34 Magnetism: An Essentially QuantumMechanical Phenomenon 34.1 Heitler and London’s Theory of the H2-Molecule; Singlet and Triplet States; the Heisenberg Model This chapter also serves a
Trang 1268 33 Time-dependent Perturbations
For f = i the Schr¨odinger equation yields the following result:
c f (t) = −1V f i(0)·ei(ω f i −ω)t − 1
Here only the linear terms in ˆV have been considered and the non-resonant
terms
∝ ˆ V ω+e+iωt , i.e., with ω → (−ω) , have also been neglected; V f i(0) stands for
1
u(0)f ˆV
ωu(0)
i
2
.
By squaring the above result one obtains
|c f (t) |2
= 1
2
V(0)
f i 2
·sin
2 (ω f i −ω)t
2
ω f i −ω2
2
This corresponds to a periodic increase, followed by a decrease, with the Poincar´e repetition time
Δt = 2π/ |ω f i − 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 ωe−iωt →
n
α=1
ˆ
V ω αei(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 n2-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
2π
2
V(0)
f i 2
γ (ω f i ) (33.5)
Trang 233.2 Selection Rules 269
In the above proof we have used the identity
lim
t →∞
⎛
⎜
⎝sin
2 (ω f i −ω)·t
2
ω
f i −ω2
2
2
· t
⎞
⎟
⎠ ≡ 2πδ(ω f i − 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
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 f i |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
the final states:
W i →K=2π V(0)
f i
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 l2=1,m l ≡0, and an
isotropic initial state, i.e., without angular dependence, i.e.,
|i ∝ Y l ≡0,m ≡0 .
Trang 3270 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 momentap iandp f :=p i+q
The wiggly line labelled ω represents the radiation quantum of frequency ω
with momentumq, which “pumps the
system” from the initial state to the
fi-nal state Energy conservation, E f =
E i+ · ω, is always obeyed
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
Trang 434 Magnetism: An Essentially Quantum
Mechanical Phenomenon
34.1 Heitler and London’s Theory of the H2-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 = ˆp
2 1
2m+
ˆ
p2 2
2m+
e2
4πε0 ·
− 1
r 1A − 1
r 2B
+
$ 1
r12
R AB − 1
r 1B − 1
r 2A
%
. (34.1)
Here r12is 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 r1 andr2 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+(r1, r2)
(symmetric w.r.t permutations of r1 andr2) 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 −(r1, r2)· χ+(1, 2) (34.2)
Trang 5272 34 Magnetism: An Essentially Quantum Mechanical Phenomenon
and singlet products
ψ si. := Φ+(r1, r2)· 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 = 0 := √1
2(α(1)β(2) + β(1)α(2)) =
1
√
2(↑↓ + ↓↑)
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
S1+ ˆS2
2
|S, M = 2S(S + 1) |S, M and
ˆ
S z
1+
ˆ
S z
2
|S, M = M|S, M , with S = 1 for the triplet states and S = 0 for the singlet 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 ±(r1, r2) := u A(r1)u B(r2)± u B(r1)u A(r2)
2(1± |S A,B |2) , (34.6) then, with
a) the so-called overlap integral
S A,B:=
d3r1(u A(r1))∗ u
B(r1) , (34.7)
b) the Coulomb integral
C A,B:= e
2
4πε0
d3r1d3r2
u ∗
A(r1)u ∗
B(r2)u B(r2)u A(r1)
r12
, (34.8)
and
Trang 634.1 Heitler and London’s Theory of the H2-Molecule 273
c) the exchange integral
J A,B:= e
2
4πε0
d3r1d3r2u
∗
A(r1)u ∗
B(r2)u A(r2)u B(r1)
r12
(34.9)
apart from minor corrections we obtain the following result:
Etriplet= constant4 +C A,B − J A,B
1− |S A,B |2 ,
Esinglet= constant4 +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=24 · C A,B |S A,B |2− J A,B
This is a positive quantity, since C A,Bis as large asO(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 Esinglet(R AB ) and Etriplet(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
Heff
ˆ
S1, ˆ S2
(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 (ˆp1, ˆ p2, r1, r2) by Hˆeff
ˆ
S1, ˆ S2
is admittedly quite subtle As a help towards understanding, see the following sketch (Fig 34.1)
Trang 7274 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 (spherichemi-cal symmetry) favors symmetrichemi-cal 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
Heff
ˆ
S A , ˆ S B
=−2 · Jeff
A,B SˆA · ˆ S B (34.12) Applied to the spin functions |S, M it yields the result
−Jeff
A,B ·
$
S(S + 1) −3
2
%
|J, M ,
since
2· ˆ S A · ˆ S B ≡SˆA+ ˆS B
2
− ˆ S A
2
− ˆ S B
2
.
With (34.11) we have:
J A,Beff =J A,B − C A,B · |S A,B |2
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 > 12.1
In particular we have seen above that the effective exchange couplings
Jeff
lm entering this model result from the interplay of Coulomb interaction and Pauli principle Actually the two entities Jeff
A,B and J A,B can have different sign For example, if J lmeff ≤ 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
Trang 834.2 Hund’s Rule Why is the O2-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 Jeff
lm ≥ 0 one obtains paramagnetic molecules
(e.g., the oxygen molecule, which is rather an exceptional case) and
nonmag-netic or ferromagnonmag-netic 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).
Paramagnetic?
The majority of diatomic molecules are diamagnetic, in accordance with
the theory of Heitler and London for the H2-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(r1)u A(r2)χ − (1, 2)
However some diatomic molecules, e.g., O2, 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 M n2+, 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 Coulomb2integral, which involves one and the same orbital:
E (i) = U AA= ( e2/(4πε0))·
d3r1d3r2|u A(r1)|2· |u A(r2)|2/r12
2 These intra-orbital Coulomb integrals are often called Hubbard integrals or screened Hartree integrals.
Trang 9276 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-negative3 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 Jeff < 0 in the preceding section.
Trang 1035 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 He4 and He3 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( |r1− r2|), 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, He4-gas becomes a liquid at 4.2 K; on further decrease of the temperature the normal fluid becomes superfluid at 2.17 K.
1 A reminder: the characteristic atomic length is the Bohr radius a0 = 0.529 ˚A
(= 0.0529 nm).