Basic Theoretical Physics: A Concise Overview P25 pps

246 28 Abstract Quantum Mechanics (Algebraic Methods) Up till now we have used Schr¨odinger’s “picture”. The transitions to the other representations, (ii) the Heisenberg or (iii) the Dirac (= interaction) representations, are described in the following. The essential point is that one uses the unitary transformations corresponding to the so-called time- displacement operators ˆ U(t, t 0 ). In addition we shall use indices S, H and I, which stand for “Schr¨odinger”, “Heisenberg” and “interaction”, respectively. a) In the Schr¨odinger picture the states are time-dependent, but in general not the operators (e.g., position operator, momentum operator,. . . ). We thus have |ψ S (t)≡ ˆ U(t, t 0 )|ψ S (t 0 ) . Here the time-displacement operator ˆ U(t, t 0 ) is uniquely defined according to the equation −  i ∂ ˆ U(t, t 0 ) ∂t = ˆ H S ˆ U(t, t 0 ) (i.e., the Schr¨odinger equation) and converges → ˆ 1fort → t 0 .Heret 0 is fixed, but arbitrary. b) In the Heisenberg picture the state vectors are constant in time, |ψ  ≡|ψ H  := |ψ S (t 0 ) , whereas now the operators depend explicitly on time, even if they do not in the Schr¨odinger picture: i.e., we have in any case: ˆ A H (t):= ˆ U + (t, t 0 ) ˆ A S (t) ˆ U(t, t 0 ) . (28.12) If this formalism is applied to a matrix representation (A H ) j,k (t):=(ψ H ) j | ˆ A H (t)(ψ H ) k  , one obtains, quasi “by the way”, Heisenberg’s matrix mechanics. c) In Dirac’s interaction picture, the Hamilton operator ˆ H(= ˆ H S ) is decomposed into an “unperturbed part” and a “perturbation”: ˆ H S = ˆ H 0 + ˆ V S , where ˆ H 0 does not explicitly depend on time, whereas V S can depend on t. One then introduces as “unperturbed time-displacement operator” the unitary operator ˆ U 0 (t, t 0 ):=e −i ˆ H 0 (t−t 0 )/ 28.3 Unitary Equivalence; Change of Representation 247 and transforms all operators only with U 0 , i.e., with the definition A I (t):=U + 0 (t, t 0 ) ˆ A S (t) ˆ U 0 (t, t 0 ) . (28.13) (A I (t) is thus a more or less trivial modification of A S (t), although the time dependence is generally different.) In contrast, the time-displacement of the state vectors in the interaction picture is more complicated, although it is already determined by (28.13) plus the postulate that the physical quantities, e.g., the expectation values, should be independent of which aspect one uses. In fact, for |ψ I (t) := ˆ U + 0 (t, t 0 )|ψ S (t) a modified Schr¨odinger equation is obtained, given by −  i ∂|ψ I (t) ∂t = ˆ V I (t)|ψ I (t) . (28.14) The unperturbed part of the Hamilton operator, ˆ H 0 , is thus “transformed away” from (28.14) in the interaction picture; but one should note that one has some kind of “conservation of effort” theorem: i.e., in (28.14) one must use ˆ V I (t) instead of ˆ V S . The formal solution of (28.14) is 9 the the following perturbation series (Dyson series): |ψ I (t)=  ˆ 1 − i   t t 0 dt 1 ˆ V I (t 1 ) +  i   2  t t 0 dt 1  t 1 t 0 dt 2 ˆ V I (t 1 ) ˆ V I (t 2 ) ∓  |ψ I (t 0 ) , (28.15) which may be also symbolically abbreviated as  T e − i  R t t 0 dt 1 ˆ V I (t 1 )  |ψ I (t 0 ) ; the symbol T is called Dyson’s time-ordering operator, because in (28.15) t ≥ t 1 ≥ t 2 ≥ . Apart from the spin, which is a purely quantum mechanical phenomenon, there is a close correspondence between classical and quantum mechanical observables. For example, the classical Hamilton function and the quantum mechanical Hamilton operator usually correspond to each other by the above- mentioned simple replacements. The correspondence is further quantified by the Ehrenfest theorem, which states that the expectation values (!) of the 9 This can be easily shown by differentiation. 248 28 Abstract Quantum Mechanics (Algebraic Methods) observables exactly satisfy the canonical equations of motion of the classical Hamilton formalism. The theorem is most simply proved using Heisenberg’s representation; i.e., firstly we have dˆx H dt = i   ˆ H, ˆx H  and dˆp H dt = i   ˆ H, ˆp H  . Evaluating the commutator brackets, with ˆ H = ˆp 2 2m + V (ˆx) , one then obtains the canonical equations dˆx H dt = ∂ ˆ H ∂ ˆp H and dˆp H dt = − ∂ ˆ H ∂ˆx H . (28.16) 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem) 29.1 Spin Momentum; the Hamilton Operator with Spin-orbit Interaction The Stern-Gerlach experiment (not described here) provided evidence for the half-integral spin of the electron 1 . As a consequence the wavefunction of the electron was assigned an additional quantum number m s , ψ ≡ ψ(r,m s ) , i.e., it is not only described by a position vector r but also by the binary variable m s = ± 1 2 , corresponding to the eigenvalues m s  of the z-component ˆ S z of a spin angular momentum. This was suggested by Wolfgang Pauli. The eigenfunctions of the hydrogen atom are thus given by the following expression, for vanishing magnetic field and neglected spin-orbit interaction (see below), with s ≡ 1 2 : u n,l,s,m l ,m s (r  ,m  s )=R nl (r  ) · Y l,m l (θ  ,ϕ  ) · χ s,m s (m  s ) . (29.1) Here the two orthogonal spin functions are a) χ 1 2 , 1 2 (m  s )=δ m  s ,+ 1 2 , which is identical to the above-mentioned two-spinor α :=↑=  1 0  , b) χ 1 2 ,− 1 2 (m  s )=δ m  s ,− 1 2 , which is identical with β :=↓=  0 1  . 1 Apart from the Stern-Gerlach experiment also the earlier Einstein-de Haas ex- periment appeared in a new light. 250 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem) Corresponding to the five “good” quantum numbers n, l, s, m l and m s there are the following mutually commuting observables: ˆ H  ≡ ˆ p 2 2m − Z e 2 4πε 0 r  , ˆ L 2 , ˆ S 2 , ˆ L z and ˆ S z . In contrast, in a constant magnetic induction B = e z B z = μ 0 H = μ 0 e z H z without the spin-orbit interaction (see below) one obtains ˆ H≡ ( ˆ p − eA) 2 2m − Z e 2 4πε 0 r − gμ B H · ˆ S  (the last expression is the so-called “Pauli term”, which is now considered). Here we have without restriction of generality: A = 1 2 [B ×r]= B z 2 (−y, x, 0) . For electrons in vacuo the factor g is found experimentally and theo- retically to be almost exactly 2 (more precisely: 2.0023. . . . Concerning the number 2, this value results from the relativistic quantum theory of Dirac; concerning the correction, 0.0023. . . , it results from quantum electrodynam- ics, which are both not treated in this volume.) The quantity μ B = μ 0 e 2m is the Bohr magneton (an elementary magnetic moment); μ 0 =4π · 10 −7 Vs/(Am) is the vacuum permeability (see Part II). The spin-orbit interaction (see be- low) has so far been neglected. Now, including in ˆ H all relevant terms (e.g., the kinetic energy, the Pauli term etc.) in a systematic expansion and adding the spin-orbit interaction as well, one obtains the following expression: ˆ H = ˆ p 2 2m − Z e 2 4πε 0 r − k(r) ˆ L  · ˆ S  − μ B H ·  g l ˆ L + g s ˆ S   + μ 2 0 e 2 8m H 2 ·  x 2 + y 2  . (29.2) Here the last term, ∝ H 2 , describes the diamagnetism (this term is mostly negligible). In contrast, the penultimate term, ∝ H, describes the Zeeman 29.2 Rotation of Wave Functions with Spin; Pauli’s Exclusion Principle 251 effect, i.e., the influence of an external magnetic field, and the temperature- dependent paramagnetism, which has both orbital and spin contributions. Generally this term is important, and through the spin g-factor g s =2one recognizes at once the anomalous behavior of the spin momentum ˆ S com- pared with the orbital angular momentum ˆ L, where the orbital g-factor g l is trivial: g l =1. The third-from-last term is the spin-orbit interaction (representing the so-called “fine structure” of the atomic spectra, see below): In fact, an electron orbiting in an electrostatic field E partially experiences this electric field, in the co-moving system, as a magnetic field H  ,cf.Part II: H  = − v ×E μ 0 c 2 . With E = Z|e| 4πε 0 r 3 r one obtains a correction to the Hamilton operator of the form −k(r) ˆ L  · ˆ S  , which has already been taken into account in (29.2). Here −k(r) is positive and ∝ 1/r 3 ; the exact value follows again from the relativistic Dirac theory. In the following sections we shall now assume that we are dealing with an atom with many electrons, where L and S (note: capital letters!) are the quantum numbers of the total orbital angular momentum and total spin mo- mentum of the electrons of this atom (Russel-Saunders coupling, in contrast to j −j coupling 2 ). Furthermore, the azimuthal quantum numbers m l and m s are also replaced by symbols with capital letters, M L and M S . The Hamilton operator for an electron in the outer electronic shell of this atom is then of the previous form, but with capital-letter symbols and (due to screening of the nucleus by the inner electrons) with Z → 1. 29.2 Rotation of Wave Functions with Spin; Pauli’s Exclusion Principle; Bosons and Fermions Unusual behavior of electronic wave functions, which (as explained above) are half-integral spinor functions, shows up in the behavior with respect to spatial rotations. Firstly, we note that for a usual function f (ϕ) a rotation about the z-axis by an angle α can be described by the operation  ˆ D α f  (ϕ):=f(ϕ − α) . 2 Here the total spins j of the single electrons are coupled to the total spin J of the atom. 252 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem) As a consequence one can simply define the unitary rotation operator ˆ D α by the identity ˆ D α := e −iα ˆ L z / ; i.e., with ˆ L z =  i ∂ ∂ϕ one obtains ˆ D α f(ϕ)=e −α ∂ ∂ϕ f(ϕ) , which is the Taylor series for f(ϕ − α). As a consequence it is natural to define the rotation of a spinor of degree J about the z-axis, say, by e −iα ˆ J z / |J, M J  . The result, e −iαM J |J, M J  , shows that spinors with half-integral J (e.g., the electronic wave functions, if the spin is taken into account) are not reproduced (i.e., multiplied by 1) after a rotation by an angle α =2π, but only change sign, because e −2πi/2 ≡−1 . Only a rotation by α =4π leads to reproduction of the electronic wave function. As shown below, Pauli’s exclusion principle is an immediate consequence of this rotation behavior. The principle describes the permutational behavior of a wave function for N identical particles (which can be either elementary or compound particles). Such particles can have half-integral spin (as elec- trons, or He 3 atoms, which have a nucleus composed of one neutron and two protons, plus a shell of two electrons). In this case the particles are called fermions. In contrast, if the particles have integral spin (as e.g., pions, and He 4 atoms) they are called bosons. In the position representation one then obtains the N-particlewavefunc- tion: ψ(1, 2, ,N), where the variables i =1, ,N are quadruplets i =(r  i , (M  S ) i ) of position and spin variables. In addition to the square integrability  d1   dN|ψ(1, 2, ,N)| 2 ! =1, with  d1 :=  M 1  d 3 r 1 , one postulates for the Pauli principle the following permutational behavior for the exchange of two particles i and j (i.e., for the simultaneous interchange of the quadruplets describing position and spin of the two particles): 29.2 Rotation of Wave Functions with Spin; Pauli’s Exclusion Principle 253 (Pauli’s relation) ψ( ,j,i, ) ! =(−1) 2S ψ( ,i,j, ) , i.e., with (−1) 2S = − 1forfermions and +1 for bosons . (29.3) An immediate consequence is Pauli’s exclusion principle for electrons: elec- trons have S = 1 2 ; thus the wavefunctions must be antisymmetric. However, an anti-symmetric product function (u i 1 (1) · u i 2 (2) · ·u i N (N)) asy vanishes identically as soon as two of the single-particle states |u i ν (ν =1, 2, ,N) are identical (see also below). In other words, in an N-electron quantum mechanical system, no two electrons can be the same single electron state. Thus, both the rotational behavior of the states and the permutational behavior of the wave function for N identical particles depend essentially on S  = S   , or more precisely on (−1) 2S , i.e., on whether S is integral (bosons) or half- integral (fermions) (the so-called “spin-statistics theorem”). In fact, this is a natural relation, since the permutation of particles i and j can be obtained by a correlated rotation by 180 ◦ , by which, e.g., particle i moves along the upper segment of a circle from r i to r j , while particle j moves on the lower segment of the circle, from r j to r i . As a result, a rotation by 2π is effected, which yields the above-mentioned sign. Admissable states of a system of identical fermions are thus linear combi- nations of the already mentioned anti-symmetrized product functions, i.e., linear combinations 3 of so-called Slater determinants of orthonormalized single-particle functions. These Slater determinants have the form ψ Slater u 1 ,u 2 , ,u N (1, 2, ,N) := 1 √ N!  P 0 @ i 1 i N 1 N 1 A (−1) P u 1 (i 1 ) · u 2 (i 2 ) · ·u N (i N ) (29.4) ≡ 1 √ N!         u 1 (1) ,u 2 (1) , , u N (1) u 1 (2) ,u 2 (2) , , u N (2) , , , u 1 (N) ,u 2 (N) , ,u N (N)         . (29.5) 3 Theoretical chemists call this combination phenomenon “configuration interac- tion”, where a single Slater determinant corresponds to a fixed configuration. 254 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem) Every single-particle state is “allowed” at most once in such a Slater deter- minant, since otherwise the determinant would be zero (→ Pauli’s exclusion principle, see above). This principle has enormous consequences not only in atomic, molecular, and condensed matter physics, but also in biology and chemistry (where it determines inter alia Mendeleev’s periodic table), and through this the way the universe is constructed. A more technical remark: if one tries to approximate the ground-state en- ergy of the system optimally by means of a single anti-symmetrized product function (Slater determinant), instead of a sum of different Slater determi- nants, one obtains the so-called Hartree-Fock approximation 4 . Remarkably, sodium atoms, even though they consist exclusively of fermi- ons, and both the total spin momentum of the electronic shell and the total spin of the nucleus are half-integral, usually behave as bosons at extremely low temperatures, i.e., for T less than ∼ 10 −7 Kelvin, which corresponds to the ultraweak “hyperfine” coupling of the nuclear and electronic spins. However in the following sections, the physics of the electronic shell will again be at the center of our interest although we shall return to the afore-mentioned point in Part IV. 4 This is the simplest way to characterize the Hartree-Fock approximation in a nut- shell. 30 Spin-orbit Interaction; Addition of Angular Momenta 30.1 Composition Rules for Angular Momenta Even if the external magnetic field is zero, M L and M S are no longer good quantum numbers, if the spin-orbit interaction is taken into account. The reason is that ˆ L z and ˆ S z no longer commute with ˆ H. However, ˆ J 2 , the square of the total angular momentum ˆ J := ˆ L + ˆ S , and the z-component ˆ J z := ˆ L z + ˆ S z , do commute with each other and with ˆ H,andalsowith ˆ L 2 and ˆ S 2 : the five operators ˆ H, ˆ L 2 , ˆ S 2 , ˆ J 2 and ˆ J z (see below) all commute with each other. In fact we have 2 ˆ L · ˆ S  ∝ δ ˆ H  ≡ ˆ J 2 − ˆ L 2 − ˆ S 2 . Thus, by including the spin-orbit interaction, a complete system of mu- tually commuting operators, including ˆ H, in a shell model, now consists of ˆ H, ˆ L 2 , ˆ S 2 , ˆ J 2 and ˆ J z . The corresponding “good quantum numbers” are: N, L, S, J and M J (no longer N, L, S, M L and M S ). In the following, the radial functions R N,L can be omitted. Then, by linear combination of the Y L,M L (θ  ,ϕ  ) · χ S,M S (M  S ) (see below) one must find abstract states Y L,S,J,M J (θ  ,ϕ  ,M  S ): Y L,S,J,M J (θ  ,ϕ  ,M  S ):=  M L ,M S c (M L ,M S ) L,S,J,M J · Y L,M L (θ  ,ϕ  ) χ S,M S (M  S ) , (30.1) which are eigenstates of ˆ J 2 and ˆ J z with eigenvalues  2 J(J +1)andM J . 1 1 Using a consistent formulation a related abstract state |ψ L,S,J,M J  is defined by Y L,S,J,M J (θ  ,ϕ  ,M  S )=θ  ,ϕ  ,M  J |ψ L,S,J,M J  := r.h.s. of (30.1) ,whichis similar to defining an abstract state function |ψ by the function values ψ(r):= r|ψ. . which is a purely quantum mechanical phenomenon, there is a close correspondence between classical and quantum mechanical observables. For example, the classical Hamilton function and the quantum mechanical. that the expectation values (!) of the 9 This can be easily shown by differentiation. 248 28 Abstract Quantum Mechanics (Algebraic Methods) observables exactly satisfy the canonical equations of motion. permutational behavior of a wave function for N identical particles (which can be either elementary or compound particles). Such particles can have half-integral spin (as elec- trons, or He 3 atoms,