. , , ' at I (a", P.)'" I -= I P;';, ã
I, t I
This implies (1.24).
The equations (1.25) are infinite in number when one deals with continuous functions (k == 1,2 ••• ). They are important because they define the matrix (a'l) aa A which represents the linear operator A in the coordinate system
.p,. The matrices which we have introduced this way to associate with every
lin~ operator and which we will call representation matrices depend essentially on the fundamental basis functions. Their components can be obtalined, according to (1.25), by forming the scalar products
<l/I, ã Al/I,) - f l/IiAl/Itd't = t f l/Itl/lJaJtd't = all:; (1.26)
(1.25) and (1.26) are constantly used in quantum theory.
The equation (1.25) may still be considered from another point of view:
as a 'CUansformatioD of axes" (compare (1.4a) and (1.2ằ). Particularly if we span the function space by another complete set of orthogonal functions C{J,
we will have by developing each epic in a series of functions t/I,:
€p" == L , .;,u"'. (1.27)
Hence if we introduce a new set of orthogonal functions the linear operators will undergo a similarity transformation (see (1.7)). Since the transformation matrix U == ("It) conserves the orthogonality of the basis functions it is a unitary matrix. One can verify that this satisfies (1.9).
Ch. 1, § 5] OPERATORS 21 The arguments of § 1.6 are still valid under the condition that we give the infinite matrices, with which we are dealing here, convenient convergence
properties. We will obtain for the expression of the transformation A in the coordil1ate system lPk'
A' = U-1AU. (1.27a)
5.2. BILINEAR FORMS. HERMITIAN OPERATORS
In n-dimensional space the scalar product (x -Ay) is a bilinear form of the components of the two vectors x and y (compare (1.10ằ).. We will establish the same product (",. Alp) supposing that our functions are developed according to (1.21):
Taking into account (1~25) we obtain
(1/1 • A(p) = f t/I* Aq>d-c = L pil/ljl/!,alk y"d-c == L Pi aUc"lh
i~ kf I i, k
(1.28)
which is an infinite bilinear form of Fourier coefficients of the .. components of the two functions 1/1 and qJ in the basis system y, ,- Since one defines an Hermitian operator in an n-dimensional space by the condition
(1/1 'A!f') = (Al/!' qằ = L f l/!:atiP'iYIl/!,d-r: = LYka~M~ (1.29)
11:, i, i I, t
from which, according to (1.28)
(1.29a) thus, the corresponding matrix is Hermitian too.
5 .. 3. REDUCTION OF AN HERMITIAN OPERATOR TO ITS MAIN AXES
Let us reason again by analogy with the n-dimensional space: we want to
find a systenl of basis vectors '" 1 , "'2 ... to span the function space such that eacl1 vector A ifr is "parallel" to t/I, or more precisely that the function Ay", the transforrnation of 1/1, by the operator A, be equal to "', multiplied by a
constant:
(1.30) The functions found in this way are the eigen/unctio1t8 or eigenvectors of the operator .t\; the numerical values eli its eigenvalues.
22 VECTOR SPACES - UNITARY GEOMETRY {Ch~ 1, § S
The solution of this problem for a space with a finite number of dimensions is relatively simplo ,at least in principle (§ 3) and can be done by purely algebraic methods. In function space, on the contrary, complications and considerable difficulties appear and rigorous discussion of the equation (1.30)1 has not been completed by the mathematicians.
The problem of finding (%, is algebraic, differential, or integral according to whether the operator A is algebraic, differential, or integral. The Schr6dinger equation (2.15) is a particular case: The Hamilton operator to which it refers is a differential operator. Discussion of physical problems presented by quantum. mechanics has lead to a better understanding of equation (1.30) since the work of Hilbert ..
Depending on the nature of the operator A and on the size of the domain
D, which restricts the variables on which the function l/I depends, we are dealing with two different cases.
1. The easiest case arises when the square summable solutions of equation (1.30) form a denumerable set, corresponding to a discontinuous and denumerable set of eigenValues a;: ttl' el2, . • ... There may correspond several ei~nfunctions till to certain constants ex" but this multiplicity is
alwayS-supposed to be finite. This is expressed by saying that the spectrum of eigenvalues of the eigenfunctions is discontinuous or discrete ..
This case generalizes in a most direct way the results obtained in n-
dime~siona1 space. It occurs most often if the domain D is finite (as e.g. the case ~f free particles confined to a box), but this condition appears to be neitber necessary nor sufficient. It happens for instanc:e in the quantum the91"Y of a harmonic oscillator that the levels form a discontinuous spectrum although the size of the domain is infinite"
It can be shown in the case of discrete eigenvalues that the eigenfunctions .; , of the operator A form a complete set of orthogonal functions which span the function space completely2. In order to prove the orthogonality of two eiaenfunctions t/I, and J/!" correspondina to two dUI'erent eigenvalues of an Hermitian operator A, we have
Multiplying the first equation by f/lt and the second by JjI f and 8ubstractiDg.
1 One ftDds a buic discussion in 1. v. NaUMANN {193l]~
• See, for example, ltlLBD.T and CoURANT (1930]. Chap .. Sand 6 or P. lvi. Mi.~ a.ad.
H. FuIlJlACH !19S3] p~ 727 and p~ ilSi>
Ch. 1, § S1 OPERATORS 23 and using
we have
(ex, - rl,t)( '" i • VI).) == 0 j~e. (t/I i • l/IlIJ == 0,
If several eigenfunctions, for example t/ll' 1/12' "'3, belong to the same eigenvalue eX we have degeneracy: every linear combination of these three functions is again an eigenfunction beloIlging to the value et:, because the three equations
after multiplication with three arbitrary constants Pi and summation win give
L 3 {JiAVt, = A L PiifJ! == rJ. L Pio/l-
t= 1 , ,
Hence one can always choose three orthogonal linear combinations among them.
In order to make use of the analogy that exists between the function space and a n-dimensional space to the largest extent, we will consider an arbitrary function", having components Pl, P2, .... in a coordinate system of orthogo- nal functions, as in equation (1.21). The scalar product (1/1 a At/I) can be written as a quadratic form in PI according to (1.28)~
(!/I ~ At/!) == L p'taikPk ~
l , l
But if the basis functions are eigenfunctions of the operator A, (1.25) can be replaced by (1.30) and we obtain
Thus the quadratic form (t/I ã At/!) is reduced to a "sum of squaresU "
2. It happens very often and particularly if the domain D is infinite t}1~t
in addition to the discontinuous spectrum we have a continuous spectrum of eigenvalues, and it may even happen that the discontinuous spectrum is not thereat all. The theory of atoms with a central force field furnishes an
example: the spectrum of energy levels is discontinuous up to a certain limit (tlie ionization potential) and becomes continuous above this\> Since the essential character of the orthogonal functions is that they fonD. a denumer-
24 VECTOR SPACES - UNITARY GEOMETRY [eh. 1, § 5
able set, obviously, one does not have the right to apply \v:ithout modification the language of linear vector spaces1•
The development in Fourier series (1.21) is then replaced at least partially (compare (1 .. 32) below) by integrals. These integrals can be considered as the limits of series of orthogonal functions.. This permits us at least to a certain extent to give a sense to the expression of orthogonal functions in the continuous case. It will be sufficient to give two examples. Let us consider first functions of a single variable x defined in an arbitrary finite or infinite domain D and the operator "multiplication by x": (1.30) has now the form
xt/l(x) = exifJ(x),
an equation which has to hold everywhere in the domain D .. It is obviously impossible to construct an analytical function which satisfies this condition, but we will imagine a function '" (J, which is everywhere zero except at the point x = eX and we would consider this as an eigenfunction of the preceding equation. In order that this has a sense one has, moreover, to demand that the integral over the product of this function and an arbitrary function f(x) is not zero. This makes it necessary that the 0/ô becomes infinite for
x = tX. We will take this integral equal to f(a):
I f(x)I/I",(x)dx = /(a).
If we permit the existence of such functions we observe that the spectrum . of ~igenvalues and eigenfunctions of the operator x is continuous in the domain D as the value of r:t, is arbitrary. Our function ",ô{x) coincides with the Dirac delta function lJ(x - ex) .. It is possible to construct analytical expressions that have this function as their limit 2 •
We have obviously
LI/I",(x)I/Iix)dX = 0,
1 The label i in (1.20) has to be replaced by a parameter E that varies continuously
f '1'; (x)'PI:(X)dT -+ f'P*(X, E)1p(x. E')dT
and we will see below that the Kronecker Ott !nust be replaced by a new ki.nd of function of the variable (E-E'). ~
2 For explicit representations see HEtTLER, [1957], section 8.
This function ,vas accepted "reluctantly by mathematicians because of its non-rigorous definition. It ,vas known, ho\\:cver, that the results obtained via the a-function could alway£ be repeateu without its use. Later a French mathematician, L. Schwartz, incorpo- rated the underlying ideas into mathematics by defining a g~neraHzed notion of the idea of a function (LIGHTHILL [1958]).
eh. t, § 5] OPERAT()R~ 25
. ,.... f . 1" h 11 b . dãff~ fj
SInce one ot trle unctIons oeCOnlCf:~ zero W tell tne Gtier vD.C' IS h erent rom zero .. So we Inay qualify the sym.bols !/lo..(X) ~~ o(x-o:) to a certain extent as orthogonal and normal function~ depen.ding on a continuous: index Ct" l"'he
"norm" '} hO~Never, is in...'inite as \V{~ can see fron):
if now Xl = x..?, the hnornl" is i)(O}.
The second example will be given \:vitli Fourjer integrahL It \ãvill be useful in the next chapter, 'Ve kn.ow that to ext.end the d01TIain uf validity of the f'ourier series to an arbitrary interval --tZ, + d (0 i.s rcal) it is 5ufficient to make a change of the variable: t = ,)Lt/a~ In ih.ts \vay we obtain for functions ljJ defined in this domain, having the periodicity 2a and satisfying certain continuity conditjons~ the 'veH-kno\vn developm.:nts'
~oc
tjJ(x) = (2a)-t L Cn exp (inn:.x:/at
(n JS an integer)~
The orthogonal basis functions are the exponentia.ls (1 ' rv I",)" ""Q} exp ( ' )nnx/ a : ' )
they form a complete set. These are the eigenfunctions of the operator --i(d/<L",')s because we have
-i(d/dx) exp (innx/a) = (n1t/a) exp (innxia)~
II has to be an integer in order that the fu.uctions are periodic over a distance 2a. The normalizing factor 1/ J2ii decreases wIlen the domain D extends.
If we let a go to infinity it is convenient to introduce~
PvL1v == cff~,i I n/a.
\Ve obtain
26 VEcrOR SPACES - UNITARY GEOMETRY [eb. 1, § 5
and if a -+ 00
t/I{x) == (2n)-+ f:: py exp (ivx)dv, p., == (2tt)-i-f:: ",(x) exp (-illx)dx.
(1.32) These are the formulas of Fourier where the index v is a continuous variã
able. The series development is replaced by an integral.
l"he formulas similar to (1.23) and (1.31) are
I +oo f+oo
('" ã "') = -00 ",ã",dx = -00 P:fJ .. dv,
and
as one can easily verify.
The pree~ding argument is not a proof but the passing to the limit <Xl can easily be justified 1• As a result we see that the eigenvalues y of the operator -i(dldx) form a continuous spectrum in the interval - 00, + 00. The corresponding eigenfunctions:
Lim .J -d11/21t exp (ivx)
.4v .... O
~ve a normalizjng factor that goes to zero if the domain D goes to infinity i.e.
in\ the limit their "amplitude" is zero.
tWe have seen in these two examples the diflculties that will occur if one
d.teDds the idea of eigenfunctions to the case of continuous spectra. It is therefore better in physical problems to avoid if possible the continuous spectrum by a limitation of the domain D. The theory of black body radiation is a well-known example of this procedure: Conowina Rayleish and Jeans one supposes the radiation enclosed in a recta.nplar box with perfectly
~waUs. The ampli~de .; of the waves is zero at the boundaries and therefore may be developed ill a Fourier series and it is not necessary to use an integral expression.
A similar procedure is used in solid state physics where the Born-von
~ periodic boundary conditions enable us to use a Fourier series also.
lti. the general case in which we rulve a juxtaposition of a continuous spes- trum and a discontinuous spectrum of eigenfunctions the last, although
1 See the standard book~ on c~~d(:Ultu (e.g. WElf,STER, [1955] p. 153.)
Ch. 1, § S1 OPERATORS 27
consisting of an infinite number of functions, does not form. a complete set.
The combination of the two spectra, however, does form a complote system.
The development of an arbitrary function 1 consists then of a sum of a series corresponding to the discontinuous spectrum, and of an integral providing the continuous spectrum.
1/1 - ~ p" I/It + f (J 1. '" J. dJ. (1.33)
a formula that includes as a particular case the series and the integrals of Fourier.
1 At least the functions that occur in physics.
Comparative table of the properties of function space and If-dimensional vector space
l
Orthogonal basis vectors
Components of a vector
I
j L.;-ngth or norm of a vector
j Scalar product
----_ .. _---
r x",,, i;xle(
i= 1
ly=:Ey/e,
I
(x' y) = 2: >:: y,
I Express,ion for one (;omponent I Xl ,~., (ej' I x)
1---ã----_._----.---:.- ---"---'- ---- -- ..
flJl f!'s ••. !f'" ...
(1.1)
~ = S fJi'{; 1,1
'f = 2: YtVô~ ,
(1.21)
{J.3) 0ã23)
(U8) (Uk)
O.22~
~ : :/;!-- c'-x = A{:::xf,t'.) ~~~ t'~(!t;~.J:1ãf
I Linear l)perators (mappings and I I - ; .,
II ) ' ; "'"" 2. (jitX~
coordinate transfonnations) h
I I 17 .. ,= .-\,t'k =~ 1: e, 0:, '.
... --- --.. ---~-l---
Bilinear forms I (x . Ay) -~" L -'(I' (ifh.;}.
I Hermitlan forms I i,k
I. --- -Iããã .---- ---.(~:~.:~:--- "-
II Reduction to main axes I Ax, = C{.Xi
E~genvalues ! X' A}f ,= :E !XiX~ XI
L=lge:l~t~~~. ___ , _~ ______ ! _________ 1 _ _ _ _ _ _ _ (x ' Ay) = (Ax -
(I A) P';'= L (,i!P~,
k
1p~ "~, Arpl: "= l~ V-',(!ik
(1,24) (1.25)
(1.28) (!.ll) I (If' • Alp} '.~ (A1p' rp), (1.29) (l. 1l a) I a:,~-. <I,k. (l29a\
-'-ã~~l~~-I-~---~~:~-=~:~~~ã . --.---~--~---
(1.12,> I ('P'A'P)'=2;(Xd3~{J, (1.30) (1.31)
N 00
'0 p-
:'"
iO') VI
CHAPTER 2
'rHE PRINCIPLES OF QUANTUM MECHANICS 1. Waves
1.1. CLASSICAL WAVES
In classical physics wave motion in a continuum is described by the three- dimensional wave equation,
(2.1) A possible elementary solution of this equation is given by
",(x, y, z, t) == A exp {-2niV (t- IXX+P;+'VZ)} (2.2)
where A is the amplitude, 11 is the frequency, c the velocity of propagation, and CI., P and y the direction cosines of the normal to the plane wavefront.
The wavelength A = c/v which we prefer to replace by the wave vector k, which has the components
k" = 21tfXV/C; k, == 2ttpv/c; kz = 2TCYVjC. (2.3) The magnitude k is 21t times the number of waves per em; the direction of Ie is the direction of the wavefront.
Each complete solution of the wave equation (in a box or in infinite ~e)
can be obtained by superposition of elementary solutions. The function that satisfies (2.1),
00
t/I(x, y, z, t) = L Ak exp i{k ã r-2?tvkt} (2.2a)
"=--00
represents an arbitrary disturbance which may be electromagnetic or elastic.
The right--hand-side of equation (2.2a) represents a Fourier series or, in case of an infinite space, a Fourier integral and either side of the equation describes at a given moment t a spatial distribution of waves. If l/I(x, y, z, t)
29
30 THE PRINCIPLES OF QUANTUM MECHANICS [eh. 2, t 1
is substantially different from zero over a localized region the function is usually called a wave packet.
If we take together all waves of the same frequency, that is of the same absolute k-value, but of different directions (2.2a) becomes
GO
Y, = L: a.t(x, y, z) exp {-21riVlt}. (2.4)
t=O
The coefficients of (2.2a) and (2.4) are generally complex because the different spectral components usually have different phases. The relative phases will determine whether we will have constructive or destructive inter- ference, one of the most striking properties of waves.
The product a: Qt(x, y, z) is the intensity of a particular spectral compo- nent Vic of the wave at a certain point x, y, z. On the other hand, if we look for the average value < "'. '" > of l/I* '" over a time T long compared to the periods 1/'1k, we have:
<1/1*1/1> = lim ! L: fTa:ase2'rl (Vr-v.)t dt ... L: a; a,(x. y, z) (2.5)
T-+~ T ",Jo ,.
or,the average intensity <"'. "'> at a point (x, y, z) is the sum over all the intensities of the different spectral components.
The expression (2.5) is obtained by integrating separately the terms in which ", ~ v" and those in which 1', = V •• The first set of integrals are oscillat- ingffunctions of T with a constant amplitude. After multiplicatibn with T-i the product goes to zero for T -+ 00. The second integral is proportional
toT hence it is unnecessary to take the limit of the product.
1.2. QUANTUM MECHANICAL WAVES
Early in the development of quantum mechanics a postulate was intro- duced that the energy was proportional to a frequency
or where II == h/2n,
E = hv E = flO)
(2.6)
(2.7) It was successively rea1ized that the nature of the wave was not always
electromagnetic and an additional postulate
(2.8)
Ch. 2, § 11 WAVES 31 which was originally suggested by electromagnetism and relativity, was proposed for matter waves.
The result is that a particle is associated with a quantum wave. In par- ticular a free particle is represented by:
{21ti }
J/! = A exp h (1'. ,.-Et) ã (2.2b)
The complementarity principle, that is the statement that electrons can have either a wave aspect or a corpuscular aspect but not both, can be clarified very beautifully by considering wave packets. Consider the following two cases ..
a) Ifwe have a single wave (2.2b), that is one frequency only, the wave is a 6'monochromatic" wave, the momentum is exactly determined by (2.8) and the position is completely washed out;
b) How'ever, we do have superpositions like (2.2a) and the possibility exists to choose the spatial distribution of
- t/I*t/I = L a:a
1e2xi(Yk-"l)t k.l
such that "'--1/1 is only different from zero in a certain region of space, say at
t == O. If we now interpret the intensity as proportional to the probability to locate a particle, we obtain in this case a reasonable determiaation for the position of the particle. This occurred at the expenseãã of a well-determined
mom~ntum since we must use not one but many differen~ k-values inã our wave ~ packet.
It}s interesting to notice that the question as to which of the two pictures is adequate is completely determined by the experiment. This is a special example of a general pattern iII: quantum mechanics viz. that the choice of the ~ave function is dictated by the experiment. This choice is called the state of the system.
If one tries to construct wave packets with the smallest possible spread in velocity and position, one is led to a Gaussian packet which has tho .property that the product of the root mean squares of the momentum an<i of the position obeys Heisenberg's uncertainty principle. Hence (2.2b) or a super- position of similar waves gives a complete and consistent description of the free particle.
1.3. THE FREE PARTICLE
Which equation has (2.2b) as a solution? At first one would be tempted to .. !uote (2.1) but it turns out that by introducing a slightly different form one
32 THE PRINCIPLES OF QUANTUM MECHANICS [eh. 2, § 1, 2
can give the formalism an analogy to classical mechanics such that one can generalize to a non-free particle, i.e. bound by some potential field, in a
natural way. .
The function (2.2b) is a solutio~ of an equation of the first order in time.
(but with an imagi~ary coefficient), i.e.:
~ V2t/! = h ot/! • (2.9)
2m i at
and by comparing this with the classical energy momentum relation
L=E 2
2m
we find the following correspondence:
f, 0 H+-+---
i at
p~-.V. Ii
1
(2.10) (2. lOa) Hence by introducing (2.10) and (2.10a) and by proclaiming that:
A physical f/UIIIUltY or obseruable is an operator applied to a wave function~
we can drop the postulates (2 7) and (2.8). The reason is that the operators (2.tt) and (2.10a) can be related to certain eigenvalues, and upon sub- stitution of the eigenfunction (2.2b) in their respective eigenvalue equations
Hop'" = Et/I; (2.11)
Popt/! = pt/! (2.lla)
one obtains (2.7) and (2.8). Since the operators are Hermitian, the eigen- values are always real (comp. Cbapter 1, § 5). There arc of course circurp.- stances in which the state is not pure (monochromatic), i.e. that a distribu- tion of eigenvalues is found and thus the physical observable is not sharply determined. The interpretation of such a case will be discussed in § S of this chapter.
2. De Scbriidinger Equation
In this section we discuss the ScbrOdinger equation, i .. e. the general wave equation for conservative mechanical systems, and also we will introduce the many-particle equation.
Ch. 2, § 2] THE SCfIRODINGER EQUATION 33
21. SCHR{)DINGER EQUATION
A conservative system can be described by a Hamiltonian. According to (2alO) we have
h OV!
- - + Hl,fI ::; O.
i ot (2.13)
The Hamiltonian can usually be separated into a kinetic and a potential energy:
1 ? (
H = - p-+ V, 2.14)
2m
hence we have the following equation, named after Schr6dinger Ii 01/1 112 2
-: -a- - -2 V t/J + V(x, y, z, t)t/J == O. (2. 13a)
1 t m
If V does not depend on t we obtain in general an infinite set of functions l/Ik{X, y, z) such that the operation H is reduced to multiplication with a real constant, i.e.
(2. IS) If we are able t~ determine Ek we find by integrating (2.13a) that t/lk is a
periodic function of the time t. Hence we call this the stationary state of the system:
.11 ( t) .11 ( )e - iBJet/1
'l'k X, y, Z, = Y'k X, y, Z • (2.16)
In this state the energy H has a well-defined value Ek • The spatial function t/I{x, y, z) is a solution of the time independent Schrodinger equation:
V2y,+ ~~ (E- V)/I = 0, (2.17)
which can be obtained by separation of variables from (2.13a), provided, of course, the potential energy does not depend on t.
After introducing the potential energy the eigenfunctions of H are no longer simultaneous eigenfunctions of Pop. This has a very good physical reason because if we would ask for the value of the momentum at a fixed E we would find that it is no longer a. constant. Hence if we apply Pop to the eigenfunction we do not expect to find a time independent distinct eigenvalue.
2.2~ THE n-PAl~"rICLE PROBLEM
It is important to realize that one has to add a new concept in order to handle two or more particles .. V is now of the form V(Xl' Yl' Zl' X2' Y2, Z2,