236 27 The Hydrogen Atom according to Schr¨odinger’s Wave Mechanics The characteristic values for the orbital angular momentum, l =0, 1, 2, 3, 2 correspond,in“chemicallanguage”,tos,p,d,f, orbitals(wealsohavethe relation l ≤ n − 1, where n(= 1, 2, 3, ) is the principal quantum number, see below). The “magnetic” quantum number m assumes the allowed values m = −l, −l +1, ,+l (→ quantization of the orbital angular momentum; we do not yet consider the spin momentum which will be treated below). In addition we need to express the Laplace operator in spherical coordi- nates: ∇ 2 f = 1 r ∂ 2 ∂r 2 (rf) − ˆ L 2 f 2 r 2 , (27.4) where 3 the square of the orbital angular momentum can be written as − ˆ L 2 2 f(θ, ϕ):= 1 sin θ ∂ ∂θ sin θ ∂f ∂θ + 1 sin 2 θ ∂ 2 f ∂ϕ 2 . (27.5) For the radial function R(r) one obtains (using an auxiliary function w(r):=r · R(r)) the following one-dimensional differential equation (N.B. this depends on l, but not on m): − 2 2m w(r) + 2 l(l +1) 2mr 2 + V (r) · w(r)=E · w(r) . (27.6) Additionally, the following boundary conditions must be satisfied: w(0) ! =0 4 and ∞ 0 r 2 |R(r)| 2 dr = ∞ 0 dr|w(r)| 2 ! =1 since d 3 r = r 2 dr dΩ and dΩ|Y lm (θ, ϕ)| 2 := π θ:=0 sin θdθ 2π ϕ=0 dϕ|Y lm (θ, ϕ)| 2 = 1). From now on we concentrate on a Coulomb potential V (r)=−|e| 2 /(4πε 0 r) , i.e., to an A/r-potential, which has already been shown to be special in clas- sical mechanics (remember that for A/r-potentials the Runge-Lenz vector is conserved (→ absence of perihelion motion of the Kepler ellipses and hyper- bolae )). 2 It can be shown algebraically that the commutation relations of angular mo- menta can also be satisfied for half-integral numbers, 1/2, 3/2, 5/2 etc., which corresponds to the possibility of non-orbital angular momenta, e.g., the spin,see below. 3 The relation (27.4) has (since ˆp 2 = − 2 ∇ 2 ) the following classical pendant: p 2 = p 2 r + r −2 [r × p] 2 = p 2 r + L 2 /r 2 ;hereL = r × p is the orbital angular momentum vector. 4 Remember the footnote on the quasi one-dimensional behavior of bound s states in a 3d quantum box in the previous Chap. 25. 27.1 Product Ansatz; the Radial Function 237 By analogy with the harmonic oscillator, reduced lengths and energies are also defined, i.e. the variables := r/a 0 and ε := −E/E R ,where a 0 = 2 4πε 0 e 2 (≈ 0.529 ˚ A(≡ 0.0529 nm)) is called the Bohr radius and E R := 2m 2 e 4 (4πε 0 ) 2 (≈ 13.59 eV) the Rydberg energy. Finally the differential equation for w() thus obtained can again be sim- plified by an ansatz separating the dominating asymptotic behavior for 1 and 1 by two factorizations, i.e. by w()=: l+1 ·F () · e − √ ε . Using this ansatz one obtains a rather simple differential equation (Laguerre’s equation)forF (), which is again solved by a power series F ()= ∞ ν=0,1,2, b ν ν . In this way one is lead to the recursion relation b ν+1 b ν =2 √ ε ·(l +1+ν) −1 (ν +1)(ν +2l +2) . (27.7) 27.1.1 Bound States (E<0) These states correspond to negative energies E (positive values of ε). As for the harmonic oscillator, the condition of square-integrability of w() , ∞ 0 d|w()| 2 ! =1, is again only satisfied if the recursion relation (27.7) terminates. Again a pos- itive integer n must exist, the “principal quantum number” (characterising the electron shell ), such that the recursion relation (27.7) results in b ν+1 ≡ 0, as soon as ν has reached the integer n − l −1. The energy E is then = − E R n 2 , i.e., it does not depend on l, although the radial function R()(= R nl ()) does. The fact that the binding energy of the H-atom only depends on the principal quantum number n (and not on the orbital quantum number l) 238 27 The Hydrogen Atom according to Schr¨odinger’s Wave Mechanics is a special property of the Coulomb potential 5 , i.e., this “accidental degen- eracy”, as it it often called somewhat misleadingly, is not valid for general V (r). The product ansatz u(r)=R nl (r) · Y lm (θ, ϕ) , however, applies generally for rotationally invariant potentials V (r), and the energy of bound states depends generally both on n and l. 27.1.2 The Hydrogen Atom for Positive Energies (E>0) This case is important inter alia because of its relation to the solar atmo- sphere and corona. The analysis corresponds to the Kepler problem, but with hyperbolic orbits, i.e., ε is now imaginary. Thus the recursion relation (27.7) no longer terminates, since √ ε =i |ε| , and one obtains radial functions R E,l (r), which are (analogously to monochro- matic plane waves) only almost square integrable, i.e., although not being square integrable themselves, they can be superposed to square integrable wave packets. The corresponding wave functions are u E;l,m (r)=Y lm (θ, ϕ) · R E,l (r) , where the radial functions can be orthonormalized as R E,l |R E ,l = ∞ 0 drr 2 R E,l (r) ∗ R E ,l (r)=δ ll δ(E − E ) . (27.8) The asymptotic behavior of the R E,l (r)forr →∞is given by R E,l (r) ∝ sin k E ·r + κ E ln(2k E r)+η l,E − lπ 2 r (27.9) with k E := 2m 2 E, κ E := 1 |ε| and with a real so-called Coulomb phase η l,E . 5 In classical mechanics the corresponding speciality is the conservation of the so- called Runge-Lenz vector for the Kepler problem (i.e., planetary motion), which (as already mentioned) corresponds to the absence of any perihelion rotation in the ideal case, i.e., for a perfect A/r-potential. 27.2 Spherical Harmonics 239 27.2 Spherical Harmonics The spherical harmonics Y l,m (ϑ, ϕ) appearing in the above product ansatz, and in (27.2) and (27.3), are eigenfunctions of the operators ˆ L 2 and ˆ L z with eigenvalues 2 l · (l +1) and m , respectively (here, for convenience, we are using the index “m” (not mass) instead of “m l ”). The facts concerning ˆ L 2 and ˆ L z are most important at this point and related to properties already known from electrodynamics, see Part II, since products r l ·Y l,m (ϑ, ϕ) are harmonic, i.e., ∇ 2 r l ·Y l,m (ϑ, ϕ) ≡ 0 , which yields a simple proof of the above relation (27.2). Concerning the second relation we note that the functions Y lm (ϑ, ϕ)are defined as Y lm (θ, ϕ)=c l,|m| ·P l,|m| (cos θ) · e imϕ , (27.10) where the normalization factors c l,|m| are not of interest (in particular, be- cause there are different conventions, distinguished by a complex factor of magnitude unity, a fact that one should consider when discovering apparent mistakes or inconsistencies in some formulae). The P l,|m| are called “asso- ciated Legendre polynomials”; for m = 0 one has the “genuine” Legendre polynomial. The fact that Y lm (ϑ, ϕ) is an eigenfunction of the z-component ˆ L z ,with eigenvalue m, follows directly from the definition (27.10), since for ˆ L z = xˆp y − yˆp x , by application to a function of r, ϑ and ϕ, we have the simple identity ˆ L z = i ∂ ∂ϕ , which can easily be proved (usually as an exercise). In addition, the relation ˆ L 2 Y lm = 2 l(l +1)Y lm , which is equivalent to ∇ 2 r l ·Y lm =0, follows easily from the expression (27.4) for the ∇ 2 operator in spherical coordinates. 240 27 The Hydrogen Atom according to Schr¨odinger’s Wave Mechanics The normalization coefficients (see below) are chosen in such a way that one obtains a complete orthonormalized system of angular functions on the surface of a sphere, i.e. Y lm |Y l m := π 0 dθ sin θ 2π 0 dϕY lm (θ, ϕ) ∗ Y l m (θ, ϕ) = δ ll δ mm , (27.11) such that a generalized Fourier expansion of an arbitrary function f(ϑ, ϕ)is possible. Important spherical harmonics are listed in the following table (we repeat that the normalization factors are not important. The bracketed terms on the r.h.s. show explicitly the relation to harmonic functions.) Y 0,0 (θ, ϕ)= 1 4π , (27.12) Y 1,0 (θ, ϕ)= 3 4π cos θ ∝ z r (27.13) Y 1,±1 (θ, ϕ)= 3 8π sin θ · e ±iϕ ∝ x ±iy r (27.14) Y 2,0 (θ, ϕ)= 5 16π (3 cos 2 θ − 1) ∝ 3z 2 − r 2 r 2 (27.15) Y 2,±1 (θ, ϕ)= 15 16π cos θ sin θ e ±iϕ ∝ z · (x ± iy) r 2 (27.16) Y 2,±2 (θ, ϕ)= 15 32π sin 2 θ e ±2iϕ ∝ (x ± iy) 2 r 2 . (27.17) A nodal theorem is also valid for the spherical harmonics: The real and imaginary parts of Y lm possess l −|m| polar zero-lines (ϑ = constant) and additionally |m| azimuthal zero-lines (ϕ = constant). The radial functions R n,l (r)haven −l −1 radial nodal surfaces. The sum of the radial, polar and azimuthal nodal surfaces of the product functions R nl (r) ·Re[Y lm (ϑ, ϕ)] (and similarly for the imaginary part) is thus n − 1, as expected. As a consequence, in the hydrogen atom, the probability of finding the electron outside a radius r 0 n ·a 0 is exponentially small, ∝ e −2r 0 /(n·a 0 ) 6 . 6 i.e., not as small as for the harmonic oscillator, where it is ∝ e −x 2 /x 2 0 ,seeabove. 28 Abstract Quantum Mechanics (Algebraic Methods) 28.1 The Harmonic Oscillator II: Creation and Destruction Operators Heisenberg was the first to recognise 1 that for a given Hamiltonian the impor- tant point is to represent the observables by algebraic entities (e.g., matrices within his formalism; operators in Schr¨odinger’s wave mechanics) which sat- isfy the canonical commutation relations (24.10) 2 . Then everything else fol- lows purely algebraically (sometimes after a certain amount of perturbation theory) without the aid of differential operators. This approach will now be used to treat the harmonic oscillator (although even the hydrogen atom was treated completely algebraically by Wolfgang Pauli 3 ,beforeSchr¨odinger invented his wave mechanics). For the harmonic oscillator the Hamilton operator ˆ H written as a function of the reduced length ξ := x/ /(mω 0 ) , see Sect. 26, and reduced momentum ˆp ξ := 1 i ∂ ∂ξ has the form: ˆ H = ω 0 2 ˆp 2 ξ + ˆ ξ 2 . The canonical commutation relations are equivalent to ˆp ξ , ˆ ξ = 1 i . 1 in his famous publication where he created matrix mechanics and in a subsequent publication with the co-authors Max Born and Pascual Jordan. 2 This generalizes the Poisson brackets of classical mechanics, see Part I. 3 See the book on quantum mechanics, [24], by W. D¨oring, G¨ottingen 1962, who also presents Pauli’s algebraic approach for the H atom. 242 28 Abstract Quantum Mechanics (Algebraic Methods) By complex rotation one now introduces the following operators, which are called the destruction operator and the creation operator, respectively: ˆ b := 1 √ 2 ˆ ξ +iˆp ξ and ˆ b + := 1 √ 2 ˆ ξ − iˆp ξ . (28.1) They are mutually adjoint to each other, ˆ b + ψ 1 ψ 2 = ψ 1 | ˆ bψ 2 (if | ˆ bψ 2 = 0); i.e. they are clearly non-Hermitian, but instead they have a number of remarkable properties, inter alia: If |ψ n (n =0, 1, 2, ) is a (normalized) eigenstate of ˆ H then one has for n =0, 1, 2, ,andforn =1, 2, 3, , respectively: b + |ψ n = √ n +1|ψ n+1 and ˆ b|ψ n = √ n|ψ n−1 , (28.2) as well as ˆ b|ψ 0 =0. Thus ˆ b + increases the number of excited quanta by 1, whereas ˆ b decreases this number by 1, which explains their names. The Hamilton operator itself can be written as: H = ω 0 2 · ( ˆ b + ˆ b + ˆ b ˆ b + )=ω 0 · ˆ b + ˆ b + 1 2 . (28.3) In (28.3) we have used the commutation relation 4 ˆ b, ˆ b + =1 (which follows from [ˆp, ˆx]= i ). The number operator ˆn := ˆ b + ˆ b is Hermitian, with eigenvalues n =0, 1, 2, To prove the above statements we firstly calculate |Φ := ˆ b + ˆ b ˆ b + |ψ n and find with the canonical commutation relation ˆ b ˆ b + =1+ ˆ b + ˆ b that |Φ = ˆ b + 1+ ˆ b + ˆ b |ψ n . Since |ψ≡|ψ n is already a (normalized) eigenstate of the number operator ˆn = ˆ b + ˆ b with eigenvalue n, we find that the state |ψ := b + |ψ n must also be eigenstate of ˆn, viz with eigenvalue n +1. 4 The operator ˆ b is thus also not unitary, since for unitary operators ˆ U one would have ˆ U ˆ U + = 1, i.e., without commutator bracket. 28.2 Quantization of the Angular Momenta; Ladder Operators 243 The length of |ψ is calculated with ψ |ψ = ˆ b + ψ n ˆ b + ψ n = ψ n | ˆ b ˆ b + ψ n , and with the canonical commutation relations we find (1 + n) ψ n |ψ n (≡ 1+n) . Thus ||ψ | must be √ n + 1. In addition, if one assumed that the number operator ˆn had eigenvalues different from the non-negative integer numbers, one would be led into contradictions. 28.2 Quantization of the Angular Momenta; Ladder Operators One can proceed similarly with the angular momenta. Firstly, it follows from the canonical commutation relations for ˆ r and ˆ p that for the orbital angular momentum L := ˆ r × ˆ p canonical angular momentum commutation relations are valid, i.e., ˆ J x , ˆ J y =iJ z (etc., by cyclic permutation of the three Cartesian components of an angular momentum vector operator ˆ J, which correspond to Hermitian operators ˆ J k , with k = x, y, z). In the following, the algebra of the angular commutation relations is as- sumed to apply more generally, i.e., not only for orbital moments, J = L, where it can be derived from the canonical commutation relations for ˆ r and ˆ p, but without derivation right from the beginning. For example, on purely mathematical reasoning we consider the possibility that there may be, for- mally or not, other angular momenta or quasi-momenta 5 than those of the orbital motion. The generalization of what is known for the harmonic oscillator is based on the following facts: the ladder operators, ˆ J ± := ˆ J x ± i ˆ J y , increase (or decrease) the eigenvalue of ˆ J z by one unit of from m · to (m ± 1) · , as far as possible, so that the following statements are true and can be proved purely algebraically (→ exercises): 5 e.g., the so-called isospin. 244 28 Abstract Quantum Mechanics (Algebraic Methods) a) The operators ˆ J 2 and ˆ J z commute with each other, and are thus simul- taneously diagonalizable. b) The orthonormalized abstract eigenvectors |ψ J,M J of ˆ J 2 and ˆ J z (i.e., of this simultaneous diagonalization) satisfy the relations ˆ J 2 |ψ J,M J = 2 J(J +1)|ψ J,M J , (28.4) with integral or half-integral (!) value of J (= 0, 1 2 , 1, 3 2 , ), while ˆ J z |ψ J,M J = M J |ψ J,M J , (28.5) with M J = −J, −J +1, ,J . c) Under these constraints (and the usual phase conventions concerning square roots of a complex number) one has ˆ J ± |ψ J,M J = J(J +1)−M J (M J ± 1)|ψ J,M J ±1 , (28.6) i.e., ˆ J ± |ψ J,± J =0. Thus, in addition to the orbital angular momentum (J =0, 1, 2, ),al- gebraically a second kind of angular momentum J = 1 2 , 3 2 , should exist, which has no classical analogue. This comprises (in addition to other cases) the so-called intrinsic or spin angular momentum of an electron 6 , for which J = 1 2 . In fact, for J =1/2 the above-mentioned canonical angular momentum commutation relations can be implemented as follows: ˆ J = 2 σ , with the Pauli matrices σ x = 0 , 1 1 , 0 ,σ y = 0 , −i i , 0 ,σ z = 1 , 0 0 , −1 . (28.7) The corresponding states (complex two-component “spinors”) can be gen- erated by the following basic states: ↓:= ψ 1 2 ,− 1 2 = 0 1 , ↑:= ψ 1 2 , 1 2 = 1 0 . (28.8) 6 For other particles the spin may be different. 28.3 Unitary Equivalence; Change of Representation 245 28.3 Unitary Equivalence; Change of Representation All the above statements are not only independent of the basis functions used in the calculations but are also independent of the dynamical “aspect” used in the formulation of quantum mechanics. One usually begins with the first of the following three “aspects” (or “pic- tures”, or “representations”) (i) the Schr¨odinger aspect, (ii) the Heisenberg aspect, and (iii) the interaction (or Dirac) aspect. Changes in the basis functions and/or “representations” 7 are implemented by means of unitary operators ˆ U. These are special linear operators defined on the total Hilbert space 8 , which can be interpreted as (complex) rotations of the Hilbert space, since they leave scalar products invariant. We thus assume that for all |ψ, |ψ 1 and |ψ 2 in Hilbert space the fol- lowing equalities are valid: |ψ = ˆ U|ψ or ψ 1 |ψ 2 = ˆ Uψ 1 ˆ Uψ 2 = ψ 1 ˆ U + ˆ Uψ 2 = ψ 1 |ψ 2 . (28.9) For unitary transformations one therefore demands that ˆ U + ˆ U = ˆ U ˆ U + = ˆ 1 , or ˆ U + = ˆ U −1 . Such operators, if they do not depend on time, can always be written as ˆ U =e i ˆ B with self-adjoint (Hermitian plus complete) ˆ B. The transition from position functions to momentum functions, or the transitions between various matrix representations, can be written in this way. However, unitary opera- tors can also depend nontrivially on time. For example, the generalized gauge transformation (24.16), which was treated in an earlier section, corresponds to a (generally time-dependent) unitary transformation. In addition to the invariance of all scalar products a unitary transforma- tion also preserves the quantum mechanical expectation values, probability statements, commutation relations etc., i.e., all measurable physical state- ments. It is only necessary that all observables ˆ A appearing in these state- ments are transformed covariantly with the states, i.e. as follows: |ψ→|ψ := ˆ U|ψ , ˆ A → ˆ A := ˆ U + ˆ A ˆ U. (28.10) It is easy to show that such transformations conserve all expectation val- ues: ψ ˆ Aψ ≡ ψ ˆ A ψ . (28.11) 7 ,,Darstellungs- bzw. Bildwechsel” in German texts 8 We remind ourselves that usually the Hamilton operator ˆ H is not defined in the total Hilbert space, since the functions ψ must be differentiable twice. In contrast, exp [−i ˆ H · t/], by a subtle limiting procedure, is defined in the total Hilbert space. . unitary transformation. In addition to the invariance of all scalar products a unitary transforma- tion also preserves the quantum mechanical expectation values, probability statements, commutation. relations etc., i.e., all measurable physical state- ments. It is only necessary that all observables ˆ A appearing in these state- ments are transformed covariantly with the states, i.e. as. where he created matrix mechanics and in a subsequent publication with the co-authors Max Born and Pascual Jordan. 2 This generalizes the Poisson brackets of classical mechanics, see Part I. 3 See