268 33 Time-dependent Perturbations For f = i the Schr¨odinger equation yields the following result: c f (t)=− 1 V (0) fi · e i(ω fi −ω)t − 1 ω fi − ω . (33.3) Here only the linear terms in ˆ V have been considered and the non-resonant terms ∝ ˆ V + ω e +iωt , i.e., with ω → (−ω) , have also been neglected; V (0) fi stands for u (0) f ˆ V ω u (0) i . By squaring the above result one obtains |c f (t)| 2 = 1 2 V (0) fi 2 · sin 2 (ω fi −ω)t 2 ω fi −ω2 2 2 . (33.4) This corresponds to a periodic increase, followed by a decrease, with the Poincar´e repetition time Δt =2π/|ω fi − ω| , which is extremely long near a resonance of the denominator. Thus, with a source of radiation consisting of n uncorrelated “radiators” of (almost) the same frequency ω α ≈ ω, e.g., ˆ V ω e −iωt → n α=1 ˆ V ω α e i(r(α)−ω α t) with random phases r(α), one obtains the n-fold result of (33.4) (if the fre- quencies are identical). In contrast, if the radiation were coherent (e.g., laser radiation), one would obtain the n 2 -fold result. However in that case it makes no sense to interrupt the time-dependent perturbation series, as we did, after the lowest order. In fact, at this point the transition from coherent and reversible quantum mechanics to incoherent and irreversible behavior occurs, as in statistical physics (→ Part IV). Thus, if one has a continuum of sources of incoherent radiation, i.e., with ω α → dω α · γ (ω α ) , then one obtains as transition rate W i→f (≡ transition probability i → f divided by the time t): W i→f := lim t→∞ |c f (t)| 2 t = 2π 2 V (0) fi 2 · γ (ω fi ) . (33.5) 33.2 Selection Rules 269 Intheaboveproofwehaveusedtheidentity lim t→∞ ⎛ ⎜ ⎝ sin 2 (ω fi −ω)·t 2 ω fi −ω2 2 2 ·t ⎞ ⎟ ⎠ ≡ 2πδ(ω fi − ω) . The matrix elements have been incoherently averaged, as expressed by the ‘bar’ in (33.5). Equation (33.5) describes transitions from a discrete energetically lower level i to an energetically higher level f by induced absorption of continuous radiation of frequency ω with an incoherent density γ (ω). See also Fig. 33.1 below. Conservation of energy, ω = ω f −ω i , is explicitly given by the δ-function in the above formal correspondence. By permutation of f and i and the simultaneous replacement ω → (−ω) one obtains almost the same (33.5) for the induced emission of radiation. But there is also a spontaneous emission of radiation, which has an emission rate ∝|ω fi | 3 . This fact makes it hard (since spontaneous emission should be avoided) to obtain the necessary occupation of a high-energy level for X-ray lasers. A formula similar to (33.5) is also obtained for incoherent transitions from a discrete state i into a continuum K with the (continuous) density f (E)of the final states: W i→K = 2π V (0) fi 2 f (E i + ω) . (33.6) Such formulae are called Fermi’s “golden rules”. An induced absorption process is illustrated in Fig. 33.1 by means of a so- called Feynman diagram. The corresponding induced emission process would instead have an outgoing wiggly line to the right. Concerning translation in- variance (which does not apply to defective or amorphous solids) the related momentum conservation provides an example for the presence (and conse- quences) of selection rules (see below). 33.2 Selection Rules Selection rules arise naturally from Fermi’s “golden rules”. The “selection” refers to the (squared) matrix elements appearing in the “golden rules”, and refers essentially to their predicted vanishing or nonvanishing due to charac- teristic symmetry arguments. To give a simple but typical example we consider a perturbation with so-called σ symmetry, i.e., ˆ V ∝ z,thus∝ cos ϑ, i.e., ∝ Y l 2 =1,m l ≡0 ,andan isotropic initial state, i.e., without angular dependence, i.e., |i∝Y l i ≡0,m l ≡0 . 270 33 Time-dependent Perturbations Fig. 33.1. Feynman diagram for an induced absorption process. The solid lines with symbols i and f correspond to an initial state |ψ i and a final state |ψ f , e.g., to an atom with energy lev- els E i and E f (>E i ) and to particles or quasi-particles propagating from the left with momenta p i and p f := p i +q. The wiggly line labelled ω represents the radiation quantum of frequency ω with momentum q, which “pumps the system” from the initial state to the fi- nal state. Energy conservation, E f = E i + · ω,isalwaysobeyed As a consequence the matrix element f|cos ϑ|i is then only nonvanishing, if |f∝Y l f ≡1,m l ≡0 , i.e., for l f − l i ≡ 1 ,m f ≡ m i (= 0) . Further selection rules for other cases, e.g., for perturbations with π ± - symmetry, i.e., ˆ V ∝ (x ±iy) , or for multipole radiation beyond the dipole case, are obtained analogously, corresponding to the addition rules for angular momenta. 34 Magnetism: An Essentially Quantum Mechanical Phenomenon 34.1 Heitler and London’s Theory of the H 2 -Molecule; Singlet and Triplet States; the Heisenberg Model This chapter also serves as preparation for the subsequent section on the interpretation of quantum mechanics. Firstly we shall treat the hydrogen molecule according to the model of Heitler and London, which is a most- important example. The Hamilton operator for the two electrons is ˆ H = ˆ p 2 1 2m + ˆ p 2 2 2m + e 2 4πε 0 · − 1 r 1A − 1 r 2B + 1 r 12 + 1 R AB − 1 r 1B − 1 r 2A . (34.1) Here r 12 is the distance between the two electrons; R AB is the separation of the two nuclei, which can be assumed to be at fixed positions, because the mass of the nuclei is ≈ 2000 times larger than that of the electrons; and r 1B is the distance of the first electron from nucleus B, etc The sum in braces, i.e., the last four terms on the r.h.s., can be considered as a perturbation of the first four terms. Since the Hamilton operator does not depend on the spin and is per- mutationly symmetrical w.r.t. 1 and 2, the eigenfunctions can be written as products of position functions and spin functions, and they must have a well- defined parity w.r.t. permutations of the position variables r 1 and r 2 . Furthermore, due to Pauli’s exclusion principle, both the position and spin functions must have complementary permutation behavior, i.e., a symmetric position function Φ + (r 1 , r 2 ) (symmetric w.r.t. permutations of r 1 and r 2 ) must be multiplied by an an- tisymmetric spin function χ − (1, 2), and vice versa. This leads to so-called triplet products ψ tr. := Φ − (r 1 , r 2 ) ·χ + (1, 2) (34.2) 272 34 Magnetism: An Essentially Quantum Mechanical Phenomenon and singlet products ψ si. := Φ + (r 1 , r 2 ) ·χ − (1, 2) . (34.3) The names singlet and triplet are self-explanatory, i.e., there are three or- thonormalized triplet spin functions χ + (1, 2): |S =1,M =+1 := α(1)α(2) =↑↑ |S =1,M =0 := 1 √ 2 (α(1)β(2) + β(1)α(2)) = 1 √ 2 (↑↓ + ↓↑) |S =1,M = −1 := β(1)β(2) =↓↓ , (34.4) but only one orthonormal singlet spin function χ − (1, 2): |S =0,M =0 := 1 √ 2 (α(1)β(2) −α(2)β(1)) = 1 √ 2 (↑↓ − ↓↑) . (34.5) The functions are simultaneously eigenfunctions of the relevant operators for the total angular momentum, i.e. ˆ S 1 + ˆ S 2 2 |S, M = 2 S(S +1)|S, M and ˆ S z 1 + ˆ S z 2 |S, M = M|S, M, with S =1forthetriplet states and S =0forthesinglet state (→ ex- ercise). (What has been said in Section 30.1 about the addition of angular momenta and the so-called Clebsch-Gordan coefficients can be most easily demonstrated with this problem.) Approximating the Φ ± by ground state functions of the hydrogen atom: Φ ± (r 1 , r 2 ):= u A (r 1 )u B (r 2 ) ±u B (r 1 )u A (r 2 ) 2(1 ±|S A,B | 2 ) , (34.6) then, with a) the so-called overlap integral S A,B := d 3 r 1 (u A (r 1 )) ∗ u B (r 1 ) , (34.7) b) the Coulomb integral C A,B := e 2 4πε 0 d 3 r 1 d 3 r 2 u ∗ A (r 1 )u ∗ B (r 2 )u B (r 2 )u A (r 1 ) r 12 , (34.8) and 34.1 Heitler and London’s Theory of the H 2 -Molecule 273 c) the exchange integral J A,B := e 2 4πε 0 d 3 r 1 d 3 r 2 u ∗ A (r 1 )u ∗ B (r 2 )u A (r 2 )u B (r 1 ) r 12 (34.9) apart from minor corrections we obtain the following result: E triplet = constant + C A,B − J A,B 1 −|S A,B | 2 , E singlet = constant + C A,B + J A,B 1+|S A,B | 2 . (34.10) Here not only the numerator but also the denominator is important; viz,sur- prisingly the triplet product is energetically higher(!) than the singlet product by an amount given by ΔE =2 · C A,B |S A,B | 2 − J A,B 1 −|S A,B | 4 . (34.11) This is a positive quantity, since C A,B is as large as O(10) eV, while J A,B is only O(1) eV, such that because of the rather large value of |S A,B | 2 the energy difference ΔE is unexpectedly > 0. For the ground state of the hy- drogen molecule, the quantity ΔE is in fact of the order of magnitude of O(10) eV. (The actual functions E singlet (R AB ) and E triplet (R AB ) for the hy- drogen molecule can be found in all relevant textbooks and should be sketched; approximate values for the equilibrium separation and the dissociation energy of the hydrogen molecule can be ascertained from textbooks of experimental molecular physics.) Because of the complementary permutation behavior of position and spin functions these results, which are due to the interplay of the Coulomb in- teraction and the Pauli principle, can also be obtained by an equivalent spin operator ˆ H eff ˆ S 1 , ˆ S 2 (see below, (34.13)), as introduced by Dirac. This spin operator, which acts on the spin factor attached to the position function, is just an equivalent descrip- tion, replacing the genuine effects of Coulomb interaction plus Pauli principle, and contained in the ansatz for the two-electron function by complementary products, (34.2) and (34.3), together with the two-electron Hamilton operator H of (34.1). The above replacement of H( ˆ p 1 , ˆ p 2 , r 1 , r 2 )by ˆ H eff ˆ S 1 , ˆ S 2 is admittedly quite subtle. As a help towards understanding, see the following sketch (Fig. 34.1). 274 34 Magnetism: An Essentially Quantum Mechanical Phenomenon Fig. 34.1. A schematic aid to understanding the Heitler-London theory. The chemi- cal bonding between s-states (spherical symmetry) favors symmetrical permutation behavior of the position variables (whereas the Coulomb repulsion acts against it) with a strong overlap between the nuclei (sketched as triangles). The spin function, which (in its permutational behavior) is complementary to the position variables, must be antisymmetric, i.e., a singlet. Thus one has diamagnetism as the usual state of diatomic molecules. Paramagnetism (or ferromagnetism in a solid) is only obtained if the overlap is strongly reduced (e.g., by non-s-symmetry, by Coulomb repulsion, or by “merging” of the nuclei; see the next section) Apart from a constant, the Dirac spin operator yields the same energy spectrum as before. The operator is ˆ H eff ˆ S A , ˆ S B = −2 · J eff A,B ˆ S A · ˆ S B . (34.12) Applied to the spin functions |S, M it yields the result −J eff A,B · S(S +1)− 3 2 |J, M , since 2 · ˆ S A · ˆ S B ≡ ˆ S A + ˆ S B 2 − ˆ S A 2 − ˆ S B 2 . With (34.11) we have: J eff A,B = J A,B − C A,B ·|S A,B | 2 1 −|S A,B | 4 . (34.13) The natural generalization of equation (34.12) to systems of many atoms, i.e., a sum over A and B of terms similar to (34.12), is called the Heisenberg model of magnetism. One can show that it also applies to systems with S> 1 2 . 1 In particular we have seen above that the effective exchange couplings J eff lm entering this model result from the interplay of Coulomb interaction and Pauli principle. Actually the two entities J eff A,B and J A,B can have different sign. For example, if J eff lm ≤ 0 (although J A,B , as can be shown (e.g., [24], 1 To obtain further insight one should see the following section on Hund’s rule, which shows that within an atom there can be a strong ferromagnetic coupling between certain intra-atomic spins. 34.2 Hund’s Rule. Why is the O 2 -Molecule Paramagnetic? 275 p. 392), is always ≥ 0), one obtains diamagnetic molecules (this is the most common case, e.g., the nitrogen molecule) and nonmagnetic or antiferromag- netic solids. In contrast, for J eff lm ≥ 0oneobtainsparamagnetic molecules (e.g., the oxygen molecule, which is rather an exceptional case) and nonmag- netic or ferromagnetic solids. At this point quantum mechanics comes fully into play in all details and leads to important phenomena (magnetism, superconductivity,. . . ), which cannot be explained by classical physics. For example, Bohr and van Leeuwen proved many years before the discovery of quantum mechanics that magnetism cannot be explained solely by orbital angular moments (e.g., by Amp`ere’s cur- rent loops; see Part II). 34.2 Hund’s Rule. Why is the O 2 -Molecule Paramagnetic? The majority of diatomic molecules are diamagnetic, in accordance with the theory of Heitler and London for the H 2 -molecule, since in most cases the outer electrons of a diatomic molecule only have a single orbital at their disposal. According to the Pauli principle this orbital can at most be occupied by two electrons with opposite spins, e.g., ψ = u A (r 1 )u A (r 2 )χ − (1, 2) . However some diatomic molecules, e.g., O 2 , turn out to be paramagnetic, which does not comply with the Heitler and London theory. If two or more orthogonal, energetically degenerate orbitals can be occupied, e.g., two π ± - molecular orbitals in the case of the oxygen molecule or two or more of the five 3d-orbitals in the case of a 3d-ion as Mn 2+ , then the two electrons can choose between the following three possibilities. (i) Both electrons occupy the same orbital (due to the Pauli prin- ciple this is only possible with opposite spins); or (ii) and (iii) they occupy different orbitals. This is possible either in a singlet state (case (ii)) or in a triplet state (case (iii)). Hund’s rule for the dominance of the configuration with maximum possible multiplicity states that of these possibilities case (iii) is favored: Case (i) is excluded on energy grounds because of the large Coulomb re- pulsion of the electrons; this is roughly characterized by a Coulomb 2 integral, which involves one and the same orbital: E (i) = U AA =(e 2 /(4πε 0 )) · d 3 r 1 d 3 r 2 |u A (r 1 )| 2 ·|u A (r 2 )| 2 /r 12 2 These intra-orbital Coulomb integrals are often called Hubbard integrals or screened Hartree integrals. 276 34 Magnetism: An Essentially Quantum Mechanical Phenomenon (note A instead of B as the second index!). In contrast we have E (ii),(iii) = C A,B ± J A,B , where in general U AA is significantly larger than C A,B , which (on the other hand) is five to ten times larger than J A,B . Case (iii), i.e., the triplet state, is thus favored over case (i) and case (ii); the latter, because the overlap integral S A,B vanishes in the present case, S A,B = u A |u B ≡0, such that here the energy difference between the triplet state (iii) and the singlet state (ii) is given by the direct (here called Hund’s rule) exchange integral J A,B ; this integral (as already stated) is always non-negative 3 .Furthermore,as an intra-atomic integral the Hund’s rule exchange interaction may be signifi- cantly stronger than the inter-atomic exchange integrals appearing in Heitler- London molecules. 3 For the oxygen molecule the “Hund’s rule exchange” is J A,B ≈ +0.1 eV(i.e. > 0), as opposed to J eff A,B < 0 in the preceding section. 35 Cooper Pairs; Superconductors and Superfluids The so-called super effects (“superconductivity” and “superfluidity”, and re- cently (2004) also “supersolids”) are typical quantum mechanical phenomena. The following will be now discussed here: (i) conventional superconductivity, (ii) the superfluidity of He 4 and He 3 and (iii) so-called high-temperature superconductivity. i) Conventional metallic superconductors, e.g., Pb, have critical tempera- tures T s below roughly 10 to 20 K; for T<T s the electric current (a) flows without energy losses, and (b) (below a critical value) the magnetic induction is completely expelled from the interior of the sample by the action of supercurrents flowing at the boundary (the so-called Meissner effect, see Part II). It was discovered (not all that long ago) that the characteristic charge for these effects is not = e, but = 2e, i.e., twice the elementary charge. In fact, for the temperature range considered, in the electron liquid so- called “Cooper pairs” form, i.e., pairs of electrons which “surround” each other – metaphorically speaking – at a large radius R between ≈ 50 and ≈ 1000 ˚ A (= 100 nm). 1 The wave function of such a Cooper pair thus (i) consists of a (spheri- cally) symmetrical position factor Φ(|r 1 − r 2 |), typically an s-function; as a consequence, the remaining spin factor, (ii), must be antisymmetric. The result is so-called singlet pairing. Viewed as compound particles these Cooper pairs thus behave roughly as bosons; thus they can condense into a collective state, the so-called pair condensate, which moves without resistance through the host system. En- ergetically the electrons use – in a highly cooperative manner – a small interaction effect, which results from the fact that the host system can be slightly deformed (so-called electron-phonon interaction). This has been known since 1957 (i.e., approximately half a century after the experimen- tal discovery(!)); 1957 was the year of the formulation of the so-called BCS theory, named after Bardeen, Cooper and Schrieffer, [25].) ii) Under normal pressure, He 4 -gas becomes a liquid at 4.2K;onfurther decrease of the temperature the normal fluid becomes superfluid at 2.17 K. 1 A reminder: the characteristic atomic length is the Bohr radius a 0 =0.529 ˚ A (= 0.0529 nm). . its permutational behavior) is complementary to the position variables, must be antisymmetric, i.e., a singlet. Thus one has diamagnetism as the usual state of diatomic molecules. Paramagnetism. which is rather an exceptional case) and nonmag- netic or ferromagnetic solids. At this point quantum mechanics comes fully into play in all details and leads to important phenomena (magnetism,. singlet state (ii) is given by the direct (here called Hund’s rule) exchange integral J A, B ; this integral (as already stated) is always non-negative 3 .Furthermore,as an intra-atomic integral the