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 pro
Trang 1246 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, t0) 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, t0)|ψ S (t0)
Here the time-displacement operator ˆU (t, t0) is uniquely defined according
to the equation
−
i
∂ ˆ U (t, t0)
∂t = ˆH S U (t, tˆ 0) (i.e., the Schr¨odinger equation) and converges→ ˆ1 for t → t0 Here t0 is fixed, but arbitrary
b) In the Heisenberg picture the state vectors are constant in time,
|ψ ≡ |ψ H := |ψ S (t0) , 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, t0) ˆA S (t) ˆ U (t, t0) (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 = ˆH0+ ˆV S ,
where ˆH0 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 (t, t ) := e−i ˆ H0(t −t0)/
Trang 228.3 Unitary Equivalence; Change of Representation 247
and transforms all operators only with U0, i.e., with the definition
A I (t) := U0+(t, t0) ˆA S (t) ˆ U0(t, t0) (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 U0+(t, t0)|ψ S (t)
a modified Schr¨odinger equation is obtained, given by
−
i
∂ |ψ I (t)
∂t = ˆV I (t) |ψ I (t) (28.14) The unperturbed part of the Hamilton operator, ˆH0, 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 ˆV I (t) instead of ˆ V S
The formal solution of (28.14) is9 the the following perturbation series (Dyson series):
|ψ I (t) =
ˆ−i
t
t0
dt1VˆI (t1)
+
i
2 t
t0
dt1
t1
t0
dt2VˆI (t1) ˆV I (t2)∓ |ψ I (t0) , (28.15)
which may be also symbolically abbreviated as
T e −i Rt t0 dt1VˆI(t1)
|ψ I (t0) ;
the symbolT is called Dyson’s time-ordering operator, because in (28.15)
t ≥ t1≥ t2≥
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
Trang 3248 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 = pˆ2 2m + V (ˆ x) ,
one then obtains the canonical equations
dˆx H
dt =
∂ ˆ H
∂ ˆ p and
dˆp H
dt =− ∂ ˆ H
Trang 429 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 electron1 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 zof 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)
χ1,1(m
s ) = δ m
s ,+1 ,
which is identical to the above-mentioned two-spinor
α := ↑=
1 0
,
b)
χ1, −1(m
s ) = δ m
s , −1 ,
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
Trang 5250 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
≡ pˆ
2
2m − Z e2 4πε0r ,
ˆ
L2, Sˆ2, Lˆz and Sˆz
In contrast, in a constant magnetic induction
B = e z B z = μ0H = μ0e z H z
without the spin-orbit interaction (see below) one obtains
ˆ
H ≡ (ˆp − eA)2
2m − Z e2
4πε0r − gμ B H · Sˆ
(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= μ0e
2m
is the Bohr magneton (an elementary magnetic moment);
μ0= 4π · 10 −7Vs/(Am)
is the vacuum permeability (see Part II) The spin-orbit interaction (see be-low) has so far been neglected
Now, including in ˆH 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 = pˆ
2
2m − Z e2 4πε0r − k(r) Lˆ
·
ˆ
S
− μ B H ·
g l L + gˆ s Sˆ
+μ
2e2
Here the last term,∝ H2, describes the diamagnetism (this term is mostly
negligible) In contrast, the penultimate term, ∝ H, describes the Zeeman
Trang 629.2 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= 2 one recognizes at once the anomalous behavior of the spin momentum ˆS
com-pared with the orbital angular momentum ˆL, where the orbital g-factor g lis
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
μ0c2 .
With
E = Z |e|
4πε0r3r
one obtains a correction to the Hamilton operator of the form
−k(r) Lˆ · Sˆ ,
which has already been taken into account in (29.2) Here−k(r) is positive
and∝ 1/r3; 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 coupling2) 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
Trang 7252 29 Spin Momentum and the Pauli Principle (Spin-statistics Theorem)
As a consequence one can simply define the unitary rotation operator ˆD αby the identity
ˆ
D α:= e−iα ˆ L z /; i.e., with Lˆz=
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 He3atoms, 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
He4 atoms) they are called bosons.
In the position representation one then obtains the N -particle wave func-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 :=
M1
d3r1,
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):
Trang 829.2 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=− 1 for fermions 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 i1(1)· u i2(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 tor j , while particle j moves on the lower
segment of the circle, fromr j tor 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 combinations3 of so-called Slater determinants of orthonormalized single-particle functions These Slater determinants have the form
ψSlateru1,u2, ,u N (1, 2, , N )
:= √1
N !
P
0
@ i1 i N
1 N
1 A
(−1) P u1(i1)· u2(i2)· · u N (i N) (29.4)
≡ √1
N !
u1(1) , u2(1) , , u N(1)
u1(2) , u2(2) , , u N(2)
, , ,
u1(N ) , u2(N ) , , u N (N )
3 Theoretical chemists call this combination phenomenon “configuration interac-tion”, where a single Slater determinant corresponds to a fixed configuration
Trang 9254 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 approximation4
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
Trang 1030 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 ˆL zand ˆS zno longer commute with ˆH However, ˆ J2, the square
of the total angular momentum
ˆ
J := ˆ L + ˆ S ,
and the z-component
ˆ
J z:= ˆL z+ ˆS z ,
do commute with each other and with ˆH, and also with ˆ L2and ˆS2: the five operators ˆH, ˆ L2, ˆS2, ˆJ2and ˆJ z (see below) all commute with each other In fact we have
2 ˆL · ˆ S∝ δ ˆ H≡ ˆ J2− ˆ L2− ˆ S2.
Thus, by including the spin-orbit interaction, a complete system of mu-tually commuting operators, including ˆH, in a shell model, now consists of
ˆ
H, ˆ L2, ˆS2, ˆJ2 and ˆJ 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,Lcan 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 statesY 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 J2 and ˆJ z with eigenvalues 2J (J + 1) and M J.1
1 Using a consistent formulation a related abstract state |ψ L,S,J,M J is defined
byY L,S,J,M J (θ , ϕ , M S ) = , ϕ , M J |ψ L,S,J,M J := r.h.s of (30.1) , which is
similar to defining an abstract state function|ψ by the function values ψ(r) :=
... shall now assume that we are dealing withan atom with many electrons, where L and S (note: capital letters!) are the quantum numbers of the total orbital angular momentum and total... “configuration interac-tion”, where a single Slater determinant corresponds to a fixed configuration
Trang 9254... Part IV
4 This is the simplest way to characterize the Hartree-Fock approximation in a nut-shell
Trang 10