256 30 Addition of Angular Momenta By straightforward, but rather lengthy methods, or more generally by group theory 2 , this can be shown to be exactly possible if J is taken from the following set: J ∈{L + S, L + S − 1, ,|L −S|} . Furthermore, M J must be one of the 2J + 1 integral or half-integral values −J, −J +1, , +J,andM J = M L + M S . The coefficients in (30.1) are called Clebsch-Gordan coefficients.(N.B. Unfortunately there are many different conventions in use concerning the formulation of the coefficients.) 30.2 Fine Structure of the p-Levels; Hyperfine Structure Without the spin-orbit interaction the p-levels (l = 1) of the outermost elec- tron of an alkali atom are sixfold, since including spin one has 2 ×(2l +1)= 6 states. Under the influence of the spin-orbit interaction the level splits into a fourfold p 3 2 -level, i.e., with j = l + 1 2 , with positive energy shift ΔE,andatwofold p 1 2 -level, i.e., with j = l − 1 2 , with an energy shift which is twice as large (but negative) −2ΔE; the “center of energy” of the two levels is thus conserved. This so-called fine structure splitting gives rise, for example, to the well- known “sodium D-lines” in the spectrum of a sodium salt. Analogously to the “fine structure”, which is is based on the coupling of ˆ L and ˆ S of the electrons, δ ˆ H fine ∝ k(r) ˆ L · ˆ S , there is an extremely weak “hyperfine structure” based on the coupling of the total angular momentum of the electronic system, ˆ J, with the nuclear spin, ˆ I,tothetotal angular momentum of the atom, ˆ F := ˆ J + ˆ I . We have δ ˆ H hyperfine = c ·δ(r) ˆ J · ˆ I , where the factor c is proportional to the magnetic moment of the nucleus and to the magnetic field produced by the electronic system at the location of the nucleus. In this way one generates, e.g., the well-known “21 cm line” of the neutral hydrogen atom, which is important in radio astronomy. 2 See e.g., Gruppentheorie und Quantenmechanik, lecture notes (in German), [2] 30.3 Vector Model of the Quantization of the Angular Momentum 257 30.3 Vector Model of the Quantization of the Angular Momentum The above-mentioned mathematical rules for the quantization of the angular momenta can be visualized by means of the so-called vector model,whichalso serves heuristic purposes. Two classical vectors L and S of length L · (L +1) and S · (S +1) precess around the total angular momentum J = L + S , a vector of length J ·(J +1), with the above constraints for the admitted values of J for given L and S. On the other hand, the vector J itself precesses around the z-direction, where the J z -component takes the value M J . (At this point, a diagram is recommended.) 31 Ritz Minimization The important tool of so-called “Ritz minimization” is based on a theorem, which can easily be proved under general conditions. For example, let the Hamilton operator ˆ H be bounded from below, with lower spectral limit E 0 (usually the true energy of the ground state); then one has for all states ψ of the region of definition D ˆ H of ˆ H , i.e., for almost all states ψ of the Hilbert space (a −∀ψ): E 0 ≤ ψ ˆ Hψ ψ|ψ “a−∀ψ . (31.1) If the lower spectral limit corresponds to the point spectrum of ˆ H, i.e., to a “ground state” ψ 0 and not to the continuous spectrum, the equality sign in (31.1) applies iff ˆ H|ψ 0 = E 0 |ψ 0 . The theorem is the basis of Ritz approximations (e.g., for the ground state), where one tries to optimize a set of variational parameters to obtain a minimum of the r.h.s. of (31.1). As an example we again mention the Hartree-Fock approximation, where one attempts to minimize the energy expectation of states which are represented by a single Slater determinant instead of by a weighted sum of different Slater determinants. Generally the essential problem of Ritz minimizations is that one does not vary all possible states contained in the region of definition of ˆ H, but only the states of an approximation set T , which may be infinite, but is nonetheless often too small. Usually even in the “infinite” case the set T does not contain thetruegroundstateψ 0 ,andonedoesnotevenknowthedistanceofT from ψ 0 . The main disadvantage of the Ritz approximations is therefore that they are “uncontrolled”, i.e., some intuition is needed. For example, if one wants to obtain the ground states of (i) the harmonic oscillator, ψ 0 (x) ∝ e − x 2 2x 2 0 , 260 31 Ritz Minimization and (ii) the H-atom, ψ n≡1 (r) ∝ e − r a 0 , by a Ritz minimization, the respective functions must be contained in T ; x 0 (= 2mω 0 )anda 0 2 ma 2 0 ! = e 2 4πε 0 · a 0 are the characteristic lengths. For the n-th shell of the hydrogen atom the characteristic radius is r n ≡ n · a 0 . 32 Perturbation Theory for Static Problems 32.1 Formalism and Results Schr¨odinger’s perturbation theory is more systematic than the Ritz method. Let there exist a perturbed Hamilton operator ˆ H = ˆ H 0 + λ · ˆ H 1 , with a real perturbation parameter λ. The unperturbed Hamilton operator ˆ H 0 and the perturbation ˆ H 1 shall be independent of time; furthermore, it is assumed that the spectrum of ˆ H 0 consists of the set of discrete eigenstates u (0) n with corresponding eigenvalues E (0) n . The following ansatz is then made: u n = u (0) n + λ ·u (1) n + λ 2 ·u (2) n + , and analogously : E n = E (0) n + λ · E (1) n + λ 2 ·E (2) n + . (32.1) Generally, these perturbation series do not converge (see below), similarly to the way the Taylor series of a function does not converge in general. In particular, non-convergence occurs if by variation of λ the perturbation changes the spectrum of a Hamilton operator not only quantitatively but also qualitatively, e.g., if a Hamilton operator which is bounded from below is changed somewhere within the assumed convergence radius R λ into an operator which is unbounded. In fact, the perturbed harmonic oscillator is non-convergent even for apparently “harmless” perturbations of the form δ ˆ H = λ · x 4 with λ>0 . The reason is that for λ<0 the Hamiltonian would always be unbounded (i.e., R λ ≡ 0). However, even in the case of non-convergence the perturbational results are still useful, i.e., as an asymptotic approximation for small λ, as for a Tay- lor expansion, whereas the true results would then often contain exponentially small terms that cannot be treated by simple methods, e.g., corrections ∝ e − constant λ . 262 32 Perturbation Theory for Static Problems With regard to the results of Schr¨odinger’s perturbation theory one must distinguish between the two cases of (i) non-degeneracy and (ii) degeneracy of the unperturbed energy level. a) If u (0) n is not degenerate, one has E (1) n = u (0) n ˆ H 1 u (0) n , and E (2) n = − m(=n) u (0) m ˆ H 1 u (0) n 2 E (0) m − E (0) n , (32.2) as well as u (1) n = − m(=n) u (0) m u (0) m ˆ H 1 u (0) n E (0) m − E (0) n . (32.3) Hence the 2nd order contribution of the perturbation theory for the energy of the ground state is always negative. The physical reason is that the perturbation leads to the admixture of excited states ( ˆ= polarization) to the unperturbed ground state. b) In the degenerate case the following results apply. If (without restriction) E 0 1 = E (0) 2 = = E (0) f , then this degeneracy is generally lifted by the perturbation, and the val- ues E (1) n ,forn =1, f,aretheeigenvalues of the Hermitian f × f perturbation matrix V i,k := u (0) i ˆ H 1 u (0) k , with i, k =1, ,f . To calculate the new eigenvalues it is thus again only necessary to know the ground-state eigenfunctions of the unperturbed Hamilton operator. Only these functions are needed to calculate the elements of the pertur- bation matrix. Then, by diagonalization of this matrix, one obtains the so-called correct linear combinations of the ground-state functions, which are those linear combinations that diagonalize the perturbation matrix (i.e., they correspond to the eigenvectors). One can often guess these cor- rect linear combinations, e.g., by symmetry arguments, but generally the following systematic procedure must be performed: – Firstly, the eigenvalues, E(≡ E (1) ), are calculated, i.e. by determining the zeroes of the following determinant: P f (E):= V 1,1 − E, V 1,2 ,V 1,3 , , V 1,f V 2,1 ,V 2,2 − E,V 2,3 , , V 2,f , , , V f,1 ,V f,2 ,V f,3 , ,V f,f − E . (32.4) 32.2 Application I: Atoms in an Electric Field; The Stark Effect 263 – Secondly, if necessary, a calculation of the corresponding eigenvector c (1) := (c 1 ,c 2 , ,c f ) (which amounts to f −1 degrees of freedom, since a complex factor is arbitrary) follows by insertion of the eigenvalue into (f − 1) indepen- dent matrix equations, e.g., for f = 2 usually into the equation (V 1,1 − E) · c 1 + V 1,2 · c 2 =0. (The correct linear combination corresponding to the eigenvalue E = E (1) is then: f i=1 c i ·u (0) i , apart from a complex factor.) Thus far for the degenerate case we have dealt with first-order perturbation theory. The next-order results (32.2) and (32.3), i.e., for E (2) n and u (1) n ,are also valid in the degenerate case, i.e., for n =1, ,f,ifinthesumoverm the values m =1, ,f are excluded. 32.2 Application I: Atoms in an Electric Field; The Stark Effect of the H-Atom As a first example of perturbation theory with degeneracy we shall consider the Stark effect of the hydrogen atom, where (as we shall see) it is linear, whereas for other atoms it would be quadratic. Consider the unperturbed energy level with the principal quantum num- ber n = 2. For this value the (orbital) degeneracy is f =4 u (0) i ≡ u (0) n,l,m ∝ Y 00 , ∝ Y 10 , ∝ Y 1,+1 and ∝ Y 1,−1 . The perturbation is λ ˆ H 1 = −eF · z, with the electric field strength F . 1 With z = r · cos θ one finds that only the elements V 21 = V 12 ∝Y 00 |cos θY 10 of the 4 × 4 perturbation matrix do not vanish, so that the perturbation matrix is easily diagonalized, the more so since the states u (0) 3 and u (0) 4 are not touched at all. 1 It should be noted that the perturbed Hamilton operator, ˆ H = ˆ H 0 − λ · z,is not bounded, so that in principle also tunneling through the barrier to the result- ing continuum states should be considered. However the probability for these tunneling processes is exponentially small and, as one calls it, non-perturbative. 264 32 Perturbation Theory for Static Problems The relevant eigenstates (correct linear combinations) are u (0) ± := 1 √ 2 u (0) 1 ± u (0) 2 ; they have an electric dipole moment e · u (0) ± zu (0) ± = ±3e · a 0 , which orients parallel or antiparallel to the electric field, where one obtains an induced energy splitting. Only for H-atoms (i.e., for A/r potentials) can a linear Stark effect be obtained, since for other atoms the states with l = 0 are no longer degen- erate with l = 1. For those atoms second-order perturbation theory yields a quadratic Stark effect; in this case the electricdipolemomentitselfis“in- duced”, i.e., ∝ F. As a consequence the energy splitting is now ∝ F 2 and the ground state of the atom is always reduced in energy, as mentioned above. 32.3 Application II: Atoms in a Magnetic Field; Zeeman Effect In the following we shall use an electronic shell model. Under the influence of a (not too strong) magnetic field the f(= 2J +1)- fold degenerate states |ψ N,L,S,J,M J split according to their azimuthal quantum number M J = −J, −J +1, ,J . In fact, in first-order degenerate perturbation theory the task is to diagonalize (with the correct linear combinations of the states |M J := ψ (0) N,L,S,J,M J ) the f × f-matrix V M J ,M J := M J | ˆ H 1 |M J induced by the Zeeman perturbation operator ˆ H 1 := −μ B H z · ˆ L z +2 ˆ S z . It can be shown that this matrix is already diagonal; i.e., with the above- mentioned basis one already has the correct linear combinations for the Zee- man effect. In linear order w.r.t. H z we obtain E N,L,S,J,M j = E (0) N,L,S,J − g J (L, S) · μ B ·H z ·M J . (32.5) 32.3 Application II: Atoms in a Magnetic Field; Zeeman Effect 265 Here g J (L, S)istheLand´efactor g J (L, S)= 3J(J +1)−L(L +1)+S(S +1) 2J(J +1) . (32.6) In order to verify (32.6) one often uses the so-called Wigner-Eckart theo- rem, which we only mention. (There is also an elementary “proof” using the vector model for the addition of angular momenta.) 33 Time-dependent Perturbations 33.1 Formalism and Results; Fermi’s “Golden Rules” Now we assume that in Schr¨odinger’s representation the Hamilton operator ˆ H S = ˆ H 0 + ˆ V ω exp(−iωt)+ ˆ V −ω exp(+iωt) (33.1) is already explicitly (monochromatically) time-dependent. Due to the “Her- micity” of the Hamilton operator it is also necessary to postulate that ˆ V −ω ≡ ˆ V + ω . Additionally, it is assumed that the perturbation is “switched-on” at the time t 0 = 0 and that the system was at this time in the (Schr¨odinger) state ψ (0) S i (t):=u (0) i e −iω (0) i t . Here and in the following we use the abbreviations ω (0) i := E (0) i / and ω fi := ω (0) f − ω (0) i . We then expand the function ψ S (t), which develops out of this initial state (i =“initial state”; f =“final state”) in the Schr¨odinger representation, as follows: ψ S (t)= n c n (t) · exp −iω (0) n t ·u (0) n (r) . (33.2) (In the “interaction representation” (label: I), related to Schr¨odinger’s representation by the transformation |ψ I (t) := e i ˆ H 0 t/ |ψ S (t),weobtain instead : ψ I (t)= n c n (t) · u (0) n (r), and also the matrix elements of the perturbation operator simplify to (ψ I ) n (t)| ˆ V I (t)|(ψ I ) i (t) =e i(ω n −ω i −ω)t · u (0) n ˆ V ω |u (0) i + , where + denotes terms, in which ω is replaced by (−ω).) 1 1 For Hermitian operators we can write ψ| ˆ A|φ instead of ψ| ˆ Aφ. . the Quantization of the Angular Momentum The above-mentioned mathematical rules for the quantization of the angular momenta can be visualized by means of the so-called vector model,whichalso serves. the basis of Ritz approximations (e.g., for the ground state), where one tries to optimize a set of variational parameters to obtain a minimum of the r.h.s. of (31.1). As an example we again mention. ground state is always negative. The physical reason is that the perturbation leads to the admixture of excited states ( ˆ= polarization) to the unperturbed ground state. b) In the degenerate case