Table Of Content1
The Laws of Quantum Mechanics
Introduction
The statistical character of physical states and Hamilton’s formalism of classical
mechanics form a fundament of quantum theory. We begin our discussion from the
description of states with an example of the phenomenon of the polarization of
light.Letthelightwaves,togetherwithseparatephotonsofthislightbeam,possess
a particular polarization. We pass such a beam through a plate of tourmaline; on
passing this crystal through unpolarized light, on the back of the plate we discover
waves having the electric-field vector parallel to the optic axis of the crystal. If the
electric-field vector in our beam is perpendicular to the optic axis, then as a result
the entire absorption becomes observable. If the light is polarized at angle α to the
axis,onlyafractionequaltocos2αfromtheinitialbeampassesthroughthecrystal.
From the point of view of classical optics, these facts are trivial. The question
arises, however, in the case of separate photons, whether each photon is polarized
at angle α to the axis. The answer is simple: if we pass photons one by one from
our beam, we discover that one photon is entirely transmitted, whereas another is
entirely absorbed; the probability of observing a particular photon from the beam
is equal to cos2α, and the probability of its absorption is sin2α. As a principle of
quantum theory, one might thus apply the next device. Each photon can be repre-
sented in a state with polarization that is parallel to the axis or perpendicular
to the axis. A particular superposition of these states produces the necessary state
for the beam with polarization. In the result of an experiment, photons jump from
anuncertainstatetoastatewithaconcretepolarization(cid:1)thosethatpassandthose
thatbecomeabsorbed.
The same condition occurs for the interference of photons. If an initial beam
becomes split into two components, each photon with a particular weight enters
partly into each component beam. As we have observed, however, that a particular
photon is entirelyin one component, it is at once precluded from being in the other
component beam. A priori we may characterize a physical system with states of a
particular number that have a statistical character. Quantum mechanics requires
that each photon interferes only with itself during the interference of the two com-
ponents. An electromagnetic wave and a photon are two descriptions of light. The
same condition, as we see further, applies for physical particles with which one
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2 QuantumTheoryofAnharmonicEffectsinMolecules
might also associate individual wave fields. In this sense, the individuality empha-
sizesthestabilityofallmaterial(cid:1)electrons,protonsandsoon.
Let us generalize the facts above. What should we understand about the state of
the system (cid:1) a motion, a rest, an interaction? These concepts exist in classical
mechanics. Something similar holds in quantum mechanics, but it is less determi-
nate. What is the meaning therein? If the system is presumably in one state, we
must consider that it is partly in another state, so that its real state represents the
superposition of all possible states that have non-zero probabilities. As a classical
analogue of the expression of this principle, one might apply a wave packet, for
which a complicated wave motion is resolvable into Fourier components; through
this analogy, quantum mechanics is generally called wave mechanics. As a result,
this principal idea yields a new theory (cid:1) a theory of probabilities or amplitudes of
physicalstates.
For states in quantum mechanics, as far as practicable, we use Dirac’s notation.
In this case, to each state we ascribe a ket vector j?i, inside of which might
appear letters, words, numbers and other symbols. Keep in mind that in classical
mechanics a vector is also applied to describe motion, but it is Euclidian there,
whereas here Hilbert’s type prevails. Vectors jA i, jA i,... that belong to a Hilbert
1 2
space might be added together and might be multiplied by arbitrary complex num-
bersc ,c ,...,asaresultofwhichweobtainanothervector
1 2
jAi5c jA i1c jA i1?:
1 1 2 2
This vector, which is expressible in a form of linear combination of others, is line-
arly dependent on them. Like a Euclidean space, the systems of linearly indepen-
dent vectors are therefore of special interest. Each physical state of interest is
expressible as an expansion in terms of these system vectors. Conversely, any such
statemightdescribeaconcretestateofaphysicalsystem.Itisimportantthatapro-
cedure of multiplying the vector by the number gives no new state; for instance,
jAi and 2jAi describe one and the same state. The principle of superposition in
quantum mechanics has an important significance; considering the concrete physi-
calproblems,wegenerallyappealtothispostulate.
Let us now consider Hamilton’s formalism, which we will review briefly with
regard to methods of classical mechanics. It is remarkable that the equations of the
old theory can be borrowed with a somewhat altered meaning to construct the new
mechanics. Lagrange’s function of a mechanical system represents a function of
generalizedcoordinatesq,theirtemporalderivativesq_ (generalized velocities)and
i i
timet:
=5=ðq;q_;tÞ:
i i
By definition, the momentum is p 5@==@q_ and the force is F 5@=/@q. The
i i i i
energyofthesystemequals
X
H5 pq_ 2=:
i i
i
TheLawsofQuantumMechanics 3
Ð
Lagrange’s function = is such that integral t2=dt has a minimum; this condition
t1
leadstotheEuler(cid:1)Lagrangeequation
(cid:2) (cid:3) (cid:2) (cid:3)
d @= @= dp
2 50 i 2F 50
dt @q_ @q dt i
i i
if@=/@q 50,p isaconstantofmotionandq isacycliccoordinate.
i i i
There exists, however, an alternative method to describe a mechanical system
that employs the language of coordinates and momenta. To convert to variables q
i
andp,weapplyaLegendretransformation:
i
0 1
dH5d(cid:4)P pq_ 2=(cid:5)5 2@=1 P @2@=dq 2@=dq_ 1p dq_ 1q_ dpA
i i i i @q i @q_ i i i i i
P i P i
5 2@=2 p_ dq 1 q_ dp:
i i i i i i
Consequently,
@H @H @= @H @=
5q_; 52 52F 52p_; 52 :
@p i @q @q i i @t @t
i i i
Here, H is Hamilton’s function; this description is called a Hamiltonian formalism.
One sees that this method possesses great symmetry. Moreover, it is convenient
that H represents the total energy of the system. For instance, for interacting parti-
cles,theenergycompriseskineticandpotentialcontributions:
X
p2
H5 i 1Vðq ;q ;...Þ;
2m 1 2
i i
in which m is the mass of particle i and V is the potential energy of interaction of
i
theparticles.Inthiscase,Lagrange’sfunctionhasaform
Xmq_2
=5 i i 2Vðq ;q ;...Þ:
2 1 2
i
In Hamilton’s formalism, physical quantity f is represented as a function of the
coordinates, momenta and time: f(q,p,t). Its total derivative with respect to time
i i
hasaform
df @f X@f @q X@f @p @f X@f @H X@f @H @f (cid:6) (cid:7)
5 1 i 1 i 5 1 2 (cid:3) 1 f;H :
dt @t @q @t @p @t @t @q @p @p @q @t
i i i i i i i i i i
4 QuantumTheoryofAnharmonicEffectsinMolecules
Here,
(cid:2) (cid:3)
(cid:6) (cid:7) X @f @H @f @H
f;H 5 2
@q @p @p @q
i i i i i
is a Poisson bracket. For instance, p_ 5fp;Hg and q_ 5fq;Hg: Poisson brackets
i i i i
play an important role not only in classical mechanics but also in quantum theory;
theythereforedeservespecial attention.
As an example, we consider the Hamiltonian of a particle in an external electro-
magnetic field, which is determined by vector potential A and scalar potential U.
The energy of this particle with charge e0 and velocity v in such a field is given
withthisexpression
e0
e0U2 A(cid:4)v;
c
inwhichcisthespeedoflight.ForLagrange’sfunction,wethushave
mv2 e0
=5 2e0U1 A(cid:4)v:
2 c
Themomentumis
@= e0
p5 5mv1 A:
@v c
Bydefinition,wewriteexpressionforHamiltonianH:
e0 mv2
p(cid:4)v2=5mv21 A(cid:4)v2=5 1e0U:
c 2
However,v!(p2e0A/c)/m,sothatfinally
(cid:2) (cid:3)
1 e0 2
H5 p2 A 1e0U:
2m c
One sees that, to proceed from the Hamiltonian of the freely moving particle to the
Hamiltonian describing the motion in the external field, one must perform a
replacement p!p2e0A/c and add a trivial static energy e0U. Elsewhere in what
follows, classical mechanics in Hamilton’s form becomes the initial point of our
researchandpromptsthecorrect formofinitialequations.
Observables and Variables
To describe states in quantum mechanics, we introduced the concept of a vector.
This definition is highly abstract; one must understand how to work with it. An
TheLawsofQuantumMechanics 5
experiment produces numerical values of physical quantities, which are involved in
classical theory. This concept fails to be usable in quantum mechanics. We cannot
directlyoperate with conventional numbers, in brief, c-numbers, oremphasize their
triviality. The language of quantum mechanics involves q-numbers. If the coordi-
nate and momentum are c-numbers in classical physics, in quantum physics they
become q-numbers. The new numbers represent a new set of dynamical variables,
namely those that we must treat. These variables are just determined in a space of
abstract vectors (cid:1) vectors of a Hilbert space. Through the action of q-numbers,
such as some operation involving quantity O of q-type on some vector jϕi, we
obtain another vector jψi. One might state that, in the simplest case, q-numbers are
convenient operators, and questions of quantum mechanics consist of extracting
observablec-numbersfromatheoryofdynamicalvariablesofq-type.
Letusdiscussthemathematicalbasisofquantummechanics.
In a separable Hilbert space, vectors jϕi, jψi,... form a countably infinite
sequence. For any pair of jϕi and jψi, the sum jϕi1jψi is determined, which is
alsoavectorandpossessesthepropertiescommutativityandassociativity:
jϕi1jψi5jψi1jϕi and jϕi1ðjψi1jχiÞ5ðjϕi1jψiÞ1jχi:
The multiplication of vector jϕi by complex number c is defined; product cjϕi
representsthevectorandhasthepropertydistributivity:
cðjϕi1jψiÞ5cjϕi1cjψi; ðc1dÞjϕi5cjϕi1djϕi:
Moreover,
1(cid:4)jϕi5jϕi and 0(cid:4)jϕi50:
Anytwovectorsjϕiandjψipossessascalarproduct
hϕjψi;
in which hϕj is a so-called bra vector that is the complex conjugate of jϕi.
Obviously, hϕjϕi$0, and hϕjϕi50 only in the case when jϕi50. If jψi repre-
sentsthesumjθi1jχi,
hϕjψi5hϕjθi1hϕjχi;
ifjψiequalsvectorjχithatismultipliedbynumberc,
hϕjψi5chϕjχi:
Finally,hϕjψi(cid:5)5hψjϕi:
For the vectors in a Hilbert space, these properties are general. As an example,
we consider a case in which as vectors jϕi, jψi,... we have ordinary functions
6 QuantumTheoryofAnharmonicEffectsinMolecules
ϕ(x), ψ(x),..., which are determined in manifold G. It is generally convenient to
apply this representation to solve concrete problems of quantum mechanics. The
propertiesofvectors,inthiscase,areperformedinsuchamanner:
ϕðxÞ1ψðxÞ(cid:1)sumϕðxÞandψðxÞinG;
cϕðxÞ(cid:1)multiplicationbyanumber;
Ð
hϕjψi5 ϕ(cid:5)ðxÞψðxÞdx(cid:1)scalarproduct:
G
Foreachvector,onemightintropdffiuffifficffiffieffiffiffiffitffihffiffiedefinitionoflengthornormthat,inthe
sense of a number, equals jjϕjj5 hϕjϕi: If jjϕjj51; vector jϕi is normalized. If
the scalar product of two vectors jϕi and jψi equals zero, i.e. hϕjψi50, these vec-
tors are orthogonal. The set of orthonormal vectors might represent a complete
basis. Considering the physical principle of superposition, we have already men-
tionedthe necessityofthe condition completeness for states. Asequence of vectors
jϕi is mathematically complete if any vector jΦi in a certain space is expressible
i
inaformoflinearcombination:
X
jΦi5 cjϕi:
i i
i
VectorsPjϕii are linearly independent only in the case in which there is no relation
oftype cjϕi50;eliminatingthecasec 50.
i i i i
Withtheaidofaconvenientoperator,onemightconvertonevectorintoanother.
pffiffiffiffiffiffi
For instance, square root ... and differentiation dð...Þ=dx are simple operators.
Not all operators, however,represent a physical interest,and amathematical opera-
tion should not be associated with a dynamical variable; only a few of them are
applicable in physics. We imply here linear operators O that play an exceptional
roleinquantummechanics.QuantityOimpliessomeruleaccordingtowhichavec-
tor,e.g.jϕi,transformsintojψi.Linearitymeansthat
Oðajϕi1bjψiÞ5aOjϕi1bOjψi;
in which a and b are c-numbers. As simple examples of linear operations, one
might undertake multiplication by an arbitrary coordinate function (cid:1) F(x)ϕ(x), or
differentiation (cid:1) dnϕ(x)/dxn. Exponentiation of a vector to some power as ϕn(x) is,
however,notalinearoperation.
Let us enumerate the general properties of linear operators. For any pair of
operatorsAandB,sumA1Bisdefined:
ðA1BÞjϕi5Ajϕi1Bjϕi;
suchasumpossessespropertiescommutativityandassociativity:
ðA1BÞjϕi5ðB1AÞjϕiandAjϕi1ðB1CÞjϕi5ðA1BÞjϕi1Cjϕi:
TheLawsofQuantumMechanics 7
ThemultiplicationofoperatorAbycomplexnumbercisdetermined:
ðc(cid:4)AÞjϕi5cðAjϕiÞ:
ThereisdeterminedtheproductofoperatorsA(cid:4)Bwithpropertiesdistributivity
AðB1CÞjϕi5ABjϕi1ACjϕi;
associativity
ðABÞjϕi5AðBjϕiÞ
and,generally,non-commutativity
ABjϕi6¼BAjϕi:
Theprincipalrolebelongstothecommutatoroftwooperators
½A;B(cid:6)5AB2BA52½B;A(cid:6);
obviously,[A,A]50.Forinstance,ifvectorjϕiisfunctionϕofvariablex,
(cid:9) (cid:10) d dϕ
d=dx;x ϕðxÞ5 ðxϕÞ2x 5ϕðxÞ;
dx dx
wherefrom
½d=dx;x(cid:6)51:
If equation Ajϕi5jψi is solvable with regard to jϕi, such that there exists a rela-
tion of type jϕi5Bjψi, operator B, which is equal, by definition, to A21, is called
reciprocaltoA.So
ðA21AÞjϕi5A21ðAjϕiÞ5A21jψi5jϕi;
i.e.A21A51,andalsoAA2151,hence[A,A21]50.
Theproductofthesameoperatorsyieldsaconceptofpowernofanoperator:
Anjϕi5AðAðA...ðAjϕiÞ...ÞÞ;
inparticular,ifA5d/dx,thenAn5dn/dxn.Withtheaidofanexponentiationopera-
tion,onemightdeterminefunctionfofanoperator:
XN fðiÞð0Þ
fðAÞ5 Ai:
i!
i50
8 QuantumTheoryofAnharmonicEffectsinMolecules
Forinstance,weconsiderfunction
XN αn dn
eαðd=dxÞ5 ;
n!dxn
n50
onactingonϕ(x),wehave
XN αndnϕ
eαðd=dxÞ ϕðxÞ5 5ϕðx1αÞ:
n! dxn
n50
Operatoreα(d/dx)thusshiftstheargumentoffunctionϕ(x)byquantityα.
Furthermore,ifthereexistsanequation
Ajϕi5ajϕi;
in which a is a c-number, quantities a represent eigenvalues of operator A and jϕi
are its eigenfunctions. Let us draw an important conclusion. Suppose that A and B
arecommutativeoperators,then
BðAjϕiÞ5BðajϕiÞ and AðBjϕiÞ5aðBjϕiÞ:
One sees that vector Bjϕi is an eigenvector of operator A, and there must exist a
relationoftype
Bjϕi5bjϕi;
in which b is a c-number. Thus, if [A,B]50, A and B have simultaneously a com-
pletesystemofeigenvectors(eigenfunctions).
LinearoperatorAinsomebasiscanberepresentedwithamatrix.Thiscondition
is easy to understand if we suggest that we have a complete system of vectors jϕi,
i
andarbitraryvectorjψiisexpressibleinaformofthisexpansion
X
jψi5 cjϕi;
i i
i
inwhichc arecoefficients.OnactionbyoperatorAonjψi,wehave
i
X
Ajψi5 cAjϕi;
i i
i
wherefrom
X X
hϕ jAjψi5 chϕ jAjϕi(cid:3) cA :
k i k i i ki
i i
TheLawsofQuantumMechanics 9
The complete set of matrix elements A forms a matrix representing the linear
ki
operator. To coefficients c one might ascribe a physical meaning of amplitudes of
i
the states; then jcj2 is the probability of state jϕi, and a sum of all probabilities
i i
equalsunity:
X
hψjψi5 jcj251:
i
i
Forinstance,unitoperatorIisdefinedwithequationIjϕi5jϕiandmightberepre-
sentedwithunitmatrixδ ,inwhichδ 51andδ 50fork6¼i.
ki kk ki
Weconsideranexpressionfordiagonalmatrixelement,
X
hψjAjψi5 c(cid:5)c hϕjAjϕ i:
i k i k
ik
Ifjϕianda areeigenvectorsandeigenvaluesofoperatorA,
i i
X
hψjAjψi5 jcj2a
i i
i
that represents the mathematical expectation value of quantity A. In state jψi, the
expectationvalueofadynamicalvariable(operator)isthusdeterminedbythediag-
onalmatrixelement
X
hAi5hψjAjψi5ðinparticularÞ jcj2a:
i i
i
Inquantumtheory,theexpectationvaluesbelongtoaclassofobservablequantities.
Accordingtoadefinition,
hϕjAjψi5hψjA1jϕi(cid:5);
1
to every linear operator A one might determine Hermitian conjugate operator A .
Oneseesthat
ðA1Þ15A;ðA1BÞ15A11B1 and ðABÞ15B1A1:
IfA5A1,Aiscalledtheself-adjointorHermitianoperator;inthiscase,
hϕjAjψi5hψjAjϕi(cid:5):
Hermitian operators play an important role in quantum mechanics. For instance, we
consideradynamicalvariablethatisdescribedwithoperatorA.Supposefurthermore
that, in some state jϕi, our variable equals a certain c-number a; then Ajϕi5ajϕi,
and,consequently,hϕjAjϕi5a.IfoperatorAisHermitian,
hϕjAjϕi5hϕjA1jϕi(cid:5)5hϕjAjϕi(cid:5);
10 QuantumTheoryofAnharmonicEffectsinMolecules
thus, a5a(cid:5), and the eigenvalues of A are real numbers. In physics, the dynamical
variablesinsomearbitrarystatesmusthaveonlyrealvalues.Theconditionofhermi-
tivityis,therefore,generallynecessarytoascribesomeoperatortoaphysicalquantity.
Another important consequence deserves attention. Let a and a0 be eigenvalues
ofoperatorAwithcorrespondingvectorsjϕiandjϕ0i.Then
Ajϕi5ajϕi and hϕ0jAjϕi5ahϕ0jϕi:
Ontheotherside,ifAisaHermitianoperator,
hϕ0jA5a0hϕ0j and hϕ0jAjϕi5a0hϕ0jϕi:
Weseethat
ða2a0Þhϕ0jϕi50:
Thus, if a6¼a0, hϕ0jϕi50; two eigenvectors of a Hermitian dynamical variable
belongingtovariouseigenvaluesareorthogonal.
The Conditions of Quantum Theory
Comparing dynamical variables with linear operators, we understand that one might
scarcelysucceedtopreserveintheiroriginalformtheequationsofclassicalmechan-
ics.Thoseoperatorsgenerallyfailtoconformtothecommutativeconditions.Because
ofthisobstacle,wecannotbuildquantumtheoryusingonlyexperimentalrelationsfor
physicalquantities:wemustinvokeadditionalrelationsonq-numbers(cid:1)thequantum
conditions.Theseconditionsaregenerallyexpressiblethroughthecommutatorsofthe
particularvariables.Foreverysuchpair,thereexistsacertaincommutator.Thedeter-
minationofallnecessarycommutatorsisanindispensableconditionapriori,without
whichitmightbeimpossibletofindasolution.Thereisnowaytowriteallconditions
in a unified manner; in some way, they are individual. One might, however, reveal
somesimilaritieswithclassicaltheory.Itturnsoutthatthepropertiesofcommutators
closelyresemblethoseofclassicalPoissonbrackets.
We consider a Poisson bracket for variables A and B, which are functions of
canonicalcoordinatesq andmomentap,
i i
(cid:2) (cid:3)
X @A@B @A @B
fA;Bg5 2 :
@q @p @p @q
i i i i i
Ifonevariableisconstantnumberc,
fA;cg50:
If we exchange quantities A and B within the braces, the sign is automatically
reversed:
fA;Bg52fB;Ag:
