends on ℓ, R(r) must also depend on ℓ, thus Rnℓ(r) is the radial part of ψ, and it will generally be an explicit function of two (not 3) quantum numbers, n and ℓ. Usually n specifies the number of radial nodes and ℓ the number of angular nodes, but a special numbering convention for Hydrogen (and hydrogenic ions) causes a slight distortion of this rule. 2 5.73 Lecture #28 28 - 3 The radial equation, when the explicit differential operator form of pr2 is derived and inserted, has the form ⎧ ⎤ ⎫ ⎡ ! 2ℓ ℓ +1 2 2 ⎡ ⎤ 1 d ! ⎪ − r + ⎢ + V (r)⎥ ⎪⎪ ⎪ ⎢ 2 ⎥ 2 ⎥ ⎬ Rnℓ (r) = Enℓ Rnℓ (r) ⎢ 2µr ⎨ ⎣ 2µ r dr ⎦ ⎦ ⎣ ⎪ ⎪ Tr Vℓ ⎪⎩ ⎪⎭ 1 It is customary and informative to simplify this equation by replacing Rnℓ (r) by unℓ (r) r 1 Rnℓ (r) = unℓ (r) r ( * * * Insert ) the resulting equation looks simpler the resulting volume element looks simpler the behavior as r → 0 seems more familiar 1 u (r) in place of Rnℓ (r) and then multiply through on left by r, we get r nℓ ⎡ ! 2 d 2 ! 2 ℓ(ℓ + 1) ⎤ + V (r) − E − + nℓ ⎥ u nℓ (r) = 0 ⎢ 2 µ dr 2 2 2 µr ⎣ ⎦ This looks like ordinary 1-D Schrödinger Equation. Boundary condition: unℓ (r) → 0 as WHY? Because for all ℓ > 0, Vℓ(0) → ∞. r→0 exactly as if V(r) = ∞ for r ≤ 0, but of course r < 0 is impossible, so we had better be careful about behavior of unℓ(r) and Rnℓ(r) as r → 0 3 5.73 Lecture #28 28 - 4 Note also that the volume element is dV = r 2 sinθdrdθdφ * * and the r-dependence of the integrand is Rn′ℓ′ Rnℓ r 2dr = un′ℓ′ (r)unℓ (r)dr the r2 factor is cancelled. So the volume element looks just as in 1D problems Return to special situation as r → 0. Why do we care? It turns out that only s-orbitals have Rns(0) ≠ 0 and that in ESR one measures “Fermi-contact” hyperfine structure which is related to the spin-density at specific nuclei. It is a direct measure of the ns atomic orbital character in each molecular orbital! CTDL, p. 781: What is the worst possible divergence of Rnℓ(r) as r → 0? For r → 0, Rnℓ(r) will be dominated by rs where the exponent |s| is as small as possible. This is the most strongly divergent part of Rnℓ(r), which is all we need to be concerned with as r → 0. Let Rnℓ ~ Crs, where this is a good approximation as r → 0. Plug this definition into the Schrödinger Equation d2 d2 s+1 s−1 ( )( ) rR (r) = Cr = s +1 s Cr nℓ dr 2 dr 2 "2 1 d 2 Tr = − 2µ r dr 2 "2 ( " 2ℓ ( ℓ +1) s−2 # s−2 )( ) HRnℓ (r) = − C s +1 s r + Cr + V (r)Cr s − EnℓCr s = 0 2µ 2µ * ⎡ As r → 0 V(r) rarely diverges ⎤ ⎥ ⎢ ⎢ more rapidly than 1/r , thus ⎥ ⎥⎦ ⎢⎣ V(r)C r s gives r s-1 . 1 if V (r) ∝ r Then, in the limit r → 0, the coefficients of the rs–2 term (i.e. the most rapidly divergent term) must be = 0 –(s + 1)s + ℓ(ℓ + 1) = 0 *This excludes the stronger divergence of the centrifugal barrier term in Vℓ(r), which is included separately. 4 5.73 Lecture #28 28 - 5 Satisfied if s = ℓ or s = –(ℓ + 1) explore the second possibility: s(s + 1) = (–ℓ – 1)(–ℓ – 1 + 1) = –(ℓ + 1)(–ℓ) = ℓ (ℓ + 1) If ! = ℓ, the $ → 0 form of '!ℓ($) = *$ ℓ which is OK for all ℓ. If ! = −(ℓ + 1), the $ → 0 form of '!ℓ($) = *$ #(ℓ%&) diverges for all r. In other words Rnℓ (r) → r ℓ OR (if s= − (ℓ + 1)) well behaved at r → 0 1 r ℓ+1 as r → 0 disaster even if ℓ=0 Actually, both of these possibilities satisfy the differential equation in ! the limit ! → 0 for $(!) = (known as the Coulomb — or H atom " Hamiltonian), but the one that diverges as ! → 0 cannot satisfy the ! → 0 boundary condition for the H atom, but it turns out we will need both types of solution. ** “Regular” and “Irregular Coulomb” wavefunctions – we will return to these later in the context of Quantum Defect Theory. So, for now, we insist that Rnℓ (r) → r ℓ as r→0 Rns (0) ≠ 0 special situation for R ns(r) Rnℓ>0(0) = 0 unℓ (0) = 0 for all ℓ" (no special case for uns(r)!) 5 5.73 Lecture #28 28 - 6 For Hydrogen "2ℓ ( ℓ +1) e2 − Vℓ (r) = + 2 r 2µr q2 e ≡ 4πε0 2 µ= Vℓ(r) 0 memp me + mp ≈ me r 0 Vℓ(r) for ℓ > 0 V0 (r) What do we know from our study of 1-D problems? WKB ψ envelope ∝~ p−1/ 2 ASK QUESTIONS: What are the qualitative shapes of unℓ(r) in innermost and outermost lobes? # of nodes, placement of nodes, degeneracy, behavior at inner and outer turning points, location of inner and outer turning points inner vs. outer part of unℓ (r) — where is the extra h of action 2 (associated with tunneling into nonclassical region)? ⎡ recall ⎢⎣ ∫ r> ( E ) r< ( E ) λ h = 2 2p(r) p(r) dr = 2h (n + 1/ 2) ⎤ ⎥⎦ 6 ⎛ ⎛ 1⎞ ⎞ J = p(r) dr = h n + acquired " ∫ ⎜⎝ ⎜⎝ 2 ⎟⎠ ⎟⎠ 5.73 Lecture #28 4 ℜ Enℓ = − 2 n Vℓ (r± ) = Enℓ Find that At turning point e µH ℜH = 2" 2 (m )(m ) µ = !m e− H ℜ " 2 ℓ ( ℓ + 1) e2 − 2= − 2 r± 2µr± n − − Solve for r± as function of n and ℓ r = a ⎡ n ± n ( n − ℓ ( ℓ + 1)) ⎤ ⎣⎢ ⎦⎥ 2 ± 1/2 2 0 ⎡ ⎛ ℓ ℓ +1 r± = a0n 2 ⎢⎢1± ⎜ 1− n2 ⎢⎣ ⎝ ( )⎞ 1/2 ⎤ ⎟ ⎠ 28 - 7 a0 = me− + m p+ ℜ ! 2 ℓ ( ℓ +1) − r± e 2 2µ = n2 2µr±2 2µℜr±2 = ! 2 ℓ ( ℓ +1) − r± e 2 2µ n2 Use Quadratic formula to find r±(n) !2 e2me Bohr radius when ℓ ≪ n, where are r+ and r– ? (see table on page 28-12) ⎥ ⎥ ⎥⎦ Use this equation for the turning points to construct qualitatively correct cartoons of Rnℓ(r) in crucial regions. surprising systematic degeneracy etc. 3s ———— 3p ———— 2s ———— 2p ———— 3d ———— Enl Enℓ 1s ———— Because of the pattern of degeneracies, we use n to label degenerate groups rather than number of radial nodes ℜ n2 hence n is not # of radial nodes. Enℓ = − 7 p+ e 5.73 Lecture #28 28 - 8 orbital # of radial nodes 1s 0 2s 2p 1 0 3s 3p 3d 2 1 0 (because it is lowest energy solution to the = 1 ℓequation) (because it is the lowest energy solution toℓthe = 1 equation) # radial nodes = ( n − ℓ ) − l # angular nodal surfaces = ℓ total # nodes is n − 1. What is the degeneracy? n degeneracy 1 1 2 1 + (2ℓ + 1) = 4 = 22 3 1 + 3 + 5 = 9=3 2 ! n2 n n-scaling of r σ two limits σ < –1 determined near: inner turning point outer turning point scaling: ~ n–3 Bohr model rnℓ = a0n2 (see argument for inner lobe on next page) r σ ∝ a0σ n 2σ see table on page 11 vs. σ>0 Expectation values of r� vs. transition moments and off-diagonal matrix elements of r�. Stationary phase gives the essential insight! usually occurs in first (inner-most) lobe 8 5.73 Lecture #28 28 - 9 inner region IP 3% of IP n=6 En = − ℜ 109737cm−1 = ≈ 3000cm−1 36 36 n=1 p2 Kinetic Energy = T = ≫ IP in the “inner region.” 2µ variation of T from n = 6 to n = ∞ < 3% variation of p < 1.5% ⎫ ⎪⎪ independent of n. Because p is large and ⎬ fractional change of p vs. n is negligible. λ ⎪ r = is the location of innermost node ⎪⎭ 2 – e comes into core region fast and leaves fast — ∆t same for all n −1 2 (v λ ) time inside = v is velocity fraction of time inside core? ⎞ one period ⎛ h Same as probability inside core. ⎜⎝ E − E ⎠⎟ n n+1 E = –ℜ n 2 deBroglie λ ~ h p ⎤ ⎡ h λ ⎢ p ⎥ ⎥ 2⎢ ⎢⎣ v = p m ⎥⎦ 4mℜ probability in inner lobe = ≈ 2 3 h pn 3 2ℜ n probability of finding e– inside core µ n–3 ! 9 n En+δ 2 − En−δ 2 = 2δℜ n3 n-independent (approximately for n = 6 to n = ∞) 5.73 Lecture #28 28 - 10 fraction of time inside core region ∝ n−3 ~ amplitude of �nℓ ∝ n–3/2 inside core region This is the basis of all Rydberg scaling ⎧⎪1st node does not shift with n ⎫ inner lobe ⎨ Astonishingly important! −3/2 ⎬ amplitude in first lobe scales as n ⎭ ⎩⎪ ( ) All n,n′ matrix elements of r σ , where σ < −1 scale as nn′ −3/2 ! what happens at σ = −1 ? Some matrix elements scale this way even when σ>0. McQuarrie, page 223 1/2 ⎡ ( n − ℓ − 1)! ⎤ ⎛ 2 ⎞ ℓ+3/2 ℓ − r na0 2 ℓ+1 ⎛ 2 ⎞ Ln+1 ⎜ re Rnℓ (r) = − ⎢ 3⎥ ⎜⎝ na ⎟⎠ ⎝ na0 ⎟⎠ ⎢⎣ 2n ⎣⎡( n + ℓ )!⎤⎦ ⎦⎥ 0 normalization exponential → 0 as r → ∞ associated Laguerre functions (polynominals) Regular and Irregular Coulomb functions (E ≤ 0) unℓu(r) rR(r) ℓ (r) ≡ rR(r) n regular f(E,ℓ,r) ∝ r→0 rℓ+1 g(E,ℓ,r) ∝ r→∞ decreasing exponentially u(ν,ℓ,r)sin πν – v(ν,ℓ,r)e ν, which is an increasing exponential except when ν is a positive integer. Need some other way to satisfy r → ∞ boundary condition when ν is not an integer iπ proper behavior as r → 0 irregular increasing exponentially r–(ℓ) ν ν+1/2) –u(ν,ℓ,r)cos πν + v(ν,ℓ,r)eeiπ( , which , which blows up. * u(ν,ℓ,r) is an increasing exponential as r → ∞ * v(ν,ℓ,r) is a decreasing exponential as r → ∞ (seeGallagher, Gallagher,Rydberg page 16)Atoms, page 16) (see iπ( +1/2) bad news! to behave properly as r → ∞ we need to take linear combination of the f and g functions. Possible only at certain values of ν (we sometimes call this n*) 10 5.73 Lecture #28 28 - 11 T.F. Gallagher, Rydberg Atoms, page 25 1 ⎡⎣3n 2 − ℓ ( ℓ +1)⎤⎦ 2 r r n2 ⎡⎣ 5n 2 +1 − 3ℓ ( ℓ +1)⎤⎦ 2 2 1r 1 n2 1 r2 1 n ( ℓ +1 / 2 ) 1 r3 1 n ( ℓ +1)( ℓ +1 / 2 ) l 1r 4 1 r6 3 3 3n 2 − ℓ ( ℓ +1) 2n 5 ( ℓ + 3 / 2 )( ℓ +1)( ℓ +1 / 2 ) ℓ ( ℓ −1 / 2 ) 35n 4 − 5n 2 ⎡⎣ 6ℓ ( ℓ +1) − 5⎤⎦ + 3( ℓ + 2 )( ℓ +1) ℓ ( ℓ −1) 8n 7 ( ℓ + 5 / 2 )( ℓ + 2 )( ℓ + 3 / 2 )( ℓ +1)( ℓ +1 / 2 ) ℓ ( ℓ −1 / 2 )( ℓ −1)( ℓ − 3 / 2 ) Note! all r σ σ < −1 scale as n−3! σ>0 scale as n ! 11 2σ 5.73 Lecture #28 28 - 12 Outer and Inner Turning Points Orbital 1s 2p 3d 4f 5g 10s 10p 10d 10f r+(n,ℓ)/Å 1.058 3.612 7.508 12.696 19.137 r–(n,ℓ)/Å 0.00 0.624 2.012 4.232 7.313 105.8 105.3 104.2 102.5 0.00 0.529 1.587 3.280 ℜ e2 " 2 ℓ(ℓ + 1) = E = − Vℓ ( r± ) = − + nℓ r± 2µr±2 n2 in atomic units: − 1 1 ℓ(ℓ + 1) = − + r± 2r±2 2n2 12