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Chapter 4 Orbital Angular Momentum and the Hydrogen Atom Chapter 4 Orbital angular momentum and the hydrogen atom If I have understood correctly your point of view then you would gladly sacrifice the simplicity [of quantum mechan- ics] to the principle of causality. Perhaps we could comfort ourselves that the dear Lord could go beyond [quantum me- chanics] and maintain causality. -Werner Heisenberg responds to Einstein In quantum mechanics degenerations of energy eigenvalues typically are due to symmetries. The symmetries, in turn, can be used to simplify the Schr¨odinger equation, for example, by a separation ansatz in appropriate coordinates. In the present chapter we will study rotationally symmetric potentials und use the angular momentum operator to compute the energy spectrum of hydrogen-like atoms. 4.1 The orbital angular momentum According to Emmy Noether’s first theorem continuous symmetries of dynamical systems im- ply conservation laws. In turn, the conserved quantities (called charges in general, or energy and momentum for time and space translations, respectively) can be shown to generate the respective symmetry transformations via the Poisson brackets. These properties are inherited by quantum mechanics, where Poisson brackets of phase space functions are replaced by com- mutators. According to the Schr¨odinger equation i~∂tψ = Hψ, for example, time evolution is generated by the Hamiltonian. Similarly, the momentum operator P~ = ~ ~ generates (spatial) i ∇ translations. A (Hermitian) charge operator Q is conserved if it commutes with the Hamil- tonian [H, Q] = 0. This equation can also be interpreted as invariance of the Hamiltonian 77 CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 78 1 H = Uλ− HUλ under the unitary 1-parameter transformation group of finite transformations Uλ = exp(iλQ) that is generated by the infinitesimal transformation Q. The constant of motion of classical mechanics that corresponds to rotations about the origin is the (orbital) angular momentum L~ = ~x ~p. The corresponding operator ~ in quantum × L mechanics is ∂ ∂ y ∂z z ∂y ~ ~ − ~ = X~ P~ = (~x ~ )= z ∂ x ∂ , (4.1) L × i × ∇ i ∂x − ∂z ∂ ∂ x y ∂y − ∂x or ~ ∂ i = ǫijkXjPk = ǫijkxj . (4.2) L i ∂xk There is no ordering ambiguity because X and P commute for j = k. In addition to the j k 6 orbital angular momentum ~, which is familiar from classical mechanics, quantum mechanical L point particles can have an intrinsic angular momentum, the spin ~, which will be the subject S of the next chapter. The sum of all spins and orbital angular momenta of a system will be called the total angular momentum ~. J 4.1.1 Commutation relations The canonical commutation relation [Xi, Pj]= i~δij implies [ , X ]= ǫ [X P , X ]= i~ǫ X (4.3) Li j ikl k l j ijk k and [ , P ]= ǫ [X P , P ]. = i~ǫ P (4.4) Li j ikl k l j ijk k The form of these results suggests that all vector operators Vj (i.e. operators with a vector index) should transform in the same way. Indeed, we will find that the (axial) vector Lj transforms as [ , ]= i~ǫ . (4.5) Li Lj ijkLk To show this we use the identity ε ε = δ δ δ δ , (4.6) ikl imn km ln − kn lm i.e. the sum over a common index i of a product of ε tensors is 1 if the free index pairs kl ± and mn take the same values, with the sign depending on the cyclic ordering, and vanishes otherwise. We thus find [ , ] = ε [ , X P ]= i~ε (ε X P + ε X P ) (4.7) Li Lj jlm Li l m jlm ilk k m imk l k = i~ ((δ δ δ δ )X P +(δ δ δ δ )X P ) . (4.8) ji mk − jk im k m jk li − ji lk l k CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 79 Since the terms with δji cancel this agrees with i~ε = i~ε ε X P = i~(δ δ δ δ )X P , (4.9) ijkLk ijk klm l m il jm − im jl l m which completes the proof of (4.5). Since angular momenta in different directions do not commute they cannot be diagonalized simultaneously. If we choose as our first observable we expect that the combination 2 + 2, Lz Lx Ly which is classically invariant under rotations about the z-axis, commutes with . This is Lz indeed true, but it is more useful to use the completely rotation invariant 2 = 2 + 2 + 2 as L Lx Ly Lz the second generator of a maximal set of commuting operators. 2 is obviously Hermitian and L [ , 2] = [ , ] + [ , ]= i~ǫ ( + ) = 0. (4.10) Li L Li Lk Lk Lk Li Lk ikr LrLk LkLr A similar calculation shows that [ , P 2] = [ , X2] = 0, so that the kinetic energy commutes Li Li with (and hence also with 2). Li L Angular momentum conservation [ i,H] for rotationally symmetric Hamiltonians H = 2 L P + V (r) with r = x2 + y2 + z2 now already follows from the commutation of with any 2m Li 2 function of X , but letp us check this explicitly in configuration space, ~ ∂ ~ xl ∂ [ j,H] = [ j, V (r)] = εjklxk V (r)= εjklxk V (r) = 0, (4.11) L L i ∂xl i r ∂r where we used the chain rule for V (r(x)) with ∂r = xl and the operator rule ∂xl r ∂ ∂ ∂ ∂ [ ,A(x)] ψ(x)= A(x)ψ(x) A ψ(x)= A(x) ψ(x), (4.12) ∂xl ∂xl − ∂xl ∂xl or [∂ ,A(x)] = ∂ A(x) + A(x)∂ A(x)∂ = ∂ A(x), i.e. commutation of ∂ with an xi xi xi − xi xi xi operator yields the partial derivative of that operator. 4.1.2 Angular momentum and spherical harmonics We will first derive the relation 1 ∂2 1 2 ∆= r L (4.13) r ∂r2 − r2 ~2 between 2 and the Laplacian, which will help us reduce the Schr¨odinger equation to an ordinary L radial differential equation after separation of the angular coordinates. Hence we first evaluate 2 in configuration space, L 2 = = ~2ǫ x ∂ ǫ x ∂ = L LiLi − ijk j k ilm l m = ~2(δ δ δ δ )x ∂ x ∂ = − jl km − jm kl j k l m = ~2(x ∂ x ∂ x ∂ x ∂ )= − j k j k − j k k j = ~2(x ∂ + x x ∂ ∂ 3x ∂ x x ∂ ∂ )= − j j j j k k − j j − j k k j = ~2(x x ∂ ∂ 2x ∂ x x ∂ ∂ ). (4.14) − j j k k − j j − j k k j CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 80 Next we transform to spherical coordinates ~x =(r sin θ cos ϕ, r sin θ sin ϕ, r cos θ), hence x2 + y2 y r = x2 + y2 + z2, θ = arctan , ϕ = arctan . (4.15) p z x p In particular x ∂ i = e(r), x x = r2, x ∂ = r , (4.16) r i j j j j ∂r so that we obtain ∂ ∂2 2 = ~2(r2∆ 2r r2 ), (4.17) L − − ∂r − ∂r2 or 2 1 2 1 1 2 ∆= ∂2 + ∂ L = ∂2 r L , (4.18) r r r − r2 ~2 r r − r2 ~2 which establishes (4.13). Recalling the formula for the Laplace operator in spherical coordinates 1 ∂2 1 1 ∂ ∂ 1 ∂2 ∆= r + (sin θ )+ (4.19) r ∂r2 r2 sin θ ∂θ ∂θ sin2 θ ∂ϕ2 from Mathematical Methods in Theoretical Physics [Dirschmid,Kummer,Schweda] we conclude 1 ∂ ∂ 1 ∂2 2 = ~2 (sin θ )+ . (4.20) L − sin θ ∂θ ∂θ sin2 θ ∂ϕ2 Using the chain rule ∂ ∂ ∂ ∂ =(∂ r) +(∂ θ) +(∂ ϕ) (4.21) i i ∂r i ∂θ i ∂ϕ ~ ∂ ∂ for z = (x y ) one can check that L i ∂y − ∂x ~ ∂ = . (4.22) Lz i ∂ϕ The common eigenfunctions for the angle-dependent part 2/~2 of the Laplace operator and −L for i /~ = ∂/∂ϕ are again known from the Mathematical Methods in Theoretical Physics. Lz They are the spherical harmonics [German: Kugelfl¨achenfunktionen] m 2l + 1 (l m)! (m) imϕ m Ylm(θ, ϕ)=( 1) − Pl (cos θ) e =( 1) Yl,∗ m(θ, ϕ) (4.23) − s 4π (l + m)! − − with the associated Legendre functions l+m (m) 1 2 m d 2 l m (l + m)! ( m) P (ξ)= (1 ξ ) 2 (ξ 1) =( 1) P − (ξ) (4.24) l 2ll! − dξl+m − − (l m)! l − (0) For m = 0 they reduce to the Legendre polynomials Pl(ξ)= Pl (ξ). These results are obtained by a the separation ansatz Ylm = Θ(θ)Φ(ϕ). The eigenvalues for the angular momenta become 2 Y = ~2l(l + 1) Y , Y = ~mY , (4.25) L lm lm L3 lm lm CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 81 with l a nonnegative integer and m Z obeying l m l. The quantization conditions and ∈ − ≤ ≤ the ranges of the eigenvalues follow from termination conditions for the power series ansatz and from single-valuedness at ξ = cos θ = 1. ± Figure 4.1: The eigenvalue ~2l(l + 1) of 2 is (2l + 1)–fold degenerate L m Figure 4.2: Polar plots of Yl versus Θ in any plane through the z-axis for l = 0, 1, 2. | | m m m Note the equality Yl = Yl− , which follows from Yl,m∗ =( 1) Yl, m. | | | | − − CHAPTER 4. ORBITAL ANGULAR MOMENTUM AND THE HYDROGEN ATOM 82 The completeness of the spherical harmonics enables us to solve the stationary Schr¨odinger equation for rotation invariant potentials by a separation ansatz u(~x)= Rλ(r)Ylm(θ, ϕ) with ~2 1 ~2l(l + 1) H R (r)= λ R (r), with H = ∂2 r + + V (r). (4.26) l λ λ l −2m r r 2mr2 The energy eigenvalues λ are (2l+1)-fold degenerate due to the magnetic quantum number m.
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