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Fourier Transforms
8.1 INTRODUCTION
When the method of separation of^ variables^ is^ applied^ to^ certain^ partial
differential equations^ of^ mathematical^ physics,^ integrals^ of^ the^ forni rb r(a) :^ I f(r)x1a, 1^ a^ (E^ ti
often occur. The function^ F(a) is^ said^ to^ be^ the^ integral transfsrrn^ of y'{"r}
by the kernel K(a,, x). The kernels^ associated^ with^ Fourier,^ Lapiace,^ Fouriei-
Bessel (Hankel), and Mellin^ transforms^ are
F(u): j" (^) f -f(x)e''*
dx (Fourier^ transfcrm)
F(a): !* f{)r-". a^ (Laplace transfbrm)
F(a) :^ ['o f G)xt,{ai ax^ (Fourier-Bessel^ transform)
(o -.J
(8.3)
(8"4)
r\0 ,J"^ /i.r).r" (^) ' ,i.'^ (8.5i ara
(Mcllin transform)"
FoURIER TnlNsronlras Cxlp. 8
The Fourier and Laplace transforms are the most often used in^ mathematical physics.
8.2 THEORY OF FOURIER TRANSFORMS
8.2.1 Formal Development of
the Complex Fourier Transform The form of the complex Fourier series (see Chapter 7) is
where
To make the (^) transition / (^) - -, we introduce a new variable which is by
(8.7)
defined
t_ ftlt
":-T
where (Lk1n)/:^1 since Ln: l.^ Hence^ we^ may in the following forms:
f(x): i C{k)etk*^ Ak^ (8.e)
and
I^ f(x)c-Ik*^ dx
we obtain
f(x)e-tk* dx
(8. lo)
where (^) C(k) :^ lc,ln.lf
(8.1 l)
and
f(*) :^ l^ *c1t7"'o' at.^ (8.12)
There are several ways of defining Fourier transforms, but^ the differences among the various forms are not^ significant.^ To put^ Eqs. (8.11)^ and (8.12)^ in the modern form (the^ form most^ often^ used^ by^ authors), we let^ F(k)^ - -tfrcf-D and obtain
"f(x) :^ n=U_*rn'""'" (-/^ <^ x^ I^ l)
,,: (^) fi
['_,-f{t)r-''nx/t
6rv.
(8.6)
write Eqs. (8.6) and (8.7)
C,&: * I'
we let / --+^ co,
c(k): *r
If (^) f (x)^ is an even function of x^ [/(x) :^ .f (--x)1,^ then^ we^ see^ from^ Eq.^ (8.17) that the cosine transform is equal to^ the^ Fourier transform.^ If'^ however,^ /(x) is an oddfunction of x[f(x): (^) -f(-x)], then^ we^ see^ from^ Eq.(8'17) that the sine transform is equal to the Fourier^ transform provided^ -iF(k) is replaced bV (^) 4(/c).
The one-dimensional Fourier transforms^ of^ a^ function^ of^ two^ and^ three
independent variables are given,^ respectively,^ by
230 Founlrn TneNsronMs
Note that the Fourier transform^ of/(x)^ [Eq. (8.13)]^ may
F(k): hf _f(x)e'k'dx : hJ __,f(")t.os^
kx (^) f i sinkxf dx-
F(k, y)^ :^ h t-__tr*,^
y)etk* dx
f (x,^ v) :^ h I---rn, v)-tk*^ dk and F(k, y, 4 : h l-__tO,^
y, (^) z)etk' dx
f (x,^ y,^ i : h |-_-rn, y,^ z)e-'k* dk.
The corresponding cosine and sine transform pairs^ are
F"(k, y)^ :^ ^l+ (^) [-, tU,y) cos kx^ dx
f (x,^ v) :^ l+ (^) I*" o"u,^ v) cos^ kx^ dk
F"(k, y):^ J I-" tU,y)^
sin kx ctx
.f(x, y)^ :^ ^l+ I-,^
r"n, y) sin kx dk.
to Eqs. (8.20)^ and^ (8.21) are^ valid^ for
Cslp. 8
be written as
(8.17)
(8.1 (^) 8)
(8. le)
(8.20)
(8.21)
F"(k,l, z) and
and
Equations similar F"(k, y,^ z).
Sec.8.2 Theory^ of Fourier^ Transforms
8,2.3 Multiple-Dimensional Fourier^ Transforms
The one-dimensional transform theory may^ be^ extended^ in^ a^ natural^ way^ to
the cases of two-^ and^ three-dimensional^ Fourier^ transforms.^ In^ equation
form, the two- and three-dimensional^ Fourier transform pairs^ are,^ respec-
tively,
-f (x, y)s't"'*aYt^ dx^ dY t8.22)
F(a, B){tt"'*Ptt dudf
and F(u, (^) f,f : dpJ:- (^) [- [- -rr-.!,
z)ei6"pv'v4 dx ttv^ ttz^
/R,3\
-f (x,^ v, 4 :^ eh"J:- J:- J-- tt'^ B'^ v)e^
i'"'*p"n"t du df dv'
The generalization to^ the^ case^ of^ an arbitrary^ number^ cf^ dimensions^ is straightforward.
8.2.4 The Transforms^ of^ Derivatives
Fourier transforms, cosine^ transforms, and^ sine^ transforms can^ c.tflten^ be;
used to transform a^ differential equation (ordinary^ or^ partial)^ wtrich^ describes a complicated physical^ problem^ into^ a^ simpler equation (algebraic or ordinar;"'
differential) that^ can^ bL^ easily^ solved.^ The required^ solution^ of^ the^ original
differential equation is^ then obtained by finding the^ inverse^ transform of^ the
solution of the simpler equation^ (in^ transform^ space).^ In^ order^ to^ use^ th*
transform method to^ solve^ first-^ and^ second-order^ differential^ equations'
the transforms of flrst-^ and^ second-order derivatives are^ needed.
We now develop the^ transforms^ of^ first-^ and^ second-order^ elerivatives"
The Fourier transforms of^ first-^ and^ second-order^ derivatives^ wili^ be reprs-
sented by^ F(1)(k) and^ -ts(2)(k),^ respectively.^ That^ is^ to^ say,
pr,(k) :
h I-__ory),'r'^
u,. qu.2+.i
On integrating Eq.^ (8.2a)^ by parts,^ we^ obtain
F,,,(k): (^) -L{ f1)e,u,l (^) - ,k^ f J'tx)e'k'dxtl J2n1"''- l- 'J-" (^) )
231
F(u, (^) f): (^) *r "r ^ f(x,v): (^) +f (^) -f -
ik-* l-I
N /)-r.e^ J^ I-6\
f(x)e'k'd.r
{a 1<\
Sec.8.2 Theory^ oJ^ Fourier^ Transforms^233
(8.3 (^) 3)
(8.34)
(8.3 s)
Ftz,(k) :^ ^l+ I^ "n#
sin kx dx
- k2F"(k)^ (8.32j for (^) f(x) -+^0 and/'(x) *^0 as n^ -> oo' The choice of using the cosine or^ sine^ transform^ is dictated by the^ given
boundary conditions at the lower limit.^ The^ use^ of^ the^ cosine^ transform^ to
remove (transform) a first-order derivative^ term^ in^ a^ differential^ equation
requires a knowledge ofl(0), but a knowledge^ of/(O)^ is^ not^ required^ wher-i
the sine transform is used to remove a^ first-order^ derivative^ term.^ A^ kncwl-
edge of/'(0) is^ required^ to^ remove^ a^ second-order^ derivative term^ in^ a dift'er-
ential equation if^ the^ cosine^ transform is^ used.^ However,^ only^ a kngwledge
of/(0) is needed to^ successfully^ use a^ sine^ transform to^ remove^ a^ second-order
derivative term in a differential equation.
The relations for Fl')(k, y),^ Ft')(k,y),^ F!')(k,7),^ and^ F',')(k,y)^ are:
Ftu(k, (^) r) :^ l+ I-"A*D
cos kx^ dx
: l "l1kf(0)Y7t
: (^) kF,(k, (^) t> (^) - tf (^) ]Xo, O for (^) f(x, y)^ -' 0 as^ Jr^ -+^ oo'
t-T ^/|f'(o'il
Ft'(k,D: l+
I-"t#;Y) coskx dx
: (^) -k2F"(k, y) (^) -
for (^) f (x, y)^ - 0 and df (x,^ y)l|x^ '-- 0^ as^ n^ -> oo,
Ft (k,^ r) :^ J+ I-JJ*D
sin kx dx
: (^) -kF"(k, y) for (^) f (x,y) *^0 as x ---)^ oo,^ afld
F"''(k,il (^) - ^l+ (^) I-"a:##ilsin,rx r,
k2 F.,(k,^ y)
:4;kr(o'Y)-^ t-T
for (^) f(x, y) *^0 and d/(x, y)l|x'-^0 as x --+ oo.
(8.36)
FouRlER TRnnsronlts^ Cnlp.^8
(8.37)
(8.38)
8.2.5 The^ Convolution Theorem
In linear^ response^ theory,^ the^ general^ equation^ for^ the^ one-dimensional transform, Eq. (8'1),^ takes^ the^ form
F(a):lu,f{t)x{a,ia* : (^) J:_ K@ (^) - x)f(x)dx
where K(a^ - x)^ is called the^ response^ of^ the^ linear^ system,^ /(x)^ is^ the^ input (signal) to the linear t;J;,^ uni^ r(n)^ is^ the output^ (signal)'^ If^ K(a^ -^ x)^
: d(a (^) - x),^ then r(a): (^) J_-rf" - x)f(x)dx
whichisconsistentwiththepropertiesoftheDiracdeltafunction.Inthis
latter case, K(u^ - x)^ is^ called^ the^ impulse^ response^ of^ the^ system'^ When Eq'
(8.3?) is written^ in^ the^ form
l (^) l- rG F(x): f ^ g:-t-N-'.J.' - le6)^ dE
it is^ called^ the^ one-dimensional convolution^ (faltung-folding)^
integral of
two integrable functions^ il"l^ uta^ g(x)'^ The coiresponding two-dimensional form is f s:f^ ^f -f(x-t;x -^ f)sc'fid€dP'
Let F(k) and G(k)^ be^ the^ Fourier transforms^ of/(x)^ and^ g(x),^ respectively.
For these^ functions, the^ convolution^ integral^ becomes
-+ i-^ r@^ -r)g(o^ d(:hiI- -tn'lhJ-'{r')'-*"-n
dkidc
1zn J^ _*-
: (^) * f (^) -F(k)e-'k'
ou f (^) -s(oe'r.
d(
: :: f- F(k)G(k)e-tk' dk'^ (8'39) Jzn J^ _-,
In obtaining^ Eq. (8.39),^ we^ have^ tacitly^ assumed^ that^
the process^ of^ inter-
changing the order^ .ii",.r.",'"n^ is vaiid.^ The result^
in Eq. (8.39) is^ known
as the convolution^ theore-"f*^ fou'i"r^ transforms'^ It^ means^
that the^ Fourier transform (inverse^ ";;;i;t"D of^ the product F(k)G(k)' the right-hand side
Founten TuNsronus Cslp.
Note that both/(x) and F(k) are Gaussian distribution functions^ with^ peaks
at the origin.^ The^ standard deviation,^ width,^ is defined^ as^ the range^ of^ the
variable x (or k)^ for^ which the function^ /(x) [or F(k)]^ drops^ by^ a^ factor^ of e-t/2 :^ 0.606^ of^ its maximum value.^ For^ /( x)^ :^ Ne-"*n,^ the^ standard devia-
tion is^ given^ by
o*:-T:m,-^ A:c^ I For
F(k): ]:r-lvn",r/ zd,
the standard deviation is^ given^ bY
or:i-n/2e.^ Lk^ /=-
Note that A,x Ak^ :^ (z|Jfr)Q,Jfi1:^ 4.^ If^ u^ --+^0 (small), then^ Ax^ -'^ oo and Ak (^) -' 0. For d, --+ @ (large),^ Ax^ --^ 0 and^ Ak -> oo (see^ Fig.^ 8.1).
Figure 8. Exeupln 8.2 Find the Fourier transform^ for^ the^ box^ function/(x)^ where
i(") :^ {; f;i::,=^ ^
Solution By use of^ Eq.^ (8.13), we^ obtain
F(k): )r l-^ _trx)e'k*^
dx
: fo
I eik'dx
)_"
Sec.8.
A sketch ofl(x) and
JQ)
Theory of Fourier^ Transfornts
: l:lt)" */2nLik )^ " : l2sinka ^,17-E-'
F(k) is given in Fig.^ 8.2.
Figure 8. Exaupu 8.3 Find^ the^ cosine^ and^ sine^ transforms^ of^ /(x)^ :^ €-xe
Soltdion Using Eq.^ (8.15),^ wt^ find^ that
F"(k) :^ ^11 I-,t'.)
cos kx r/'v
/1 I" -^l Y 'lft (^) Joe*cosk-rd.r tTl |^ .
The corresponding sine^ transform^ is^ obtained^ by^ means^ of'^ Eq.^ {8^ 1{;!'
We obtain
F,(k) :^ ^17 l-"to)
sir, kx ctx
: tzf*^ _,. ^l +^ J ,'-'sin
A--r dr
la( *^ 't. -^l n (^) \l+k'l
Exauplr 8.4^ By^ the^ Fourier transform^ rnethod, develop^ the^ formal^ solu- tion of the differential cquation which^ characterizes^ the^ nrotion of^ ir dampeii harmonic oscillator.
Sec,8.
where
Theory of Fourier Transhrms
g(x, t)-> 0 and (^) X-*-- 0 (as (^) x -,---+ (^) *[oo)
e(x,o): .f(*) and^ u#1,=,:^ o
Solution Let
@(k, t): h I-__rO,t)eik*^
dx.
On taking the Fourier transform of the wave equation, we obtain I l- d2Q-",*,a":L (^) I 9i .- Jfr )--^ il'"^ u'^ J2n dtt^ J^ -,,9(x'
t)e'k' dx'
By use of Eq. (8.28), we see that the above equation reduces to
-kza(k,D: #t#a
where g(x, l)--+0 and |pldx ---0^ for x-- *-. In (^) other words, there is
no wave motion at _f oo^ (the ends of a very long string). The solution of the
above ordinary differential equation (since the derivative with respect to
only one independent variable (^) appears) is Q(k' t): cretkat (^) t cre-'g".
The given^ initial conditions will now be used to evaluate the constants
ct and cr. From g(x,0) -:.f(x),^ we obtain
o(k, 0)^ : h [-__rO,o)s-i*" dy I (^) i- t/ (^) --.
- (^) -: 1 11ap-ik'^ d.Y ^/zftJ -* : (^) F(k)
:c1lc2'
The second initial condition gives
61r, o;^ :^ h f-^ _(#),-"e'ikx^
ctx : : (^) ikucr (^) - ikucr.
Fountpn Tn,cNsronl,ts^ Cnlp.
Therefore the solution^ in^ k-space^ is
@(k' r) :^ FU<) 'tt'* + Ff) '-'o"'
The required expression for^ g(x,l),^ in^ this^ simple case'^ may^ be^ obtained
directly by taking the^ Fourier transform of^ @(k,^ t)'^ We^ have
ll- eG, t): mJ __ a(u,^ t)e-tk*^ dk
: #"{+ J^^ -{'{o)''r')e'ikx
dk (^) + (^) +J-- rotol"-'o'1e-'n' dk\
or QG, t)^ :^ if@ (^) - at)^ +^ tfG *^ at)'
The above procedure^ for^ finding^ g@,t)^ from (D(k,l)^ leads^ to^ the^ desired
result only in relatively^ simple^ cases.^ IJse^ of^ the^ convolution^ theorem^ in^ the
expression for the^ inverse^ transform^ is^ the^ general procedure^ for^ finding
Q6, t)^ from^ O(k,^ /), that^ is,
s(x, t)^ :^ hJ-_.t0, t)e-ik'^ dk
:,JfrI"'I^! (^) f- oglr,kD.)e-ikx dk (^) + I (F(k)e-,k,,)e-,0, -- tl
- rmt.f-- rror.^ G^ -^ o^ d( (^) + .l- (^) -ro)s-@ -^ o *j'
Here we need^ the^ original forms^ (in^ x-space)^ of^ eikat^ and^ e-ikut^ to^ obtain
g*(x-OandS-(x-6).Toobtaintheseforms,wetaketheinversetrans-
fOrms Of e,ku, and^ e-ikuc.^ Fot^ errr,, we^ find
g(x): hf _(eikot)e-ik^
dk
: -L f^ (^) "-,0,,-,,,41. Jzn )^ -*
- Jfr^ 6(x^ - ot)- Similarly, we^ find that g-(x) (^) - ./fr,6G t ut).
242 FountPn^ Tn'cNsronl\as
By use of^ Eq.^ (8.28),^ the^ above^ equation^ reduces^ to
Since
T(k, t) :^ fs-ok't '
T(k,0): 1 :
h [--f{)"'0.^
o*
: (^) F(k)
Cn,lp. 8
-k2r(k,r>: (^) +'J*!
whereZ(x,')-0anddT|dx_+0asx--'*-(thetemperatureiszeroat the ends of the^ very long^ rod).^ The solution^ of^ the^ above^ equation^ is
the solution in^ k-sPace becomes
T(k, t) :^ F(k)e-"k".
The object is^ to^ develop^ an^ expression^ for 7(x,^ t)'^ Here^ we^ need^ the inverse
transform of^ F1k\s-"o",^ which can be obtained^ by^ use^ of^ the convolution
theorem, that^ is,
T(x, t): hf- -toO,
"tcztls-it"x dk
: _L^ i- rrorr., _ o d(.
J27t J^ _-
We need the^ original^ form (in^ x-space)^ of^ the^ second^ function'^ S@^ - 0'
It is just^ the Fourier transform of^ e-dk'zt^ which^ becomes
g(x) : j" (^) f
-(s-'t"'r)s-tt'*
dk'
lf uz :^ 2k2ot^ where ctk^ :^ dul,J1ot,^ then
g(x) '- (^) rJm^1 Ji* _-."^ t'-u'i2)s-ttu/'/-2dt"^ du
l-e-"/to'.
J2ot
Sec.8.2 Theory of Fourier Transforms
The expression for T(x,t) in terms of the^ convolution^ integral now becomes
r(x' t) :^ l+(h) J*-^ *r {tt'-^ "'^
1)! i ao'^ ct
: =-! f* fcl'*'c'ad d('
tJTtot J _*
The specific form^ of^ /(x), the^ initial^ temperature^ distribution,^ must of^ course
be given before the^ above integral^ can be evaluated.
Exauprr 8.7. By use of the three-dimensional Fourier transform method,
solve Poisson's equation for the electrostatic potential function,
Y'z$(r): __'+
Solulion In three-dimensions, we write
o(k): (^) # !_o{i,'"'o"
where d3r :^ dx dy^ dz (l'riple^ integral).^ The Fourier^ transform^ of^ @(k) is given (^) by
0(,): (^) #f-_ r,ut-,k',^ d3k.
On taking^ the Fourier^ transform^ of^ both^ sides^ of^ Poisson's equation,^ we
obtain
-k,o(k): -ry
o(k): {G;
where (^) f(r) -> 0 and d$ldr (^) -, for r --^ *-. The expression for P(k) is
P(k). dV [,.,p(r'),'u',t','.
Sec.8.3 The Wave Packet^ in^ Quantum Mechanics^ 24s
We conclude this chapter with^ a^ detailed discussion^ of^ the^ role^ of^ Fourier transforms in^ the^ foundation of^ quantum^ (wave) mechanics.
8.3 THE WAVE PACKET^ IN^ QUANTUM^ MECI{ANICS
8.3.L Origin of the Problem: Quantization of^ Energy
Prior to 1900,^ physicists^ assumed^ that^ the^ laws governing^ the^ macrorvorld,
together with^ the^ appropriate statistical^ considerations, were^ valid^ in^ the
microworld. This^ approach^ always^ led^ to^ an unsatisfactory^ theory of^ black-
body radiation (Wien's and^ Rayleigh-Jeans'^ radiation^ laws)'
The development of^ quantum theory had^ its^ origin^ in^ the^ inability of
classical physici^ (mechanics, electromagnetism,^ optics,^ and^ thermodynamics)
to account for the^ experimentally observed energy^ distribution^ (energy^ vs.
frequency or wavelength) in^ the^ continuous spectrum^ of^ black-body radia-
tion. In short, one is^ required^ to^ explain^ on^ an atomistic^ basis^ the^ color^ of
light that an object emits when^ it^ is heated^ to^ a^ certain temperature.^ A^ cor-
,..t th.ory of black-body radiation^ was developed^ by^ Planck^ (1858-1947)
in 1900. This theory requires^ that^ the radiated^ energy be^ quantized,^ that^ is,
E:hv, where Eis the energy ofthe^ radiation,^ v^ is^ the^ frequency^ ofthe radiation, and^ ft^ is Planck's constant. However, Planck^ attributed^ the^ quan-
tization phenomenon to^ the radiating^ object.
tn t905, Einstein (1s79-1955)^ further^ developed^ the^ quantization^ of
energy concept^ by^ (1)^ assuming^ that the^ quantization^ phenomenon was^ a
property of the radiation itself, (2)^ assuming^ that^ the quantization^ process
applied to^ both^ absorption and^ emission^ of^ radiation,^ and^ (3)^ using^ the
quantization concept to develop a correct theory^ of^ the photoelectric^ effect'
Bohr (1885-1962),^ in^ 1913, used a^ mixture of^ classical physics (mechanics
and electromagnetism) and quantization^ of^ energy^ concept^ to^ formulate^ a
satisfactory theory for^ the^ observed^ spectrum^ of^ the hydrogen atom. Bohr's
theory is based on^ the^ following^ three^ postulates:
- The electron in^ the hydrogen^ atom^ moves^ about^ the nucleus (the^ proton)
in certain circular orbits^ (stationary^ states)^ without^ radiating^ energy.
- The allowed stationary^ states^ are such^ that L: mur -- nh^ (n:^ 1,2,3,^ .)
where Z is the angular momentum^ of^ the electron,^ r^ is
"he
radius of^ the orbit,m is^ the^ mass^ of^ the electron,^ h:^ hl2n,a^ is the^ speed^ of^ the^ elec-
tron, and n is the principal^ quantum number.
246 FoURIER^ TRANSFoRMS^ Cn.lp.^8
- When the electron makes a^ transition from^ a^ state^ of^ energy E,^ to^ a^ state of energy E1,^ where^ E;)^ E7'^ electromagnetic^ radiation^ (photons)^ is emitted from^ the hydrogen^ atom.^ The frequency^ of this radiation^ is^ given by
Bohr's theory was gencralized^ by^ wilson^ and^ Sommerfeld^ and^ applied
to other atoms rvith lirnited^ success.^ By^ 1924,^ it^ was^ clcar^ that^ a ncw thcory
was needed to explain thc^ basic^ properties^ of^ atoms^ ancl nroleculcs^ in^ a^ sys-
tematic manner.
8.3.2 The Development of a^ New^ Quantum Theory
In an historic paper^ (1925)^ "On^ a^ Quantum Theoreticai Interpretation^ of
Kinematical and Mechanical Relations" which led^ to^ the^ development^ of
matrix mechanics, Heisenberg^ (1901-^ ) introduced^ a system^ of^ mechanics in which classical concepts of^ mechanics were^ drastically^ revised. Heisenberg
assumed that atomic theory should^ emphasize^ the^ observable quantities
rather than the shapes of electronic orbits^ (Bohr's^ theory).^ This theory^ was
rapidly developed by means of matrix^ algebra.
Parallel to the advancement of^ matrix^ mechanics,^ Schrildinger^ initiated
(1926) (^) a new line of study which evolved into^ wave mechanics. Wave mechan- ics was inspirecl^ by^ de Broglie's^ (1892-^ ) wave^ theory^ of^ matter,^ r1^ :
hf p,^ where p^ ir^ tn"^ momentum^ of^ a^ particle^ and^ ,2'^ is the^ wavelength^ of^ its
asiociated wave (matter^ wave). Schriidinger^ (1887-1961)^ introduced^ an
equation of motion, the^ Schriidinger wave equation,^ for^ matter^ waves and
proved that wave mechanics was mathematicallyequivalent to^ matrix^ mechan-
ics. However, the^ physical^ meanirrg^ of^ wave^ mechanics^ was^ not^ clear^ at
lirst. Schrtjdinger first^ considercd^ the de Broglie^ wave as^ a^ physical entity
(the particle, electron, is actually a wave).^ This interpretation^ soon^ led to
clifficulty since a wave^ may^ be^ partially^ reflected and^ partially^ transmitted^ at
a boundary, but an electron cannot^ be^ split^ into^ two^ parts^ for^ transmission
and reflection. This difficulty^ was removed^ by^ Born^ (1882-1970)^ who^ pro-
posed a statistical interpretation of^ the^ de^ Broglie^ waves^ which^ is^ now
lenerally accepted.^ The^ new^ theory^ based^ on the^ statistical interpretation^ was
very rapidly^ developed^ into^ a^ general^ coherent^ system^ of^ mechanics (called
quantum mechanics).
8.3.3 A Wave Equation for^ Particles:^ The Wave Packet
In developing a^ wave equation^ for^ particles,^ Schrodinger^ knew^ that^ (1)
Hamilton had established an analogy^ between^ the^ Newtonian^ mechanics^ of
E'-Er h
