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SELECTED TOPICSOF QUANTUM MECHANICS
Jacek KarwowskiInstytut Fizyki, Uniwersytet Mikołaja Kopernika, Torun
April 15, 2010
BRIEF HISTORY
1
• Max Karl Ernst Ludwig Planck 1858-1947 – Black body radiation,quanta of energy 1900, the Nobel Prize 1918.
• Albert Einstein 1879-1955 – Photoelectric effect ⇒ quantum structureof radiation 1905, Nobel Prize 1921
• Sir Ernest Rutherford 1871-1937 – Atomic nucleus 1911, Nobel Prize1908 (for his investigations for the chemistry of radioactive substances)
• Niels Hendrik Bohr 1885-1962 – "Old quantum theory" 1913, NobelPrize 1922 (for his investigations of the structure of atoms, and of theradiation emanating from them)
• Arnold Sommerfed 1868-1951 – The unification of the quantum theoryof Bohr with the special theory of relativity 1916.
2
• Louis-Victor Pierre Raymond Prince de Broglie 1892-1987 – for hisdiscovery of the wave nature of electrons awarded the nobel Prize in1929.
• Werner Karl Heisenberg 1901-1976 – Quantum mechanics 1925, theuncertainty principle, Nobel Prize (for the creation of quantummechanics) 1932
• Erwin Schrödinger 1887-1961 – Wave equation 1926, Nobel Prize (forthe discovery of new productive forms of atomic theory) 1933
• Wolfgang Pauli 1900-1958 – Quantum theory of many-particlesystems, the Pauli principle 1925, Nobel Prize (for the discovery of theExclusion Principle, also called the Pauli Principle) 1945
• Paul Adrien Maurice Dirac 1902-1984 – The unification of quantummechanics and special theory of relativity 1928, Nobel Prize 1933(shared with Schrödinger)
3
• Enrico Fermi 1901-1954 – Statistics of identical particles 1934, solidstate physics, artificial radioactivity, Nobel Prize (for his demonstrationof the existence of new radioactive elements produced by neutronirradiation, and for his related discovery of nuclear reactions broughtabout by slow neutrons) 1938
• Richard Feynman 1918-1988, Julian Schwinger 1918-1994, ShinichiroTomonaga 1906-1979 – Quantum electrodynamics 1949, Nobel Prize(for their fundamental work in quantum electrodynamics, with deepploughing consequences for the physics of elementary particles) 1965.
• Tsung Dao Lee 1926, Chen Ning Yang 1922 – the violation of theparity conservation law in weak interactions, Nobel Prize 1957 (fortheir penetrating investigation of the so-called parity laws, which hasled to important discoveries regarding the elementary particles).
4
• Hans Albrech Bethe 1906 – Nucleosythesis, Nobel Prize 1967 (for hisdiscoveries concerning the energy production in stars)
• Murray Gell-Mann (Czerniowce) 1929 – The classification ofelementary particles, quark model, quantum chromodynamics, NobelPrize 1969
• Sheldon Lee Glashow (Bobrujsk) 1932, Abdus Salam 1926-1996,Steven Weinberg 1933 – unified theory of the electromagnetic andweak interactions, Nobel Prize 1979
• John Henry Schwarz 1941, Michael Green 1946, Edward Witten 1951– quantum theory of gravitation, string theory, grand unification (TOE)
5
• John Stewart Bell 1928-1990 - Bell inequalities; Alain Aspect 1947 –the experimental proof of the completeness of quantum mechanics 1981
• Robert Floyd Curl 1933, Sir Harold Walter Kroto (Krotoschiner,Bojanowo) 1939, Richard Errett Smalley 1943-2005 - the discovery offullerenes, nanostructures, Nobel Prize 1996 (chemistry)
• Eric Allin Cornell 1961, Carl Edwin Wieman 1951, Wolfgang Ketterle1957 - Bose-Einstein condensation (the phenomenon predicted bySatyendra Nath Bose 1894-1974 and Albert Einstein in 1924), theNobel Prize 2001
• Quantum features of the Hall effect: Klaus von Klitzing 1943 –quantum Hall effect Nobel Prize 1985; Robert Betts Laughlin 1950,Horst Ludwig Störmer 1949, Daniel Chee Tsui 1939 – fractional Halleffect, Nobel Prize (for their discovery of a new form of quantum fluidwith fractionally charged excitations) 1998.
6
THE POSTULATES
7
POSTULATE I
A quantum system is associated with a linear vector space H.
• The elements of H: kets |〉. (e. g. |A1〉, |A2〉).
• The basis: orthonormal, in general infinite.May be discrete or continuous or both.An arbitrary vector in H may be represented as
|A〉 =∑
n
An|n〉 +
∫
C
A(ξ)|ξ〉dξ,
• Vectors dual to the kets: bras 〈| (e.g 〈A1|, 〈A2|)
〈A| =∑
n
A∗n〈n| +
∫
C
A∗(ξ)〈ξ|dξ.
8
• The scalar product of A by B is the complex conjugate of the scalarproduct of B by A: 〈A|B〉 = 〈B|A〉∗.
• The orthonormality of the basis:
〈n|n′〉 = δmn′ , 〈ξ|ξ′〉 = δ(ξ − ξ′).
• The completeness of the basis: the closure relation:
I =∑
n
|n〉〈n| +∫
C
|ξ〉dξ〈ξ|,
Note: |n〉〈n| ≡ Pn
is a projector on the one dimensional space of vector |n〉.
9
POSTULATE II
Each state of the quantum system is described by a normalized ket in H.
A quantum state may be identified with a point on a unit sphere in H or,alternatively, with a direction in H
The linearity of H implies that a superposition of two different states is alsoa state of the quantum system, i.e., if |ψ1〉 and |ψ2〉 describe quantum states,then also
|ψ〉 = c1|ψ1〉 + c2|ψ2〉is a state of this system.
Conclusion:equations which determine the evolution of a quantum system are linear.
10
POSTULATE III
Each dynamical variable Ω is represented in H by a Hermitian operator Ω.
• The correspondence principle: The relationship between the operator(observable) and the dynamical variable is the same as in the classicalmechanics.Example: if x⇒ x, px ⇒ px then
T ⇒ T =p2
2m, kx2 ⇒ kx2
r = (x1, x2, x3), p = (p1, p2, p3)
L⇒ L = [r × p]
11
• Each observable has a complete set of eigenvectors and realeigenvalues. Its eigenvalue problem
Ω|ωn〉 = ωn|ωn〉,
defines a complete basis |ωn〉.
• If the eigenvectors of Ω are taken as the basis in H we say that we havethe Ω representation.
• A vector space H spanned by the basis of the representation Ω isreferred to as the Ω space, e.g. the coordinate space, the momentumspace, etc.
12
• The closure relation gives the resolution of the identity
I =∑
n
|ωn〉〈ωn|,
Therefore for a vector |A〉 =∑
n
|ωn〉 〈ωn|A〉
and for an observable Ξ =∑
mn
|ωm〉〈ωn|Ξ|ωm〉〈ωn|.
• Matrix element of the operator Ξ:
Ξmn = 〈ωn|Ξ|ωm〉
• Spectral decomposition of Ω in its own representation:
Ω =∑
n
|ωn〉ωn〈ωn|.
13
Conclusions:
• In a given representation Ω, a vector |A〉 is represented by a set ofnumbers An = 〈ωn|A〉 labeled by one index, i.e. a column matrixcomposed of the numbers An, n = 1, 2, . . ., is the representative of |A〉.
• An operator Ξ is represented by a set of numbers Ξmn = 〈ωm|Ξ|ωn〉labeled by two indices, i.e. a square matrix composed of elementsΞmn, m,n = 1, 2, . . . is the representative of Ξ.
• A transition from one representation to another is equivalent to atransition from one orthonormal basis to another orthonormal basis inthe same space; it is described by a unitary transformation performedon all the representatives of vectors and operators. Such atransformation changes the numerical values of the matrix elements,however it does not change the physical content of the theory.
14
“Measurement postulates” defining links between the formalismand the physical reality modeled by this formalism.
POSTULATE IV
Each result of a measurement of Ω is equal to one of the eigenvalues of Ω.
POSTULATE VThe probability P (ωn) that ωn is the result of a measurement of Ω
performed on a quantum system in a state |A〉 is equal to
P (ωn) = 〈A|Pn|A〉 = 〈A|ωn〉〈ωn|A〉.
15
Conclusions:
• An observable Ω has a precisely defined value in its eigenstate only:P (ωn) = δkn only if |A〉 = |ωk〉.
• Two different dynamical variables may have precisely defined values ina given state, only if this state is simultaneously an eigenstate of thecorresponding observables. This is possible if these two operatorscommute.
16
THE EQUATIONS OF MOTION
17
THE SCHRÖDINGER PICTURE
|ψ, t〉 – a time-dependent state vectorThe evolution operator T (t, t0):
|ψ, t〉 = T (t, t0)|ψ, t0〉
moves the state vector from t0 to t.For an infinitesimal displacement in time:
T ≡ T (t0 + δt, t0) = I − i
~Hδt,
Thus, H is the generator of the infinitesimal displacement in time.If T is unitary then H is Hermitian.By analogy to the classical mechanics it is identified with the Hamiltonianof the quantum system.
18
The Schrödinger equation
From the definition of H:
|ψ, t0 + δt〉 − |ψ, t0〉δt
= − i
~H|ψ, t0〉.
If δt→ 0 then
i~d|ψ〉dt
= H|ψ〉.
This is the Schrödinger equation in the representation-independent form.
19
The initial state |ψ, t0〉 is an arbitrary normalized vector in HTherefore
i~dT
dt= HT .
If H is time-independent then
T (t, t0) = e−i~
H(t−t0).
If we select the representation in which H is diagonal, i.e. if
H |En〉 = En|En〉,
thenT |En〉 = e−
i~
En(t−t0)|En〉,
20
Conclusions:
• If H is time-independent, the quantum system would remainpermanently in its eigenstate.
• Transitions between different eigenstates of the Hamiltonian arepossible as a consequence of time-dependent perturbations (e.g. due toan external electromagnetic field).
21
For an arbitrary state:
|ψ, t0〉 =∑
n
cn(0)|En〉.
The time evolution:
|ψ, t〉 =∑
n
cn(t)|En〉 = T |ψ, t0〉
withcn(t) = cn(0)e−
i~
En(t−t0).
This is a general solution of the Schrödinger equation in the case of atime-independent Hamiltonian.
The state vector changes in time, but the distribution of the P (En)
probabilities remains constant since |cn(t)|2 = |cn(0)|2.
22
THE HEISENBERG PICTURE
One may obtain another, equivalent, mode of description of the evolution ofa quantum system by performing a time dependent unitary transformationon the vectors and operators of the Schrödinger picture.
Transformation T † defines the Heisenberg picture:
|ψH〉 = T †|ψS〉 = T−1|ψS〉
ΞH = T †ΞST ,
One can see that |ψH〉 = |ψ, t0〉 is time-independent.
Operators in the Schrödinger picture are time-independent.In the Heisenberg picture the operators are time-dependent.
23
The Heisenberg equation of the motion:
i~dΞH
dt= [ΞH , H] + i~
∂ΞH
∂t.
Conclusion:If the quantum system is isolated then an observable is time-independent(i.e. it describes a constant of the motion) if it commutes with theHamiltonian.
24
COORDINATE AND MOMENTUMREPRESENTATIONS
25
The coordinate operator
The basis vectors of the coordinate representation:
r|r〉 = r|r〉,
The spectrum is continuous and extends from −∞ to +∞ in eachdimension.
The orthonormality condition:
〈r|r′〉 = δ(r− r′).
The closure relation:I =
∫
R3
|r〉d3r〈r|.
26
The representative of a state vector |ψ, t〉 depends upon the basis vector r,i.e. it is a function of r. It is called the wavefunction:
ψ(r, t) ≡ 〈r|ψ, t〉.
Similarly,ψ(r, t)∗ = 〈ψ, t|r〉.
The normalization condition:
1 = 〈ψ, t|ψ, t〉 =
∫
R3
〈ψ, t|r〉d3r〈r|ψ, t〉 =
∫
R3
|ψ(r, t)|2d3r.
27
The momentum operator
Electron diffraction experiments and the de Broglie hypothesis say that thecoordinate representative of an eigenvector of the momentum operator is aplane wave:
〈r|p〉 =1
√
(2π~)3e
i~r·p,
wherep|p〉 = p|p〉
By differentiating with respect to r one gets
−~
i∇r〈p|r〉 = p〈p|r〉.
28
An observable which depends upon the coordinate operator only is diagonalin the coordinate representation:
〈r|f(r)|r′〉 = f(r′)δ(r− r′).
Also:
〈r|p|r′〉 =
∫
〈r|p|p〉〈p|r′〉d3p =
∫
p〈r|p〉〈p|r′〉d3p
= −i~∇r
∫
〈r|p〉〈p|r′〉d3p = −i~∇rδ(r− r′),
and similarly 〈r|p|ψ, t〉 = −i~∇rψ(r, t).
In this way we have derived the well known formula for the momentumoperator in the coordinate representation:
p = −i~∇r,
29
The Schrödinger equation
By multiplying the representation-independent Schrödinger equation by 〈r|from the left and inserting the resolution of identity between H and |ψ, t〉,we get
i~∂ψ(r, t)
∂t=
∫
H(r, r′)ψ(r′, t)dr′,
whereH(r, r′) = 〈r|H|r′〉.
30
If
H =p2
2m+ V (r),
then
〈r|H|r′〉 =
[
− ~2
2m4r′ + V (r′)
]
δ(r− r′).
In consequence
i~∂ψ(r, t)
∂t=
[
− ~2
2m4r + V (r)
]
ψ(r, t).
31
The momentum representation:The basis in H is formed by the eigenvectors of the momentum operator.The state vector in the momentum representation:
χ(p, t) ≡ 〈p|ψ, t〉.
The coordinate and the momentum representatives of the state vector arerelated by a Fourier transformation:
ψ(r, t) = 〈r|ψ, t〉 =
∫
R3
〈r|p〉χ(p, t)d3p =1
√
(2π~)3
∫
R3
ei~r·pχ(p, t)d3p.
Similarly:〈p|f(p)|p′〉 = f(p)δ(p− p′)
and〈p|r|p′〉 = −i~∇pδ(p− p′).
32
The Schrödinger equation in momentum representation
i~∂χ(p, t)
∂t=
∫
R3
H(p,p′)χ(p′, t)d3p′,
whereH(p,p′) = 〈p|H|p′〉.
33
If
H =p2
2m+ V (r),
then
i~∂χ(p, t)
∂t=
p2
2mχ(p, t) +
∫
V (p,p′)χ(p′, t)dp′,
where
V (p,p′) = 〈p|V (r)|p′〉
=
∫
〈p|r〉〈r|V (r)|r′〉〈r′|p′〉d3r d3r′
=1
(2π~)3
∫
V (r)ei~(p−p
′)rd3r.
34
Note:
In the case of the harmonic oscillator with
H =p2
2m+mω2
2r2
the Schrödinger equation in the coordinate and in the momentumrepresentation has the same mathematical structure.
In most of other cases, the Schrödinger equation in the momentumrepresentation is an integral equation in which the potential function is thekernel of the integral.
35
THE SCHRÖDINGERAND
THE DIRAC EQUATIONS
36
Invariance properties
Given an equationΩΨ = 0
and a transformation S. The transformed equation reads
Ω′Ψ′ = 0
whereΩ′ = SΩS−1
andΨ′ = SΨ.
37
The results given by a model described by this equation are independent ofthe transformation if Ω is invariant with respect to S:
SΩS−1 = Ω
i.e. if[
S, Ω]
= 0.
In such a case S is called a symmetry transformation.
The transformation S is a symmetry transformation if one cannot design anexperiment which would allow to detect whether this transformation hasbeen performed or not.
For example, a translation in three-dimensional space of a free particle or arotation of a free atom belong the symmetry operations of these systems.
38
The most universal symmetry is related to transformations between twoinertial (i.e. moving with a constant velocity relative to each other)reference frames.
Newton: “The motions of bodies included in a given space are the sameamong themselves, whether that space is at rest or moves uniformly forwardin a straight line”.
Poincaré (principle of relativity): “The laws of physical phenomena mustbe the same for a fixed observer as for an observer who has a uniformmotion of translation relative to him, so that we have not, nor can wepossibly have, any means of discerning whether or not we are carried alongin such a motion”.
39
Conclusion:
All physical theories, including quantum mechanics, should be formulatedin such a way that the results of all measurements are the same independentof whether they are taken by an observer in rest or by an observer moving inuniform translation.
This implies that the equation of the motion must be invariant with respectto a transformation from one inertial frame to another. In the non-relativisticcase this implies the invariance of the Hamiltonian.
40
Transformations of the reference frames:
Newtonian mechanics is invariant with respect to the Galileantransformations:
x′1 = x1 − ut,
x′2 = x2, x′3 = x3,
t′ = t,
The axes of both frames are parallel,The unprimed quantities: the frame is "in rest"The primed quantities:the frame is moving with a constant velocity u along the x1 axis.
41
Michelson and Morley experiment (1887):
The velocity of light measured from differentinertial reference frames is always the same.
Inconsistent with the Galilean transformation.
42
Einstein (1905): The transformation between two inertial frames whichleads to the conclusion that the velocity of light is a universal constant is theLorentz transformation
x′1 =x1 − ut
√
1 − u2/c2, x′2 = x2, x′3 = x3, t′ =
t− ux1/c2
√
1 − u2/c2
with respect to which the Maxwell equations are invariant. Mechanics hasto be formulated in such a way that it is also invariant with respect to thistransformation.
Result: the special theory of relativity.
The causality principle: In a Lorentz-invariant theory no signal can travelwith a velocity larger than that of light.
43
A requirement of invariance with respect to either Galilean or Lorentziantransformation imposes very strong restrictions on the structure of a theoryand on the form of the basic equations.
A model which is Galileo-invariant is called NON-RELATIVISTIC.
In a non-relativistic model three coordinates of a point, momentum,velocity, are three-component vectors, while time or energy are invariants(scalars). Therefore the equation of motion (the Schrödinger equation) isinvariant with respect to the Galilean transformation if the Hamiltonian isGalileo-invariant.
44
The non-relativistic (Galileo-invariant) Hamilton function of a free particle:
H0 =p2
2m.
If the particle moves in an external field characterized by
• Scalar potential V and
• Vector potential A,
the non-relativistic Hamilton function is
H =1
2m
(
p− e
cA
)2
+ V.
The correspondence principle applied to the nonrelativistic Hamiltonfunction leads to the nonrelativistic Hamiltonian operator and opens a wayto formulation of the nonrelativistic (Schrödinger) quantum mechanics.
45
Models which are Lorentz-invariant, are called RELATIVISTIC
In a relativistic model three coordinates: x1, x2, x3 and time, x4 = ict,form a four-component vector.
Also momentum and energy, current and density, are components of thesame four-vector.
Conclusion: In a Lorentz-invariant model, the space coordinates of aparticle and the time corresponding to this particle should appear on anequal footing, contrary to a non-relativistic model, where time is an absolutequantity, the same for all particles.
In particular, if an equation is of the first-order in time, it must be of thefirst-order in all coordinates.
46
In the general equation of motion
i~d|ψ〉dt
= H|ψ〉.
time plays a special role.
We say that this equation is written in a non-covariant form, i.e. its possibleLorentz-invariance is hidden and may be discovered only after analyzingsimultaneously the form of the Hamiltonian and the structure of theequation.
A Lorentz-invariant quantum mechanical equation (the Dirac equation)results from the general equation of motion if the Hamilton operator isderived, through the correspondence principle, from the relativisticHamilton function.
47
The relativistic Hamilton function of a free particle:
H0 ≡ Ep = mc2√
1 +( p
mc
)2
.
Its resolution into a power series of(
pmc
)2:
H0 = mc2 +p2
2m− p4
8m3c2+mc2O
[
(p/mc)6]
,
The Hamilton function of a particle in an external field:
H = mc2√
1 +1
m2c2
(
p − e
cA
)2
+ V.
48
Note:
No Lorentz-invariant equation describing more than one particle can bewritten in the form
i~d|ψ〉dt
= H|ψ〉.
In the case of N particles such an equation would have to depend on Nindependent time coordinates.
49
The relativistic quantum mechanics
The first Lorentz-invariant equation describing the evolution of a quantumstate was obtained by Schrödinger, Klein and Gordon.
They took a square of the operators on both sides of
i~∂ψ(r, t)
∂t= Hψ(r, t).
For the free-particle relativistic Hamiltonian they got:
−~2 ∂
2ψ(r, t)
∂ t2= (m2c4 + c2p2)ψ(r, t).
50
Puttingp2ψ = −~
24ψwe get
(2 − κ2)ψ = 0,
where2 = 4− 1
c2∂2
∂t2
andκ =
mc
~.
This equation is known as the Klein-Gordon equation. It
• Is relativistically invariant,
• May be generalized to account for an external field,
• Does not describe electrons,
• Describes spinless particles.
51
In order to derive a relativistic equation describing an electron, let usconsider the non-relativistic free-particle Hamiltonian
Ha0 =
p2
2m.
For a spin- 12 particle one may take
Hb0 =
(σ · p)2
2m,
instead, where σ are the Pauli spin matrices.
52
Due to the identity
(σa)(σb) = (a · b) + iσ · [a× b],
both equations are consistent with the correspondence principle.
However, in equation
Ha0Ψ = i~
∂Ψ
∂t
Ψ is a scalar function while in equation
Hb0 Ψ = i~
∂Ψ
∂t
Ψ is a two-component function.
53
If the particle moves in an external field the difference between these twoequations becomes even larger.
After the substitutionp → p− e
cA
one gets
Ha =1
2m
(
p− e
cA
)2
andHb =
1
2m
(
p− e
cA
)2
− e~
2mc
(
σ · B)
,
where B = ∇× A is the magnetic field operator.
The σ-dependent term was introduced to the Schrödinger equation by Paulion a purely phenomenological basis in order to account for the interactionof the electron spin magnetic moment with the external magnetic field.
54
Spin-1
2particle.
The correspondence principle applied in the same way as in Hb to
E2p − c2p2 = m2c4
gives the following quantum equation:
[E − c (σ · p)][E + c (σ · p)]Ψ = m2c4Ψ,
where Ψ is a two-component function.One may rewrite this equation in the form
Ω−Ω+Ψ = m2c4Ψ,
whereΩ± = E ± c (σ · p).
55
Dirac equation in the representation of WeylDefining
Ψr ≡ 1
mc2Ω+Ψ
Ψl ≡ Ψ
we get
Ω+Ψl = mc2Ψr,
Ω−Ψr = mc2Ψl,
i.e.
E − c (σ · p), −mc2
−mc2, E + c (σ · p)
Ψl
Ψr
= 0.
56
Taking
E = i~∂
∂t
This pair of equations may be rewritten as
i~∂Ψw
∂t= HwΨw
where
Ψw =
Ψr
Ψl
is a four-component wavefunction and
Hw =
c (σ · p), mc2
mc2, −c (σ · p)
We derived Dirac equation in the representation of Weyl.
57
Representations in the spinor spaceLet A be non-singular 4 × 4 matrix. Then the equation
i~∂Ψ
∂t= HΨ
whereH = AHw A
−1
andΨ = AΨw.
is equivalent to the original equation.The standard (Dirac-Pauli) representation is obtained if
A =1√2
I, I
I, −I
= A−1.
58
Dirac-Pauli representationIn this representation
i~∂ΨD
∂t= HDΨD
where
ΨD =
ΨL
ΨS
is the Dirac wavefunction [ΨL / ΨS are its large / small components] and
HD =
mc2, c (σ · p)
c (σ · p), −mc2
= cα · p + β mc2
with
α =
0, σ
σ, 0
, β =
I, 0
0, −I
59
External fields
In the presence of an external electromagnetic field the free-particle DiracHamiltonian has to be replaced by
HD = cα ·(
p− e
cA
)
+ β mc2 + V .
60
It is important to note that equations:
i~∂Ψ(r, t)
∂t=
p2
2mΨ(r, t)
andi~∂Ψ(r, t)
∂t=
[
cα · p + β mc2]
Ψ(r, t)
are defined in different Hilbert spaces.
The Hilbert space of the first equation is H. In this case it is spanned by theset of eigenvectors of the coordinate operator. The second equation isrepresented in a space HD which is a tensor product of H and a4-dimensional spin-space HΣ in which operators α and β are representedas 4 × 4 matrices:
HD = H⊗HΣ.
61
Non-relativistic limit
The non-relativistic Schrödinger equation may also be written as afirst-order, four-component equation.
For a stationary state, after shifting the energy scale by mc2, i.e.E −mc2 → E, the Dirac equation may be rewritten as
−E, c(σ · p)
c (σ · p), −E − 2mc2
ΨL
ΨS
= 0.
From here
−E, (σ · p)
(σ · p), −2m−Ec−2
ΨL
cΨS
= 0.
62
Lévy-Leblond equation
In the non-relativistic approximation Ec−2 = 0 and
−E, (σ · p)
(σ · p), −2m
ΨL
cΨS
= 0.
This is Lévy-Leblond equation. It is defined in the same space as the Diracequation.
The elimination of cΨS gives
(σ · p)2
2mΨL = EΨL
63
SIMPLE SOLUTIONS
64
A free particleBoth Schrödinger and Dirac Hamiltonians commute with the momentumoperator. Then, in both cases the eigenstates of the Hamiltonian may bechosen to be eigenfunctions of the momentum and represented by the planewaves. The eigenfunction of the free-particle Dirac equation, correspondingto the momentum p, may be expressed as
ψ(r) =
ψ1
ψ2
ψ3
ψ4
ei~(r·p)
√2π~
3 ,
where r-independent ψk, k = 1, 2, 3, 4 form a four-component spinor.
65
The components of the spinor may be obtained from the Dirac equation:
(mc2 −E)ψL + c (σ · p)ψS = 0,
c (σ · p)ψL − (mc2 +E)ψS = 0,
where p are eigenvalues of p and
ψL =
ψ1
ψ2
, ψS =
ψ3
ψ4
If |E −mc2| = |ε| << mc2 (non-relativistic approximation) then
|ψS| ≈∣
∣
∣
∣
(σ · p)
2mcψL
∣
∣
∣
∣
<< |ψL|.
Therefore ψL and ψS are, respectively, referred to as the large and the smallcomponents of the Dirac wavefunction.
66
The free-particle Schrödinger Hamiltonian commutes with the orbitalangular momentum operator L = r× p, but the Dirac one does not.
For a free particle, L is a constant of the motion for the Schrödingerelectron but not for the Dirac one.
The free-particle Dirac Hamiltonian commutes with
J = L +~
2Σ,
where
Σ =
σ 0
0 σ
It is defined as the total angular momentum operator of the electron,composed of the orbital part J and the spin part ~
2 Σ.
67
The eigenvalues of the Dirac Hamiltonian are:
E = ±mc2√
1 +( p
mc
)2
.
Then, the spectrum of the Dirac Hamiltonian is not limited from below.
68
One should expect that an electron in a positive energy state should fall intoa negative energy state and emit a photon with an energy which may beinfinite, since the spectrum is unlimited. In order to solve this difficultyDirac proposed that all the negative-energy states are filled under normalconditions and the Pauli exclusion principle prevents transitions to thesestates. An excitation of one negative-energy electron results in creation of ahole in the "Dirac vacuum" and of one positive-energy electron andcorresponds to the creation of an electron-positron pair. This interpretationcontains a self-contradiction: the Dirac model describes a single electron.At the same time, it is unable even to explain the behaviour of a free particlewithout assuming that it is surrounded by an infinite number of particlesoccupying the negative-energy states.
69
Dirac velocityIn the Schrödinger case:
(
dr
dt
)
S
=i
~[HS, r] =
p
m.
In the Dirac case:(
dr
dt
)
D
=i
~[HD, r] = cα.
This result is very strange in many aspects. First, the eigenvalues of αk are±1. Therefore the eigenvalues of
(
drdt
)
Dare equal to ±c.
70
Besides, different components of(
drdt
)
Ddo not commute. Then, a
measurement of one component of the Dirac velocity is incompatible with ameasurement of another one. This is also strange since differentcomponents of the momentum operator do commute. Moreover,
(
drdt
)
D
does not commute with HD and therefore the velocity is not a constant ofthe motion despite the fact that the particle is free.
71
An explanation of this fact involves an analysis of the time-dependence ofα and of the coordinate operator r. It results that both of them execute veryrapid oscillations with an angular frequency (2mc2)/~ = 1.5 · 1021sec−1.This motion, named by Schrödinger Zitterbewegung, is due to aninterference between the positive- and negative-energy components of thewavepacket describing the electron. Intuitively it may be interpreted as aconsequence of a permanent creation and annihilation of the so calledvirtual electron-positron pairs.
72
In order to explore this problem in more detail it is convenient to introducethe even and odd operators. An even operator acting on a function whichbelongs to the positive (negative) energy subspace of a free electrontransforms it into a function which belongs to the same subspace. An oddoperator transforms a positive (negative)-energy-subspace function into afunction which belongs to the complementary subspace.Operators
Π± =1
2(I ± Λ),
where
Λ =HD
Ep=c (α · p) + β mc2√
m2c4 + c2p2
is the operator of the sign of energy (its eigenvalues are +1 for the positive-and −1 for the negative-energy states), are projection operators to thepositive (Π+) and to the negative (Π−) energy subspace.
73
For example, the operator Π+ΩΠ+ acts in the free-electron positive-energyspace only. In the free particle space each operator, say Ω, may bedecomposed into its even part [Ω] and its odd part Ω:
Ω = [Ω] + Ω,
where
[Ω] =1
2(Ω + ΛΩΛ),
Ω =1
2(Ω − ΛΩΛ).
After some algebra one can see that[(
dr
dt
)
D
]
= c [α] =c2p
EpΛ,
i.e. the result corresponding to the standard definition of velocity.
74
One-electron atomAn exact quantum-mechanical treatment of a hydrogen-like atom isequivalent to solving a two-body problem with a Coulomb interaction. Inthe nonrelativistic case it may be solved exactly. By the separation of thecenter of mass the six-dimensional Schrödinger equation is split into twothree-dimensional equations. One of them describes a free motion of thecenter of mass and the other one – the relative motion of the electron andthe nucleus.The Hamiltonian which governs the relative motion:
HS =p2
2µ− Ze2
r,
where Ze is the nuclear charge,
µ =mM
m+Mis the reduced mass; m – mass of electron and M – mass of the nucleus.
75
The Hamiltonian and the orbital angular momentum operators L2 and Lz
form a set of mutually commuting operators.
The eigenvalue spectrum of HS consists of the discrete part (describing thebound states) and the continuous part (describing the ionized states). Thediscrete part of the eigenvalue problem may be written as
HSψnlm(r, θ, φ) = ESnψnlm(r, θ, φ)
where r, θ and φ are the spherical coordinate system variables and
n = 1, 2, . . . ,
l = 0, 1, . . . , n− 1,
m = −l,−l + 1, . . . , l
are the quantum numbers describing, respectively, the energy (the principalquantum number), the orbital angular momentum and its projection.
76
The eigenvalues (the energies of the bound states) in the Hartree atomicunits (a.u.: m = 1, e = 1, ~ = 1) are given by
ESn = − Z2
2n2
µ
ma.u. = −
(
Z
n
)2
RM ,
whereRM =
µ
mR∞
and
R∞ =mc2
2α2
are the Rydberg constants for the finite and infinite nuclear mass,respectively, and
α =e2
~c≈ 1
137
is the fine structure constant.
77
The eigenfunctions are labeled by three quantum numbers and the degree ofdegeneracy is equal to n2. There are two origins for this degeneracy. One,with respect to the quantum number m, is a consequence of the sphericalsymmetry of the problem (in all spherically symmetric one-electron systemsthis kind of degeneracy would appear). The second one, with respect to thequantum number l, is a consequence of a dynamical symmetry connectedwith some specific properties of the interaction potential. In the case of thenon-relativistic one-electron atom there is an additional constant of themotion arising from the commutation of the so called Runge-Lenz vectorwith the Hamiltonian. The l-degeneracy is related to the existence of thisconstant of the motion and is removed if the potential is deformed so that itloses its Coulomb character.
78
The eigenfunctions of the hydrogen-like Hamiltonian are prototypes formost of approximate one-electron functions (orbitals) used in quantumchemical calculations. They may be expressed as
ψnlm(r, θ, φ) =1
rRnl(r)Ylm(θ, φ),
where Ylm (the spherical harmonics) are the eigenfunctions of L2 and Lz
and Rnl are the radial functions.The radial functions are of the form
Rnl(r) = W ln(r)e−Zr/n,
where W ln is a polynomial of the degree n with l-dependent coefficients.
The number of nodes of this polynomial is equal to n− l.At the origin the radial part of the wavefunction r−1Rnl behaves as rl, i.e.it vanishes for r = 0 if l 6= 0.
79
The spherical harmonics are, for m 6= 0, complex and:
Yl,−m = (−1)mY ∗lm.
Their real combinations;
Ylm + (−1)mYl,−m and i [Ylm − (−1)mYl,−m]
may be expressed as1
rl
∑
abc
Cabcxaybzc,
where a+ b+ c = l.For l = 0, 1, 2, 3, . . . they are referred to as s,p,d,f ,. . .-type functions,respectively.The consecutive discrete states of the hydrogen-like atom are labeled by thequantum numbers n and l as 1s, 2s, 2p, 3s, 3p, 3d, . . ..
80
The standard Schrödinger model of a hydrogen-like atom does not containspin. Spin may be introduced by using the phenomenological Pauli model.Then the wavefunction ψnlm has to be multiplied by a two-component spinfunction α or β corresponding, respectively, to the eigenvalues + ~
2 and −~
2
of a projection Sz of the spin operator. These functions are referred to asspinorbitals. The spinorbitals may be combined according to the rules ofcoupling the angular momenta to form eigenfunctions of the total angularmomentum operator which, of course, also commutes with the Hamiltonian.However introducing spin at this level does not influence the energy of theatom, unless the atom is placed into an external magnetic field when thePauli term influences the energy (do not confuse with the Pauli relativisticcorrections which result from a reduction of the Dirac equation).
81
In the relativistic case a two-body Coulomb problem cannot be solvedexactly. Therefore the hydrogen-like atom in the Dirac theory is modeled byan electron moving in an external Coulomb field. We solve thecorresponding Dirac equation in the reference frame in which thesingularity of the potential is placed at the origin, i.e. with
V (r) = −Ze2
r
82
Neither the square nor the projection of L commutes with the DiracHamiltonian. Therefore the Dirac wavefunction can be an eigenfunction ofneither L2 nor Lz . However, in the standard representation, the large andsmall components are eigenfunctions of L2, to the eigenvalues l(l + 1) andl′(l′ + 1) respectively, where l′ = l ± 1. We introduce, apart of the totalangular momentum quantum number j = l ± 1/2 and its projectionmj = −j,−j + 1, . . . , j, the relativistic angular momentum quantumnumber
κ = ±(j + 1/2) =
l if j = l − 1/2
−(l + 1) if j = l + 1/2.
The energy spectrum consists of two continua (one extending from +mc2
up and another one extending from −mc2 down) and the discrete spectrumlocated below the ionization threshold +mc2 and depending on twoquantum numbers: n and j (or, equivalently, on n and κ).
83
The eigenvalue equation:
HDψnljmj= ED
njψnljmj,
where
EDnj −mc2 = − Z2
N(N + n)·2R∞
withN =
√
α2Z2 + n2,
n = n+ s− |κ|,and
s =√
κ2 − α2Z2.
The Dirac states are labeled by n, l and j (l corresponds to the L2
eigenvalue for the large component of the pertinent wavefunction).
84
The consecutive discrete states of the Dirac atom are:1s1/2, 2s1/2, 2p1/2, 2p3/2, 3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2, . . ..For α→ 0, i.e. in the nonrelativistic limit,
s→ κ,
n→ n,
N → n,
(EDnj −mc2) → ES
n.
By expanding the Dirac energy into a power series of (αZ)2 one gets
EDnj = mc2
[
1 − (αZ)2
2n2− (αZ)4
2n3
(
1
|κ| −3
4n
)
+O(
(αZ)6)
]
,
where the first term is the rest energy of the electron, the second one is thenon-relativistic (Schrödinger) energy and the third one is the relativistic(Pauli) correction.
85
In the Dirac spectrum, apart of shifting the energy levels relative to theSchrödinger ones, some of the degeneracies have been removed. Now theenergy levels depend upon the total angular momentum quantum number jbut, also in the Dirac theory, they do not depend upon l. Then, the energy ofthe states 2p1/2 and 2p3/2 are different while the energies of 2p1/2 is thesame as that of 2s1/2. The splitting of the energy levels due to j is called thefine structure splitting. It is relatively small for small Z but it grows veryfast with increasing nuclear charge (it is proportional to Z4).
86
Experimental measurements show that the energies of a real hydrogen-likeatom depend upon l. In particular the energies of 2s1/2 and 2p1/2 aredifferent. For the first time this splitting was measured for the hydrogenatom by Lamb and Retherford. The effect is called the Lamb shift. It maybe explained on the ground of quantum electrodynamics. Its value growsrapidly with increasing Z. For the hydrogen atom it is equal 1060 Mc =4·10−6 eV; for the uranium like atoms it is about 70 eV, i.e. by 7 orders ofmagnitude larger than in hydrogen!
87
The eigenfunctions of a spherically-symmetric Dirac Hamiltonian may berepresented as
ψnlκmj=
1
r
Gnκ(r) Φ(θ, φ)nlκmj
i Fnκ(r) Φ(θ, φ)nl′−κmj,
where l′ = l − κ/|κ| and Φ(θ, φ) are combinations of products of thespherical harmonics and the two-component spin functions constructedaccording to the rules of coupling the angular momenta.The radial functions fulfill the following eigenvalue equation:
[
mc2 −E + V (r)]
G(r) − c
[
d
dr− κ
r
]
F (r) = 0,
c
[
d
dr+κ
r
]
G(r) −[
mc2 +E − V (r)]
F (r) = 0,
where, in the case of a hydrogen-like atom, V (r) = −Ze2/r.
88
Then, the small component is related to the large one:
F (r) = c[
mc2 + E − V (r)]−1
(
d
dr+κ
r
)
G(r).
Both large and small radial components may be expressed as products ofe−ζr and a polynomial, where ζ =
√
m2c2 − E2/c2. At the origin R(r)
behaves as rs. Then, for |κ| > 1 it vanishes for r = 0. However, forκ = ±1, s < 1 and r−1R(r) is singular. This singularity is weak and forZα < 1 does not obstruct the normalizability of the wavefunctions. Therelativistic radial electron density is contracted relative to thenon-relativistic one (the Dirac hydrogen atom is "smaller" than theSchrödinger one) . Besides, it is nodeless since the nodes of the large andsmall radial components (except the r = 0 node) never coincide.
89
Singularity of the Dirac wavefunctions at the origin is a consequence of theCoulomb singularity in the potential. A more general potential may betaken in the form
V (r) = −Z(r)e2
r.
If Z(r) is constant, then we have a point nucleus and singular potential.One of the simplest and most common is a model in which a uniformnuclear charge distribution is assumed. In this case
V (r) =
− 3Z2A
(
1 − r2
3A2
)
e2, if 0 ≤ r ≤ A
−Ze2
r if r > A.
Taking the finite nuclear model we get the wavefunction without anysingularity: r−1G(r) behaves as rl and r−1F (r) – as rl+1.
90
RELATIVISTIC COVARIANCE OFTHE DIRAC EQUATION
91
Notations and basic properties
xµ = (r, ict)
pµ = −i~ ∂
∂ xµ=
(
p,−~
c
∂
∂ t
)
γµ = (−iβα, β) = (γ, β)
γ =
0 −iσiσ 0
, γ4 =
I 0
0 −I
㵠= ㆵ
γµγν + γνγµ = 2δµν
92
Dirac equation
i~∂Ψ
∂ t=
(
cα · p+ β mc2)
Ψ
Ψ =
ψ1
ψ2
ψ3
ψ4
, Ψ† = [ψ∗1 , ψ
∗2 , ψ
∗3 , ψ
∗4 ]
Ψ = Ψ†β = [ψ∗1 , ψ
∗2 ,−ψ∗
3 ,−ψ∗4 ] = Ψ†γ4
γµ∂Ψ
∂ xµ+ κΨ = 0,
∂Ψ
∂ xµγµ − κΨ = 0, κ =
mc
~
93
Continuity equation: ∂ jµ∂ xµ
= 0
jµ = iceΨγµΨ = (j, icρ) , ⇒ j = ceΨ†αΨ, ρ = eΨ†Ψ
Pauli fundamental theoremGiven two sets of 4 × 4 matrices satisfying
γµ, γν = 2δµν and γ′µ, γ′ν = 2δµν
with µ, ν = 1, 2, 3, 4, there exists a nonsingular 4 × 4 matrix S such that
SγµS−1 = γ′µ.
Moreover, S is unique up to a multiplicative constant.
Most commonly used representations: Dirac-Pauli (standard), Weyl,Majorana (γk are purely real, γ4 is purely imaginary)
94
Lorentz transformation – a rotation in the Minkowski space
x′µ = aµνxν
aµνaλν = δµλ, det |aµν | = 1, a44 > 0.
Special cases
• Rotation in the usual three-dimensional space
x′1
x′2
=
cos ω sin ω
− sin ω cos ω
x1
x2
• “Pure” Lorentz transformation – rotation in 1 − 4 plane by angle iχ
x′1
x′2
=
cosh χ i sinh χ
− sinh χ cosh χ
x1
x2
, tanhχ =v
c
95
Covariance of the Dirac equation
A′µ(x′) = aµνAν(x),
∂
∂ x′µ= aµν
∂
∂ xν
γµ∂Ψ(x)
∂ xµ+ κΨ(x) = 0 ⇒ γµ
∂Ψ′(x′)
∂ x′µ+ κΨ′(x′) = 0
Assumption: Ψ′(x′) = SΨ(x), S – 4 × 4 matrix independent of xµ
S−1γµSaµν = γν ⇒ S−1γλS = aλνγν
96
The non-relativistic Pauli model
Rzω|lm〉 = eimω|lm〉, m = −l,−l + 1, . . . , l
Rzω
∣
∣
12 ± 1
2
⟩
= e±iω/2∣
∣
12 ± 1
2
⟩
⇒ Rzω
U+
U−
=
eiω/2 0
0 e−iω/2
U+
U−
SPauli U(x) = U ′(x′)
SPauli =
cos ω2 + i sin ω
2 0
0 cos ω2 − i sin ω
2
= I cos ω2 + iσ3 sin ω
2
97
Several formulasσ × σ = 2iσ
σjσk − σkσj = 2iσl
σjσk + σkσj = 0
σjσk = iσl, (cyclic)
Σ3 =
σ3 0
0 σ3
γ1γ2 =
0 −iσ1
iσ1 0
0 −iσ2
iσ2 0
= i
σ3 0
0 σ3
= iΣ3,
(γ1γ2)2 = −I4
98
The Dirac model - rotation in 1 − 2 plane
Srot = I4 cos ω2 + iΣ3 sin ω
2 = I4 cos ω2 + γ1γ2 sin ω
2
S−1rot = I4 cos ω
2 − iΣ3 sin ω2 = I4 cos ω
2 − γ1γ2 sin ω2
Theorem: S−1rotγλ Srot = aλµγν
a =
cosω sinω 0 0
− sinω cosω 0 0
0 0 1 0
0 0 0 1
where a – matrix with elements aλµ, λ, µ = 1, 2, 3, 4.
99
The Dirac model - rotation in 1 − 4 plane
ω → iχ, cosω → coshχ, sinω → i sinhχ, γ2 → γ4.
SLor = I4 cosh χ2 + iγ1γ4 sinh χ
2
S−1Lor = I4 cosh χ
2 − iγ1γ4 sinh χ2
Theorem: S−1Lorγλ SLor = aλµγν
a =
coshχ 0 0 i sinhχ
0 1 0 0
0 0 1 0
−i sinhχ 0 0 coshχ
100
More definitions and formulas
σµν ≡ 1
2i[γµ, γν ] .
σµν = −i γµγν , µ 6= ν
σij = −σji = Σk =
σk 0
0 σk
σk4 = −σ4k = αk =
0 σk
σk 0
[γ4, σij ] = 0, γ4, σk4 = 0
γ5 ≡ γ1γ2γ3γ4 =
0 −I−I 0
γµ, γ5 = 0, γ25 = I4 [γ5, σµν ] = 0
101
SummarySij
rot = I4 cos ω2 + iσij sin ω
2
Sk4Lor = I4 cosh χ
2 − σk4 sinh χ2
S†rot = S−1
rot
S†Lor = SLor 6=S−1
Lor
Conclusion: SLor is not unitary!.But
S−1 = γ4 S†γ4 and S† = γ4 S
−1γ4
It is consistent with Ψ = Ψ†γ4.Also S−1γ5 S = γ5
102
Space inversionr′ = −r, t′ = t
a =
−1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
S−1inv γλ Sinv = aλνγν ⇒
S−1inv γk Sinv = −γk
S−1inv γ4 Sinv = γ4
⇒ Sinv = γ4 = S−1inv
Transformation of the wavefunction under space inversion: Ψ′ = γ4Ψ = Ψ†
Sinvγ5 Sinv = γ4γ5γ4 = −γ5
103
Infinitesimal rotation
ω → δω << 1 ⇒ cosω → 1, sinω → δω.
Srot = 1 +i
2Σ3 δω
r′ = r + δr ⇒
x′1 = x1 + x2δω
x′2 = x2 − x1δω
x′3 = x3
104
Ψ′(r′) = Ψ′(r + δr)
= Ψ′(r) +∂Ψ′
∂ x1δx1 +
∂Ψ′
∂ x2δx2 +
∂Ψ′
∂ x3δx3
= Ψ′(r) +
(
∂Ψ′
∂ x1x2 −
∂Ψ′
∂ x2x1
)
δ ω
= SrotΨ(r) =
(
1 +i
2Σ3 δω
)
Ψ(r)
Ψ′(r) =
[
1 +i
2Σ3 δω −
(
x2∂
∂ x1− x1
∂
∂ x2
)
δω
]
Ψ(r)
=
[
1 +
(
1
2Σ3 + L3
)
i δω
]
Ψ(r) =(
1 + i δω J3
)
Ψ(r)
L3 = i
(
x2∂
∂ x1− x1
∂
∂ x2
)
, J3 =1
2Σ3 + L3.
105
Infinitesimal rotation – conclusions
Ψ′(r) =(
1 + i δω J3
)
Ψ(r)
J3 =1
2Σ3 + L3
The change in the functional form of Ψ induced by the infinitesimal rotationconsists of two parts: the space-time independent one (associated with theoperator Σ3) and the one corresponding to the rotation in three-dimensionalspace (associated with L3). The total angular momentum operator J3 is thegenerator of an infinitesimal rotation around the third axis.
106
Bilinear covariants: ΨΓΨ
Ψ′(x′) = SΨ(x),
Ψ′(x′)† = Ψ(x)† S† = Ψ(x)†γ4S−1γ4 = Ψ(x)S−1γ4
Ψ′(x′)†γ4 = Ψ(x)S−1, Ψ′(x′) = Ψ(x)S−1
S−1γµ S = aµνγν
1. Ψ′(x′)Ψ′(x′) = Ψ(x)S−1SΨ(x) = Ψ(x)Ψ(x), − scalar
2. Ψ′(x′)γµΨ′(x′) = Ψ(x)S−1γµ SΨ(x) = aµνΨ(x)γνΨ(x)
(
ΨγµΨ)′
= aµν
(
ΨγνΨ)
, − vector
3. Ψ′(x′)γ5Ψ
′(x′) = Ψ(x)S−1γ5 SΨ(x) =
Ψγ5Ψ Lor tr−Ψγ5Ψ rot
pseudoscalar
107
Bilinear covariants – summary
Scalar ΨΨ
Vector ΨγµΨ
Antisymmetric tensor ΨσµνΨ
Pseudovector iΨγ5γµΨ
Pseudoscalar Ψγ5Ψ
108
Large and small covariants
ΨΓΨ =
(
ΨL† − ΨS†)
A 0
0 −B
ΨL
ΨS
= ΨL†
AΨL + ΨS†BΨS − large
(
ΨL† − ΨS†)
0 A
A† 0
ΨL
ΨS
= ΨL†
AΨS − ΨS†
BΨL − small
Large: I , γ4, iγ5γk, σij
Small: γk, iγ5γ4, γ5, σk4
109
Clifford algebraGeneralized quaternions - William K. Clifford (1845-1879)Questions: Can we generate more bilinear covariants of the form ΨΓΨ?
1. γµ, γ2µ = 1
2. γµγν = −γνγµ = i σµν
3. γµγνγλ = γ5γσ , µ 6= ν 6= λ 6= σ
γµγνγµ = −γν
4. γ5 = γ1γ2γ3γ4
Products of more than 4 matrices can be reduced to products of at most 4
p, q = 1, 2, 3, 4, 5 : 1 matrix I , 5 matrices γq , 10 matrices σpq
− together 16 matrices Γ
110
Γ2 = I ⇒ Γ = Γ−1
If ΓA 6= ΓB and ΓA,ΓB 6= I then
1. Tr (ΓA ΓB) = −Tr (ΓB ΓA) = 0,
2. Tr(
Γ2A
)
= 4,
3. Tr (ΓA) = 0,
Tr (ΓA) = Tr(
Γ2BΓA
)
= −Tr (ΓBΓAΓB) = −Tr(
Γ−1B ΓAΓB
)
= −Tr (ΓA)
Let Λ =16∑
A=1
λAΓA
then λA =1
4Tr (ΛΓA) .
111
APPROXIMATIONS
112
APPROXIMATE SOLVING EIGENVALUE PROBLEMS
It is customary to divide the most commonly used approximate methods ofsolving eigenvalue problems to two classes:
• Variational methods,
• Perturbational methods.
113
VARIATIONAL METHOD
114
The variational methods are most appropriate if we have a discreteeigenvalue problem:
H |ψk〉 = Ek|ψk〉, k = 1, 2, . . .
withE1 ≤ E2 ≤ E3 ≤ · · · .
The variational methods result from a very simple property of theeigenvalue problem known as the variational principle.
115
The basic theorem:For a |Φ〉 which belongs to the space spanned by the eigenvectors of H ,
K[Φ] ≡ 〈Φ|H|Φ〉〈Φ|Φ〉 ≥ E1.
The equality can be achieved only if
|Φ〉 = |ψ1〉.
The functional K[Φ] is called the Rayleigh quotient.
Conclusion: 〈Φ(q)|H|Φ(q)〉〈Φ(q)|Φ(q)〉 ≥ Eq
if |Φ(q)〉 is orthogonal to |ψi〉 for i = 1, 2, . . . , q − 1.Applications of the variational principle result in modifying a trial functionΦ so that the value of K[Φ] becomes as small as it is possible within theconstraints imposed upon Φ.
116
Different trial functions ⇒ different variational models.
A precondition for applicability of the variational method is that theeigenvalue problem we are dealing with is bounded from below.
Fulfilled in the case of a Schrödinger equation
Not fulfilled in the case of the Dirac equation.
117
In the case of the Dirac equation some preliminary steps are necessary
• Restriction of the space of the trial functions by imposing the boundaryconditions which would force them to be orthogonal to thenegative-energy solutions. In such a space the Dirac eigenvalueproblem is limited from below.
• Modification of the Dirac Hamiltonian so that it is bounded from belowfor all square-integrable trial functions and retains the interesting partof its spectrum unchanged.
118
In order to accomplish this one can
• project the Hamiltonian onto the space of the positive-energy states,
• transform it in such a way that the negative-energy continuum wouldoverlap with the positive-energy continuum,
• transform the Hamiltonian so that the large and small components aredecoupled.
119
SCHRÖDIGER EQUATION: ALGEBRAIC REPRESENTATIONSchrödinger Hamiltonian eigenvalue problem (“time-independentSchrödinger equation”):
Hψk = Ekψk
A basis set expansion of the trial function
Φ =
K∑
n=1
Cn φn, 〈φn|φm〉 = δmn,
leads to the algebraic approximation to the Schrödinger equation.K
∑
n=1
(Hmn −Ekδmn)Ckn = 0, k = 1, 2, . . . , n
The K-dimensional space HK spanned by φnKn=1 is referred to as the
model space
120
McDonald theorem
Let the model space be spanned by K basis functions and let theHamiltonian matrix eigenvalues be
E(K)1 ≤ E
(K)2 ≤ · · · ≤ E
(K)K ;
as the dimension of the model space is increased by adding an additionalbasis function, eigenvalues E(K+1)
p of the (K + 1)-dimensional problemsatisfy the inequalities:
E(K)p−1 ≤ E(K+1)
p ≤ E(K)p
and as the model space approaches completeness, the algebraic solutionsapproach the exact solutions of the Hamiltonian eigenvalue equation.Conclusion: each eigenvalue of the algebraic eigenvalue problem is anupper bound to the corresponding eigenvalue of the Hamiltonian.
121
SPECTRUM OF ONE-ELECTRON DIRAC HAMILTONIAN
mc2
mc22− mc2−
L
S
0
E E D
0D
POSITIVE−ENERGY CONTINUUM
NEGATIVE−ENERGY CONTINUUM
IONIZATION THRESHOLD
FORBIDDEN ENERGY GAPIN THE CASE OF A FREE ELECTRON
DISCRETE BOUND−STATE ENERGIES,
122
ENERGY FUNCTIONAL
Rayleigh quotient: K[Φ] =〈Φ|H|Φ〉〈Φ|Φ〉 , Φ =
caΦL
cbΦS
Dirac: K[Φ]D = W+ +√
W−2 + 2mc2T
Lévy-Leblond: K[Φ]L = T + (W+ +W−)
W± =1
2
( 〈ΦL|V |ΦL〉〈ΦL|ΦL〉 ± 〈ΦS|V |ΦS〉
〈ΦS|ΦS〉
)
∓ mc2,
T =1
2m
〈ΦL|σ · p|ΦS〉〈ΦS|σ · p|ΦL〉〈ΦL|ΦL〉〈ΦS|ΦS〉 .
123
KINETIC BALANCE
LL energy functional: K[Φ]L = T +〈ΦL|V |ΦL〉〈ΦL|ΦL〉
T =1
2m
〈ΦL|σ · p|ΦS〉〈ΦS|σ · p|ΦL〉〈ΦL|ΦL〉〈ΦS|ΦS〉 .
Kinetic balance condition: ΦS ∼ (σ · p)ΦL ⇒
T =1
2m
〈ΦL|(σ · p)2|ΦL〉〈ΦL|ΦL〉
Kinetic balance condition ⇒
variational LL and Schrödingereigenvalue problems are equivalent.
124
ALGEBRAIC REPRESENTATION: MODEL SPACE
Basis set expansion of the components of the trial function
ΦL =
KL∑
k=1
CLk φL, ΦS =
KS∑
k=1
CSk φ
S
leads to the algebraic approximation to the Dirac equation.
The kinetic balance condition implies
HΦS ⊇ (σ · p)HΦL
where HΦ – space in which Φ is expanded.This condition is necessary for the correct behaviour of the variationalprocedure applied to the Dirac equation.
125
ALGEBRAIC REPRESENTATION
Dirac equation: variation of K[Φ] leads to an algebraic(KL +KS) × (KL +KS) eigenvalue problem:
HLL − ESLL cHLS
cHSL HSS −ESSS
CL
CS
= 0,
Lévy-Leblond equation: KL ×KL matrix eigenvalue equation:
(H −ESLL)CL = 0,
H = HLL +1
2mHLSS
−1SSHSL.
126
ALGEBRAIC REPRESENTATION: SPECTRUM
EXACT SPECTRUM ALGEBRAIC REPRESENTATION
0
E E D
0
0
E E D
0
−Σ c
2− mc2 − mc2
mc2
2− mc2
mc2
− mc2
Σ c+
127
EXAMPLE: SPHERICAL SYMMETRY
Symmetry-adapted basis functions:
ΦL(r) =ϕL(r)
rAL(Ω), ΦS(r) =
ϕS(r)
rAS(Ω)
A(Ω) – exact spin-angular functions
ϕL(r) =N
∑
k=1
CLk r
lke−αk r, ϕS(r) =N
∑
k=1
CSk r
ske−βk r
128
STRUCTURE OF THE SADDLE POINTGround state of Z = 90 H-like atom, K = 1, Dirac (left), LL (right)E = E(α1, β1)|l1=s1=exact (up), E = E(l1, s1)|α1=β1=exact (down)
50 60 70 80 90 100 110 120 13050
60
70
80
90
100
110
120
130
50 60 70 80 90 100 110 120 13050
60
70
80
90
100
110
120
130
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
129
BOUNDS TO EIGENVALUES: MINIMAX PRINCIPLE
A relativistic variational principle formulated as a recipe for reaching thesaddle point on the energy hypersurface in the space of variationalparameters.
E = minL
[
maxS
〈Φ|H|Φ〉〈Φ|Φ〉
]
.
130
TWO-ELECTRON DIRAC-COULOMB EQUATION
HDC(1, 2)Ψ(1, 2) = EDCΨ(1, 2)
Dirac-Coulomb Hamiltonian:
HDC(1, 2) = HD(1) + HD(2) +1
r12
A hybrid composed of a relativistic one-electron part and a non relativistictwo-electron term. Its eigenvalues corresponding to the bound-statesolutions are embedded in a continuum spreading from −∞ to +∞ and thatthe discrete and the continuum spectra are coupled by the electron-electroninteraction. This effect is known as the Brown-Ravenhall disease and thecontinuum is referred to as the Brown-Ravenhall continuum.
131
SPECTRUM OF DIRAC-COULOMB HAMILTONIAN
Spectra of one-electron (A) and two-electron (B) Dirac Hamiltonian.
Σ−c
Σ+c
2−2mc
DE
−mc2
mc2
Σ−g
Σ+g
2−2mc
2−4mc
DE
2−2mc
2 2mcΣ c
++
Σ c−−
Σ c+−
0
E
0
E
0
0
A
B
Two-electron discrete and continuous spectra overlap.
132
ARTIFACTS OF DIRAC-COULOMB EQUATION
• Electron-electron interaction couples discrete and continuum states.Consequently, all eigenvalues of the Dirac-Coulomb Hamiltoniancorresponding to physically bound states (including, for example, theground state of helium atom) are autoionizing.
• Dirac-Coulomb Hamiltonian does not have normalizableeigenfunctions.
• Removing the ‘unphysical” continuum by a projection results in anincomplete model space.
• The Dirac-Coulomb Hamiltonian is unbounded from below.
133
IMPLEMENTATIONS
The Dirac-Coulomb equation is a basis for relativistic Hartree-Fock-typemethods referred to as Dirac-Fock or Dirac-Hartree-Fock methods.
In these methods the trial functions are represented by antisymmetrizedproducts of Dirac spinors.
The influence of the artifacts is usually removed by appropriate selection ofthe model space corresponding to a projection of the Dirac-CoulombHamiltonian to the positive energy space.
134
EXAMPLE: POTENTIAL ENERGY CURVES
Non-relativistic (left panel) and relativistic (right panel) potential energycurves for the excited state of I2. States dissociating in two 5p5, 2P iodineatoms are shown [L. Vissher et al. JCP 1995]
135
EXAMPLE: X-RAY SPECTRA
[B. Galley et al., PRA 1999]
136
COMPLEX SCALING
The method known as complex coordinate rotation (CCR) has beendeveloped to study the autoionizing states also referred to as resonances.These are the states whose discrete energies are embedded in a continuum.
Basic theorem: Bound state energies of a Hamiltonian do not changeunder the complex rotation of coordinates,
r → reiΘ,
whereas the continua move to the complex plane.
From the formal point of view discrete eigenvalues of a many-electronDirac Coulomb Hamiltonian are resonances. Therefore the complex scalingmethod may also be applied to analyzing the Dirac-Coulomb spectrum.
137
EFFECT OF COMPLEX COORDINATE ROTATION
cΣ+
cΣ−Σ
−c
DE
−mc2
mc2
0
cΣ+
2−2mc
0
E
Γ/2=−Im(z)
E=Re(z)
Θ
Standard (left) and CCR (right) spectrum of a one-electron DiracHamiltonian. Solid lines represent the positive, Σ+
c , and the negative, Σ−c ,
continua. The points in the real axis represent bound-state energies.
138
SPECTRUM OF A TWO-ELECTRON DC HAMILTONIAN
Σ−g
Σ+g
2−2mc
2−4mc
DE
2−2mc
2 2mc
Σ c−−
Σ+g
Σ−g
Σ c++
Σ c++
Σ c+−
Σ c+−
−Im(z)Γ/2=
E
0
0
Θ
Σ c−−
E=Re(z)
The same as before but for a two-electron system.
139
ALGEBRAIC CCR SPECTRUM OF Z=90 He-LIKE ATOM
−8
−6
−4
−2
0
210 −1 −2
−4mc2
−2mc2
Σ++c
Σ+g
Σ+−c
Σ−g
Σ−−c
1s2
E [104a.u.]
Γ/2 [104a.u.]−10000
−8000
−6000
−4000
−2000
0
10000
E [a.u.]
(a) (b)
Γ/2 [a.u.]
1s2
1s2s1s3s
2s2 & 2p2
Right panel: Enlargement ofthe bound-state region.(a) No complex rotation: Thediscrete and the continuumenergies are mixed together.(b) The rotated spectrum: Thecontinuum eigenvalues aremoved to the complex plane.
Basis set: 1826 Dirac spinorsDots – the computed eigenvalues; Lines – limits of continua
140
PERTURBATIONAL METHODS
141
CLASSICAL METHODSPerturbational methods are applicable if the Hamiltonian may be split ontothe unperturbed part H0 and a perturbation H ′:
H = H0 + H ′
so that solutions of the eigenvalue problem of H0 are known.Solutions of the eigenvalue problem of H may then be expanded into apower series of the perturbation using the solutions of the unperturbedproblem.
• Non-relativistic:– Brillouin-Wigner– Rayleigh-Schrödinger
• Relativistic– Direct perturbation method
142
Brillouin-Wigner theory
Let(H0 + H ′)|Ψ〉 = E|Ψ〉,
H0|Ψ0〉 = E0|Ψ0〉.and
〈Ψ0|Ψ0〉 = 1, 〈Ψ|Ψ0〉 = 1
(the intermediate normalization).Then E = E0 + 〈Ψ0|H ′|Ψ〉and
|Ψ〉 = |Ψ0〉 +1
E − H0
P H ′|Ψ〉,
where P = I − |Ψ0〉〈Ψ0|projects onto the space orthogonal to |Ψ0〉.
143
The last two equations may be iterated.The result:
E = E0 + 〈Ψ0|H ′|Ψ0〉 + 〈Ψ0|H ′ P
E − H0
H ′|Ψ0〉 + · · · .
is known as the Brillouin-Wigner expansion.Note:On both left- and right-hand-side of the perturbational expansion appearsthe exact energy E.Therefore neither E nor |Ψ〉 is here expanded terms of powers of theperturbation operator.The method, when restricted to a given order and applied to an N -electronproblem, is not size extensive, i.e. its N -behaviour is incorrect.
144
Rayleigh-Schrödinger theory
Let(H0 + H ′)|Ψ〉 = E|Ψ〉,
H0|Ψ0〉 = E0|Ψ0〉.and
〈Ψ0|Ψ0〉 = 1, 〈Ψ|Ψ0〉 = 1
(the intermediate normalization).
ThenE = E0 + 〈Ψ0|H ′|Ψ〉
and|Ψ〉 = |Ψ0〉 +
1
E0 − H0
P (H ′ −E +E0)|Ψ〉.
145
The energy expansion:
E = E0 + 〈Ψ0|H ′|Ψ0〉 + 〈Ψ0|H ′ P
E0 − H0
H ′|Ψ0〉 + · · · .
The operator
R0 =P
E0 − H0
is called the reduced resolvent.
In the basis of eigenvectors of H0 it may be represented as
R0 =∑
i6=1
|Ψ0i 〉〈Ψ0
i |E
(i)0 −E0
1
,
where Ψ01 ≡ Ψ0 and E0
1 ≡ E0.
146
In the third term of the Rayleigh-Schrödinger expansion appears theunperturbed energy E0, while in the corresponding term of theBrillouin-Wigner expansion - the exact one, E. The higher terms of theRayleigh-Schrödinger expansion are entirely different from the ones of theBrillouin-Wigner ones. Their classification may be conveniently performedby means of diagrammatic techniques (the Goldstone and the Hugenholtzdiagrams are the most commonly used).
In the case of N electron system the Rayleigh-Schrödinger approach maybe formulated in such a way that its N -dependence is correct in every orderof approximation (the formulation is size-consistent).
The size-consistent version of the Rayleigh-Schrödinger perturbationmethod is referred to as the many-body perturbation theory (MBPT).
147
Direct perturbation theory
The Dirac equation for a stationary state of an electron in an externalpotential V (the energy scale is shifted by by mc2, i.e. E −mc2 → E):
V −E, c(σ · p)
c (σ · p), V −E − 2mc2
ΨL
ΨS
= 0.
The unperturbed problem:
V − E, c(σ · p)
c (σ · p), −2mc2
ΨL
ΨS
= 0.
From here ΨS =(σ · p)
2mcΨL and
[
(σ · p)2
2m+ V
]
ΨL = EΨL
148
Dirac equation:[
H0 − β+E + β−(V −E)]
ψ = 0,
β+ =
1 0
0 0
, β− =
0 0
0 1
,
ψ =
ψL
ψS
=
ψL
0
+
0
ψS
= ψ+ + ψ−
H0 = c (α · p) + β+V − 2β−mc2,
four-component spinors ψ+ and ψ− are projections of ψ:
ψ+ = β+ψ, ψ− = β−ψ.
(H0 − β+E)ψ = 0 − the unperturbed problem,H ′ = β−(V −E) − the perturbation.
149
The direct perturbation method is appropriate for relativistic perturbationalcalculations.
It is based on the partition of the Dirac equation which defines theunperturbed problem in the same Hilbert space as the exact one.
The perturbation parameter is equal to the square of the fine-structureconstant α. Therefore the k-th order of the perturbation corresponds to aterm proportional to α2k.
150
RELATIVISTIC TWO-COMPONENTMETHODS
151
THE ELIMINATION OF THE SMALL COMPONENTS
The simplest and the most commonly used approach to an approximatedescribing relativistic effects is the method in which the small componentsof the wavefunctions are expressed by the large ones using the Diracequation and then eliminated. As a result a relativistic description based ona two-component wavefunction is obtained. By expanding the resultingequation into a power series of (E − V )/mc2 one obtains the well knownPauli approximation. The resulting Hamiltonian is strongly singular and therelativistic terms can only be used as the first order perturbations. Worksdirected towards development of a two-component relativistic theory free ofthe deficiencies of the Pauli approach resulted in formulation of numerousquasirelativistic theories.
152
THE PAULI METHOD
The Dirac equation
V ψL + c (σ · p)ψS = E ψL
c (σ · p)ψL + (V − 2mc2)ψS = E ψS,
Eliminating the small component gives
ψS =1
2mc
(
1 +E − V
2mc2
)−1
(σ · p)ψL ≡ XψL,
andH lcψL = E ψL.
H lc =1
2m(σ · p)
(
1 +E − V
2mc2
)−1
(σ · p) + V
153
QUASIRELATIVISTIC FORMULATION
In order to convert this formalism into a quasirelativistic one in which onlytwo-component wavefunctions appear one has to renormalize ψL:
1 = 〈ψ|ψ〉 = 〈ψL|ψL〉 + 〈ψS|ψS〉 = 〈ψL|I + X†X|ψL〉 = 〈ψqr|ψqr〉,
where the quasirelativistic wavefunction and Hamiltonian are defined as
ψqr = OψL,
Hqr = OH lcO−1.
with
O =(
I + X†X)
1
2
154
PAULI HAMILTONIAN
Expanding H lc and O in (E − V )/mc2 on obtains the Pauli Hamiltonian:
HP =p2
2m+ V − p4
8m3c2− s · (∇V × p)
2m2c2− ~
2
8m2c24V.
The first two terms give the Schrödinger Hamiltonian. The next terms: theeffect of change of the electron mass with velocity, the spin-orbitinteraction, a correction due to Zitterbewegung (the Darwin correction). Allthese terms are highly singular and the eigenvalue problem of HP does nothave any square-integrable solutions. However this operator may be (andhas been) used to estimate the relativistic corrections as the first-orderperturbations. By properly restricting its domain one can also use thisoperator in some variational approaches. However one should rememberthat for a Coulomb potential there exists an area around the nucleus inwhich the expansion is invalid, because
∣
∣(E − V )/mc2∣
∣ > 1. On the otherhand, in this very area the relativistic effects are the most important.
155
REGULAR APPROXIMATIONS: ZORA, FORA, etc
There exist many other ways of reducing the Dirac equation to atwo-component form. A special attention deserves a recent approach by vanLenthe. If we write
H lc = (σ · p)c2
2mc2 − V
(
1 +E
2mc2 − V
)−1
(σ · p) + V,
then H lc may be expanded in E/(2mc2 − V ). This expansion is justifiedfor Coulomb-type potentials also near the singularity. One gets
H lc = V + (σ · p)c2
2mc2 − V(σ · p)
− (σ · p)
(
c2
2mc2 − V
) (
E
2mc2 − V
)
(σ · p) + · · ·
156
In a similar way one can expand the normalization operator O. In effect, inthe lowest order of the expansion, one obtains the so called zeroth-orderregular approximated (ZORA) Hamiltonian:
Hzora = (σ · p)c2
2mc2 − V(σ · p) + V.
ZORA Hamiltonian is regular, bounded from below and can be used invariational calculations without any special restrictions.
157
In the higher order one gets the Hamiltonian in the first-order regularapproximation (FORA):
H fora = Hzora − 1
2
(σ · p)c2
2mc2 − V(σ · p), Hzora
This Hamiltonian, similarly as the Pauli one, does not havesquare-integrable eigenfunctions and in not limited from below. Therefore itcannot be used in variational calculations, unless the space of the trialfunction is properly constrained.
158
FOLDY-WOUTHUYSEN TRANSFORMATION
A large family of two-component relativistic equations may be obtained byusing different modifications of unitary transformations which decouple thelarge and small components in the Dirac equation. The best known are theFoldy-Wouthuysen and Douglas-Kroll transformations.
Foldy and Wouthuysen transformation: a systematic procedure fordecoupling the large and the small component parts of the Dirac equation toany fixed order in the fine structure constant.
159
Let us consider a unitary transformation
H fw = U HD U†
with
U =(
I + W †W)−1/2
I W †
−W I
This transformation brings the Dirac Hamiltonian to a block-diagonal form(I.E. decouples the large and the small components) if W fulfills thefollowing condition:
W =1
2mc
[
(σ · p) − W (σ · p)W]
+1
2mc2
[
V, W]
.
If this equation is solved iteratively then consecutive iterations givecorrections of consecutive orders in α. One can also represent this equationalgebraically in a model space and find matrix elements of W .
160
The "large component" part of the transformed equation:
H fwψfw = Eψfw,
where
H fw = W † V W − V + c[
(σ · p)W + W †(σ · p)]
− 2mc2
is bounded from below and its spectrum is the same as the positive-energyspectrum of HD.
The expansion of W into a power series of α in the lowest order leads to thePauli Hamiltonian.
161
DOUGLAS-KROLL TRANSFORMATION
In the case of a free particle (V = 0) the Foldy-Wouthuysen transformationmay be expressed in a finite form:
W0 =(σ · p)
mc
[
1 +
√
1 +(σ · p)2
(mc)2
]−1
If V 6= 0 then in the W0 transformed Dirac Hamiltonian the term couplinglarge and small components is proportional to
[
V, W]
. From here one candesign a decoupling procedure in which consecutive orders are proportionalto powers of V . This procedure is known as the Douglas-Krolltransformation.
162
ONE-ELECTRON PAULI HAMILTONIAN
π =(
p− e
cA
)
s =~
2σ
HP(1) = HSch(1) + hP(1)
Unperturbed Hamiltonian – non-relativistic (Schrödinger):
HSch =π2
2m+ V,
Perturbation – Pauli corrections:
hP = − e
mc
(
s · B)
− π4
8m3c2− s · (∇V × π)
2m2c2− ~
2
8m2c2∇(∇V ).
Two-electron Pauli-Breit Hamiltonian:
HBP(1, 2) = HSch(1, 2) + hP(1) + hP(2) + V B12(1, 2)
163
LIÉNARD-WIECHERT POTENTIALS
Potentials generated at r1 by a charge e located at r2 and moving withvelocity v2 (Alfred-Marie Liénard 1898, Emil Wiechert 1900)
V lw2 (r1) =
e
r12
[
1 +(v2 · r12)
cr12
]−1
=e
r12
[
1 − (v2 · r12)
cr12
]
+O
[
(v
c
)2]
andAlw
2 (r1) =v2
c2V lw
2 (r1) =e
r12
v2
c2+O
[
(v
c
)2]
164
Assuming that in a two-electron system V and A are due to the nucleus andthe second electron we get the classical (non-quantum) interaction energy oftwo electrons:
V12 =e2
r12
[
1 − v1 · v2
2c2− (v1 · r12)(v2 · r12)
2c2r122
]
+O
[
(v
c
)4]
,
According to the correspondence principle, we substitute:
v ⇒ c α.
The resulting quaantum-mechanical interacion operator (Breit correction):
V B12 =
e2
r12
[
1 − (α1 · α2)
2− (α1 · r12)(α2 · r12)
2r122
]
.
The cosecutive term describe, respectively, the instantenous(non-relativistic) interaction, correction due to two-electron magneticinteraction and correction due to retardation resulting from the finitevelocity of propagation of the interaction.
165
A relativistic Hamiltonian in which the two-electron terms areapproximated by e2/r12 is known as the Dirac-Coulomb Hamiltonian.If the magnetic and retardation corrections (Breit interactions) are included,then the Hamiltonian is called the Dirac-Breit Hamiltonian.
Since the interaction potential is correct up to (v/c)2 terms, the formulationbased on this Hamiltonian is only approximately Lorentz invariant.
The elimination of the small components from the Dirac-Breit equationleads to two-electron relativistic Breit-Pauli corrections:
• orbit-orbit interaction,
• spin-spin interaction,
• spin-other orbit interaction,
• two-electron Darwin correction.
166
BREIT-PAULI HAMILTONIAN
HBP(1, 2) = HSch(1, 2) + hP(1) + hP(2)
+ hoo(1, 2) + hd2(1, 2) + hso(1, 2) + hss(1, 2)
where
• HSch(1, 2) is the two-electron Schrödinger Hamiltonian,
• hP(j) =p
4j
8m3c2−
sj · (∇Vj × pj)
2m2c2− ~
2
8m2c24Vj , j = 1, 2
where Vj ≡ V (j), pj ≡ p(j), etc, are one-electron Pauli corrections,
• The remaining two-electron terms – Breit-Pauli corrections
167
BREIT-PAULI CORRECTIONS
• Orbit-orbit interaction:
hoo(1, 2) = − 1
2m2c2
[
(p1 · p2)
r12+
r12 · (r12 · p1) p2
r312
]
• Two-electron Darwin correction:
hd2(1, 2) =1
4m2c2(41 + 42)
1
r12
• Two-electron spin-orbit coupling:
hso(1, 2) =1
m2c2
(
[r12 × p2] · s1
r312+
[r21 × p1] · s2
r312
)
• Spin-spin interaction:
hss(1, 2) =1
m2c2
[
(s1 · s2)
r312− (s1 · r12)(s2 · r12)
r512− 8π
3(s1 · s2)δ(r12)
]
168