� � � 8.04 Quantum Physics Lecture XXIIILast time • Radial equationforgiven angular momentumeigenstate Ylm(θ,φ)with quantum numbbe l � �2 � ∂2 + 2 ∂ � + �2l(l + 1) + V(r) � Rnl(r)= EnlR(r) (23-1)−2m ∂r2 r ∂r 2mr2can be written in form of 1D SE with effective potential�2l(l + 1)Veff(r)= V(r)+ (23-2)2mr2by defining u(r)= rR(r),�2 ∂2u− 2m ∂r2 + Veff(r)u(r)= Eu(r). (23-3) • Specialization tohydrogen atom: Ze2 V(r)= −4πε0r (23-4) Define dimensionless variables: • ρ = 8m|E|r,λ = Ze2 m (23-5)�24πε0� 2|E| u��(ρ)+ �λρ − 14 − l(l ρ+ 21)� u(ρ)= 0 (23-6) • Asymptotic solutions: u(ρ)= s(ρ)ρl+1e− ρ 2, for ρ →∞,ρ → 0 (23-7) andTaylorexpansion ∞s(ρ)= akρk (23-8) k=0 leads to recursion relation aak+k 1 = �k + k 1+ �lk ++ 12(−l + λ 1)�. (23-9) • Boundary conditions for ρ →∞ require series to terminate at some k = nr (23-10) nr + l + 1= λ, (nr radial quantum number) (23-11) → Massachusetts Institute of Technology XXIII-1 � � � � � � � � � � 8.04 Quantum Physics Lecture XXIIIDefine• λ = n = nr + l + 1, (principal quantum number) (23-12)→ En = −21 mc2(Znα2)2 (23-13) not relativistic formula, only written in simple form using 2e4πε0�c → (finestructureconstant) (23-14)α = Ingeneral,the(unnormalized)polynomial ()istheassociatedLaguerrepolyno-ρs•∞322221d()dY()drR()(23-20)ψθ,φΩ||||||rrrr==nlmnlnl0 �mial, = L(2l+1)s(ρ)n−l−1(ρ), (23-15) defined as nLα n (ρ)= � n + α (−ρ)m . (23-16)n − mm! m=0 The 3D wavefunction is given by: ψnlm(r,θ,φ)= Rnl(r)Ylm(θ,φ) (23-17) unl(r) = Ylm(θ,φ) (23-18)r with 2ρu(ρ)= s(ρ)ρl+1e− 1 , (23-19) normalized such that The probability to find particle within shell[r, r + dr]isgivenby dΩ|Ynl|2r2|R(r)|2dr = |u(r)|2 dΩ|Ynl| (23-21) Degeneracy of the hydrogen spectrum Forgivenl1 all magnetic quantum numbers m have the same energy, so each l is (2l + 1) degenerate. Also, for each n = nr +l+1,the radial quantum numbernr can takeon thevalues nr = 0, 1,..., n − 1, and the l quantum number the corresponding values l = 0, 1,..., n − 1. So for given n, the total number of degenerate states is 1+ 3++ 2(n − 1) + 1= (n + 1)2 (23-22)���� ���� ··· � ��(l=0) (l=1) (l=n−1) Actually, there are twice as manystates because each electron has two spin states. Massachusetts InstituteofTechnology XXIII-2 ��8.04 Quantum Physics Lecture XXIIIFigureI: Energylevel structureofhydrogen atom. Normalized time-independent eigenstates of hydrogen 2 ψ100 = e−r/a0Y00(θ,φ)1s (23-23)(a0)3/221− 2ra0 e−r/2a0Y00(θ,φ)2s (23-24)ψ200 = (2a0)3/2 ⎫⎪ ⎛⎜ ⎛⎜ ⎞⎟ ⎞⎟ ψ211 Y11(θ,φ) 1 r ψ210 = √3(2a0)3/2 e−r/2a0 Y10(θ,φ)⎬⎪ 2p (23-25)a0⎝⎠⎝⎠ψ21−1 Y1−1(θ,φ) ⎭Massachusetts Institute of Technology XXIII-3 � 8.04 Quantum Physics Lecture XXIIISpectroscopic Notationl = 0 is called s l = 1 is called p l = 2 is called d l = 3 is called f Some useful expectation values for the hydrogen atom Given the wavefunctions R(r), we can calculate expectation values �rk� = 0 ∞ drr2+k|Rnl(r)|2 (23-26) �r� = a0[3n2 − l(l + 1)] (23-27)2Za02n222�r� = 2Z2[5n+ 1− 3l(l + 1)] (23-28) �1� Z = (23-29)ra0n2 � 1� Z2 = �� (23-30)r2 a20n3 l + 12 Lifting of degeneracy in hydrogen atom Further interactions,thatwehaveneglectedsofar,liftthedegeneracybetweens,p,dlevels. For instance,fromthe electron’spointofview,themovingproton correspondstoa current. The associated magnetic field couples to the magnetic moment associated with the spin of the electron: spin-orbit interaction. Furthermore, relativistic effects lead to energy shifts that depend on the total angular momentum J = L + S of the electron(8.05: addition of angular momenta). Also, the proton has spin that has a small magnetic moment associated with it. The interaction between the proton’s and the electron’s magnetic moments is called thehyperfine interaction,andleadstoshiftsthatdependonthetotalangular momentum F = J + I = L + S + I of theatom, where I is the spin of the proton (nucleus). While the intrinsic angular momentum (spin) of fundamental particles is always �/2, composite particles, such as nuclei, can have integer spin if the number of constituents is even. Therefore, when viewed as single particles, atoms can be bosons (integer spin) or fermions (half-integer spin), with dramatic consequences for quantum statistics and low-temperature behavior. Two identical fermions mustbe describedbyawavefunction thatis antisymmetric with Massachusetts Institute of Technology XXIII-4 8.04 Quantum Physics Lecture XXIII(a) nr = 0, l = 0, n = 1 (b) nr = 0, l = 0, n = 1(c) nr = 1, l = 0, n = 2 (d) nr = 1, l = 0, n = 2(e) nr = 0, l = 1, n = 2 (f) nr = 0, l = 1, n = 2(g) nr = 2, l = 0, n = 3 (h) nr = 2, l = 0, n = 3Figure II:Afew radialwavefunction. Displayed on the leftis thewavefunction Rnl, on the right the probability density |unl|2 = r2|Rnl|2. nr is the number of nodes in the radial wavefunction. respect to particle exchange, ψτ(r1, r2)= −ψq(r2, r1) (23-31) Massachusetts Institute of Technology XXIII-5 8.04 Quantum Physics Lecture XXIII= wavefunction vanishes for r1 fermions avoid each other. ⇒ = r2 →Bosons are described by symmetric wavefunction with repect to particle exchange ψB(r1, r2)=+ψB(r2, r1) (23-32) = Bosons are more likely to be found at same position lasers, Bose-Einstein conden⇒⇒sation, superconductivity, classical notion of fields where amplitudes can be added. Polarization of light Aclassical light field traveling alongz can be linearly polarized along x, ε(z, t)= ε0eˆxeikz−iωt , (23-33) linearly polarized along y, ε(z, t)= ε0eˆyeikz−iωt , (23-34) linearly polarized alonga directionˆe = cos θeˆx + sin θeˆy, in the xy plane, ε(z, t)= ε0ˆikz−iωt ,ee(23-35) circularly polarized 1 eˆR = (ˆex + ieˆy) (23-36) √21 eˆL = √2(ˆex − ieˆy) (23-37) εL,R(z, t)= ε0eˆL,Reikz−iωt (23-38) or, in general, elliptically polarized εˆ= cos θeˆx + eiφ sin θeˆyε(z, t) = ε0ˆikz−iωt (23-39)eeAny two orthogonal polarization (e.g.,(ˆex,ˆey), � √12(ˆex + eˆy), √12(ˆex − eˆy)�,(ˆeL,ˆeR), ...) formabasis.An arbitrary polarization canbeexpressedasa superpositionofthetwo basis polarizations. Alinear polarizer has one strongly absorbing direction of polarization (ideallyε eˆ= 0· along that direction of polarization after the polarizer), and one weakly absorbing direction (ideally: no absorption). If we call the latter the axis of the polarizer, the light behind the polarizer is linearly polarized along that axis. No light is transmitted through two crossed polarizers unless a third polarizer is inserted between them at an intermediate angle In this case the transmitted field is �1 � 11 ε0 · eˆx ·√2(ˆex + eˆy)√2(ˆex + eˆy)· eˆy = 2ε0 (23-40) Massachusetts Institute of Technology XXIII-6 8.04 Quantum Physics Lecture XXIIIFigure III: The polarization of the transmitted light is εtr = (εinc · eˆx)ˆexeikz−iωt Figure IV: Light can pass two crossed polarizers if a third polarizer is inserted between them that is oriented at an angle. and the transmitted intensity is proportional to �21εo �2. The variation of transmitted electric field with polarizer angle, ˆe = cos θeˆx + sin θeˆy for incident field along ˆx, ε = ε0eˆx is eˆxeˆ= cos θ, so the transmitted intensity varies as cos2θ Quantum mechanical description A light beam consists of photons, if we attenuate the beam to the level where only on photon passes though the polarizer at anygiven time, then because photons appear only as units, the photon is either absorbed or it is not: The probability for the photon passing the polarizer is now cos2θ (the probabililty amplitude is θ). The polarizer “measures” the polarization state of the photon: if the photon is polarized along the polarizer axis, it is transmitted, if polarized perpindicular to the polarizer axis, the photon is absorbed. Massachusetts Institute of Technology XXIII-7