Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
Подождите немного. Документ загружается.
quantum
harmonic
oscillator:
power
series
method
13.6
POWER-SERIES
SOLUTIONS
OF
SOLOES
373
integer
l. In
that
casethe
infinite
seriesbecomesa
polynomial,
the
Legendre
polynomial
encountered
in
Chapter
7.
Equation (13.31) could have been obtained by substitutiog Equation (13.28) directly
intothe
Legendre
equation.
The
roundabout
wayto(13.31)
taken
hereshowsthe
generality
of
Equation
(13.29).
With
specific
differential
equations
itis
generally
better
to
substitute
(13.28) directly.
III
13.6.5. Example. WestudiedHermitepolynomialsinChapter7inthecontextofclassical
orthogonal
polynomials.
Let us seehowtheyariseinphysics.
The
one-dimensional
time-independent
SchrOdinger
equation
fora
particle
ofmassm
ina
potential
V (x) is
1\2
d
2lj1
- 2m dx2 +
V(x)ljI
= EljI,
where
E isthetotal
energy
ofthe
particle.
Fora harmonicoscillator, Vex) =
ikx2
==
!mw
2x
2
and
m
2
w
2
2m
ljI"
-
~x2lj1
+
1\2
EljI
=
O.
Substituting ljI(x) =
H(x)
exp(-mwx
2/21\)
and then making the change of variables
x =
(I/~mw/I\)y
yietds
H"
-2yH'
+J..H =0
2E
where
J..=--1.
tu»
(13.32)
Thisisthe
Hermite
differential
equation
in
normal
form.
Weassume the
expansion
H (y) =
L~O
cnyn
whichyields
00 00
H'(y)
=
Lncnyn-t
=
L(n
+
t)c
n+
1yn,
n=l
n=O
00 00
H"(y)
=
Ln(n+l)cn+tyn-l
=
L(n+l)(n+2)c
n+
2yn.
n=l
n=O
Substituting in Equation (13.32) gives
00 00
L[(n+
I)(n
+2)c
n
+2 +
J..cnJyn
- 2 L (n +
I)cn+lyn+l
= 0,
n=O n=O
or
00
2C2
+
J..co
+
L[(n
+2)(n +3)C
n+3
+
J..cn+l
- 2(n +
l)c
n+tlY
n+l
=
O.
n=O
Setting
thecoefficients of
powers
of y
equal
tozero,we
obtain
J..
C2 = -Zco,
2(n + I) -
J..
C
n+3
= (n +2)(n +3) cn+t
forn
~
0,
374 13.
SECOND-ORDER
LINEARDIFFERENTIAL
EQUATIONS'
or,replacingn withn - 1,
(13.33)
n2:l.
2n-i.
c +2 - c
" -
(n+t)(n+2)
",
Theratiotestyieldseasily thattheseriesis convergent forall values
of
y.
Thus, the
infinite
series whose coefficients obey the
recursive
relation
in
Equa-
tion (13.33)
converges
for all y.
However,
on physical
grounds,
i.e., the
demand
that
limx--+oo
1fr(x)
= 0, the seriesmustbe
truncated.
This
happens
only if)" = 21for some
integer I (see Probtem 13.20 and [Hass 99, Chapter 13]), and in that case we obtain a
polynomial,the
Hermite
polynomialof
order
1.A consequenceof sucha
truncation
is the
quantization
of
harmonic
oscillator
energy:
quantum
harmonic
oscillator:
algebraic
method
2E
21
= A = Iiw - t
=}
E = (l +
!)Iiw.
Twosolutions
are
generated
from
Equation
(13.33), one including only even
powers
andtheotheronlyoddpowers.Theseareclearlylinearly
independent.
Thus,knowledgeof
co
and
CJ
determinesthe generalsolution of the HSOLDE of (13.32).
III
The
precedingtwo examples
show
how
certain
specialfunctions
used
in
math-
ematicalphysics are obtainedin an analyticway,
by
solvinga differentialequation.
We saw in
Chapter
12 how to
obtain
spherical harmonics
and
Legendre
polynomi-
als by algebraic methods.
It
is instructive to solve the barmortic oscillator
problem
using
algebraic methods, as the following
example
demonstrates.
13.6.6. Example. The
Hamiltonian
of aone-dimensional harmonic oscillator is
p2 1
H=
-+_mo}x
2
2m 2
where
p =
-i
tid/dx isthe
momentum
operator.
Letus
find
the
eigenvectors
and
eigenvalues
ofH.
Wedefinethe
operators
and
Usingthecommutation relation [x, p] =
i1i1,
we canshow
that
and
_
...
t t
...
H - "wa a +
2,=1.
(13.34)
creation
and
annihilation
operators
Furthermore, one can
readily
showthat
[H,
a] =
-liwa,
(13.35)
Let
ItfrE)
be the eigenvector corresponding to the eigenvalne E: H
ItfrE)
= E ItfrE},and
note that Equation (13.35) gives
Ha
ItfrE)
= (aH - liwa)
/tfrE}
= (E - Iiw)a
ItfrE)
and
Hat
ItfrE}
=
(E+Iiw)a
t
/tfrE).
Thus,a
/tfrE}
is aneigenvectorof H,witheigenvalueE
-liw,
andat
11/1
E} is aneigenvector witheigenvalueE +
Iu».
Thatis why at anda arecalledthe
raising andlowering (orcreation andannihilation) operators, respectively. Wecan
write
13.6
POWER-SERIES
SOLUTIONS
OF
SOLDES
375
By
applying
a
repeatedly,
we
obtain
states
of lowerandlower
energies.
But
there
is a
limit
to this
because
His a positive
operator:
It
cannot
havea
negative
eigenvalue.
Thus,
theremustexista ground state,
11{Io},
suchthat a
11{Io}
=
O.
Theenergyof thisgrouodstate
(orthe eigenvaluecorresponding to
11{Io»
canbe obtained-I
H
11{Io}
= (lUvata +
!1Uv)
11{Io)
=
!1Uv
11{Io)
.
Repeated
application
ofthe
raising
operator
yieldsbothhigher-level statesand
eigenvalues.
Wethus define
11{In}
by
(13.36)
where
en is a
normalizing
constant.
The
energy
of
l1/Jn}
is n units
higher
than
the
ground
state's, or
t
En = (n +
~)IUv,
whichis
what
we
obtained
inthe
preceding
example.
Tofinden, wedemandorthonormalityforthe
11{In).
Takingtheinnerproductof(13.36)
with itself, we can show(see Problem 13.21) that le
nl
2
= nlc,,_1I
2,
or len1
2
= n!leoI2,
whichfor
Icol
=1 andreal c" yieldsen =
v'ni.lt
follows,then,that
quantum
harmonic
oscillator:
connection
between
algebraic
.and
analytic
methods
In
terms
of functions and
derivative
operators,
a
Ito}
= 0 gives
(J~:
x +J
2;W
:x)
1{Io(x)= 0
withthe solution
1{IO(x)
=
eexp(-mwx
2
/2/i). Normalizing
1{IO(x)
gives
1
00
(mwx
2
) (
lin ) 1/2
I =
(1{IoI1{l0)
= e
2
-00
exp
--/i-
dx
= e
2
mw
Thus,
Wecannowwrite
Equation
(13.37)in
terms
of
differential
operators:
1{I,,(x) = _1_
(~)ti4
(t
w
x _
J/i
~)n
e-mwx'/(2n)
v'ni lin 2/i 2mw dx
Defining
anew
variable
y =
.Jmw/hx
transforms
this
equation
into
=
(mW)t/4
_1_
( _
~)n
e-Y'/2.
1{1"
lin 1],=nI Y d
-v
L.··n: Y
(13.37)
7Prom
hereon,theunit
operator
1will notbe shownexplicitly.
376 13.
SECONO-OROER
LINEAR
OIFFERENTIAL
EQUATIONS
From this, the relation between Hermite polynomials, and the solutions of the one-
dimensional harmonic oscillator as givenin the previous example, we can obtain a general
formula for H
n
(x).
In particular,
if
we note that (see
Problem
13.21)
'/2
(
d)
'/2
-i d 2
e
Y
y - -
e-
Y
=
-e
Y
-e-
Y
dy dy
and, in general,
2 ( d
)n
2 2 d
ll
2
eY /2 y _ _ e-Y /2 = (_l)ney
-e-Y
,
dy dyn
we recover the generalized Rodriguez formula
of
Chapter 7.
l1li
To end this section, we simply quote the following important theorem (for a
proof, see [Birk 78, p. 95]):
13.6.7.
Theorem.
For any choice
of
Co and
CI,
the radius
of
convergence
of
any
power series solution
y =
L~o
CkX
k
for the
normal
HSOWE
y" +
p(x)y'
+
q(x)y
= 0
whose coefficients satisfy the recursion relation
of
(13.29) is at least as large as
the smaller
of
the two radii
of
convergence
of
the two seriesfor
p(x)
and
q(x).
10 particular,
if
p(x)
and
q(x)
are analytic in an interval around x = 0, then
the solution of the normal HSOLDE is also analytic in a neighborhood of
x =
O.
13.7 SOLDEs with Constant Coefficients
The solutionto a SOLDEwith constant coefficients can always be fonnd in closed
form. 10fact, we can treat an nth-order linear differential equation (NOLDE) with
constantcoefficients with no extra effort. This briefsectionoutlines the procedure
for solving
such
an equation. For details, the reader is referred to any elementary
book on differential equations (see also [Hass 99]). The most general nth-order
linear differential equation (NOLDE) with constant coefficients can be written as
L[y]
==
yCn)
+
an-1/,,-I)
+...+
aIY'
+aoy =
r(x).
(13.38)
The corresponding homogeneous NOLDE (HNOLDE) is obtained by setting
r(x)
=
O.
Let us consider such a homogeneous case first. The solution to the
homogeneous NOLDE
L[y]
==
yCn)
+an_IY
C
n-
l)
+...+
alY'
+aOy= 0
(13.39)
characteristic
polynomial
of
an
HNOLOE
can be found by making the exponential substitotion y = e
Ax
,
which results in the
equation
L[e
Ax
]
=
(A"
+a,,_IAn-1 +...+alA +ao)e
AX
=
O.
This equation will
hold only
if
Ais a root of the
characteristic
polynomial
13.7
SOLDES
WITH
CONSTANT
COEFFICIENTS
377
which, by the fundamental theorem
of
algebra,
can
be written as
(13.40)
The Aj are the distinct (complex) roots
of
peA)
and
have multiplicity k],
13.7.1.
Theorem.
Let
(Aj
J7~1
be the roots
of
the characteristicpolynomial
ofthe
realHNOLDE
of
Equation (13.39), and let the respective roots have multiplicities
(kj
J7=1'
Then the functions
are a basis
of
solutions
of
Equation (13.39).
When
a A is complex, one can write its corresponding solution in terms
of
trigonometric functions.
13.7.2. Example. An
equation
that
isusedin both
mechanics
and
circuit
theory
is
for a, b >
O.
(13.41)
Its
characteristic
polynomial
is
pC},,)
=A
2
+
a'A
+b. whichhastheroots
Al =
!(-a+Ja
2
- 4b) and
1.2=
i(-a-Ja
2
- 4b).
Wecan
distinguish
three
different
possible
motions
depending
onthe
relative
sizesof
a andb.
(a)a
2
> 4b
(overdamped):
Herewe havetwo distinctsimpleroots.Themultiplicities are
bothone(kl = k
2
=
1);
therefore,
the
power
of x forboth
solutions
is zero
Crl
=
rz
= 0).
Let
y
==
1./
a
2
- 4b.
Then
themost
general
solution
is
Since a >
2y,
thissolution
starts
at y =
C1
+C2 att = 0
and
continuously
decreases;
so,
as t
-+
00,
yet)
-+
o.
(b)a
2
= 4b (criticallydamped): In thiscasewehaveonemultiplerootof order2(kl = 2);
therefore,
the
power
of x canbezeroor 1(rl = 0, 1).
Thus,
the
general
solution
is
yet) =
Cjte-
a
t/
2
+<oe-
al/
2
This
solution
starts
at y(0) =
Co
at t = 0,
reaches
a
maximum
(or
minimum)
at t =
2/a- colcr. and subsequently decays (grows)exponentiallyto zero.
(e)
a
2
< 4b
(underdamped):
Once
more,
we havetwo
distinct
simple
roots.
The
multi-
plicities arebothone
(kl
= k2 = 1);
therefore,
the
power
of x for
both
solutions
is zero
378 13.
SECOND-ORDER
LINEAR DIFFERENTIAL
EQUATIONS
(" =r: =0). Let'" '"
tv'4b
- 0
2.
ThenAl =
-0/2
+
ico
and
A2
=Ai. Theroots are
complex, andthemost
general
solutionis thusof the
form
yet) =
e-
atj2
(cr
COS(J)t
+
cz
sin
cu)
=
Ae-
atj2
cos(evt
+
a).
Thesolution is a
harmonic
variation
withadecaying
amplitude
A exp(
-at
/2). Notethatif
a = 0, the
amplitude
doesnot
decay.
Thatis why a is calledthedamping factor (orthe
damping
constant).
These
equations
describe
either
amechanical systemoscillating (withno
external
force)
in a viscous
(dissipative)
fluid,
or an electrical
circuit
consisting of a
resistance
R, an
inductance
L, anda capacitance C. For
RLC
circuits,a = RJL andb =
Ij(LC).
Thus,
the
damping
factor
depends
onthe
relative
magnitudes
of R andL. Onthe
other
hand,
the
frequency
depends
on all
three
elements. In
particular,
for R
~
2J
L/ c the
circuit
does not oscil-
- .
A physical systemwhosebehaviorin the absence
of
a driving force is described
by a NOLDE will obey an inhomogeneous
NOLDE
in the presence
of
the driving
force. This driving force is simply the inhomogeneous term
of
the NOLDE. The
best way to solve such an inhomogeneous
NOLDE
in its
most
general form is by
using Fourier transforms and Green's functions, as we will do in Chapter 20.
For
the particular, but important, case in which the inhomogeneous term is a product
of
polynomials and exponentials, the solution can be found in closed form.
13.7.3.
Theorem.
The NOLDE L[y1= e
Ax
S(x),
where
S(x)
is a polynomial, has
the particular solution eAxq(x), where
q(x)
is also a polynomial. The degree
of
q(x) equals that
of
S(x) unless A = Ai- a root
of
the characteristic polynomial
of
L, in which case the degree
of
q(x) exceeds that
of
S(x) by ki- the multiplicity
of
Aj.
Once we know the form
of
the particular solution
of
the NOLDE, we can find
the coefficients in the polynomial
of
the solution by substitnting in the NOLDE
and matching the powers on both sides.
13.7.4. Example. Letus
find
themost
general
solutions forthefollowingtwo
differential
equationssubjecttothe bonndarycooditions y(O) = 0 and
y'
(0) = 1.
(a)The
first
DE we wantto consideris
y" + y =
xe",
(13.42)
The characteristic polynomial
is).,2+I, whose roots areAl = i andA2 =
-i.
Thus,abasis
of solutions is {cosx, sinx}. To
find
the
particular
solutionwe note
that
A(thecoefficient
of
x in the
exponential
part
of theinhomogeneous
term)
is 1, whichis
neither
of theroots
Al and).2-Thus,the
particular
solutionis of the
form
q(x)e
X
,
whereq(x) = Ax + B is
13.7
SOLDES
WITH
CONSTANT
COEFFICIENTS
379
of degree I [same degree as that of S(x) = x]. We now substitute u = (Ax + B)e
X
in
Equation (13.42) to obtain the relation
2Axe
x
+(2A +
2B).'
=
xe",
Matching
thecoefficients, we have
2A = I
and
2A+2B=O
=}
A=!=-B.
Thus,themost
general
solution is
Y =
C1
cosx +
C2
sin x +
'i<x
- l)e
x.
Imposing the given boundary conditions yields 0 = y(O) =
Cj
- !and I = y' (0) = C2.
Thus,
y = !cos x +
sinx
+
!(x
-
l)e
X
is the
unique
solution.
(b)ThenextDEwe wantto
consider
is
y" _ y = xe",
(13.43)
Here
p(A) = A
2
- I, and the roots are At I and A2 =
-1.
A basis of solutions is
{eX,e-
X}.
To find a particular solotion, we note that S(x) = x and A= I = At. Theorem
13.7.3 then implies
thatq(x) must be of degree 2, because At is a simple root, i.e., kt =1.
We therefore try
q(x) =
Ax
2
+Bx +C
=}
u = (Ax
2
+
Bx
+C)e
x.
Taking the derivatives and substituting in Eqoation (13.43) yields two equations,
4A=
I
and
A+B=O,
whosesolution is A =
-B
=
1.
NotethatC is not
determined,
becauseCe" is asolution
of thehomogeneousDE corresponding to Equation (13.43), so when Lis appliedto u, it
eliminates the
term
C
ex.
Another
way of lookingatthe
situation
is to note
that
themost
general solution to (13.43) is of the form
The
term
C
ex
couldbe
absorbed
in creX. We
therefore
set C = 0, applythe
boundary
conditions, andfindthe
unique
solution
III
(13.44)
380 13.
SECOND-ORDER
LINEAR
DIFFERENTIAL
EQUATIONS
13.8 The WKB Method
In this section, we treat the somewhat specialized
method-due
to Wentzel,
Kramers, and Brillouin-e-ofobtainingan approximate solution to a particular type
of
second-order DE arising from the Schrodingerequationin
one
dimension. Sup-
pose we are interested in finding approximate solutions
of
the
DE
d
2
y
-2
+q(x)y
= 0
dx
in whichq
varies
"slowly"
with
respect
tox
in the sense discussed
below.
If
q
varies infinitely slowly, i.e.,
if
it is a constant, the solution to Equation (13.44)
is simply an imaginary exponential (or trigonometric). So, let us define
</>(x)
by
y =
ei¢(x)
and
rewrite the
DE
as
(<//)2
+
i</>"
- q =
O.
(13.45)
(13.46)
(13.47)
Assuming that
<//'
is small (compared to q), so that y does
not
oscillate too rapidly,
we can find an approximate solution to the DE:
<//
=
±.jii
=}
</>
= ± f
Jq(x)
dx.
The condition
of
validity
of
our
assumptionis obtained by differentiating (13.46):
1<//'1
'"
~
I
~I
«
Iql.
It follows from Equation(13.46)and the definition
of</>
that
1/
y7i
is approximately
1/
(2Jf) times one "wavelength"
of
the solution y. Therefore, the approximation is
valid
if
the change in qin one wavelength is small compared to
Iq
I.
The approximation can be improved by inserting the derivative
of
(13.46) in
the
DE
and solving for a new
</>:
. ,
(.
')
t/2
12
lq
,
zq
(</»
"'q±--
=}
</>
"'±
q±--
2y7i
2y7i
or
The two choices give rise to two different solutions, a linear combination
of
which
gives the
most
general solution. Thus,
y'"
~:(X)
{C
1
exp
[i
f
.jii
dX]
+C2 exp
[-i
f
.jii
dX]}
.
13.8
THE
WKB
METHOD
381
Equation (13.47) gives an approximate solution to (13.44) in any region in
whichthe conditionof validity holds. The method fails if
q changes too rapidly or
if
it is zero at some point
of
the region. The latter is a serious difficnlty, since we
often wish to
join
a solution in a region in which q
(x)
> 0 to one in a region in
which
q(x) <
O.
There is a generalprocedurefor deriving the so-calledconnection
formulas
relating the constants C1 and C2 of the two solutions on either side of the
point where
q(x)
=
O.
We shall not go into the details of such a derivation, as it is
not particularly illuminating.f We simply quote a particularresult that is useful in
applications.
Suppose that
q passes through zero at
xo,
is positive to the right of
xo,
and
satisfies the condition of validity in regions both to the right and to the left of
xo.
Furthermore, assume that the solution of the DE decreases exponentially to the
left of
xo.
Under such conditions, the solution to the left will be of the form
I l
xO
~exp[-
J-q(x)
dX],
-q(x)
x
while to the right, we have
(13.48)
(13.49)
2
~
I cos [l
X
Jq(x)
dx
-
::].
q(x)
xo
4
A similar procedure gives connection formulas for the case where q is positive on
the left and negative on the right
of
xo.
13.8.1. Example.
Consider
theSchrodinger
equation
in one
dimension
d
2
t 2m
dx
2
+
1<2
[E -
V(x)lt
= 0
where
V(x)
isa potentialwellmeeting the horizontal line
of
constantE atx = a and x = b,
so
that
2m
{>
0
q(x)
=
Z[E
- Vex)]
I<
< 0
if a < x < b,
if x < a or x > b.
Thesolution thatis
bounded
totheleft of a mustbeexponentially
decaying.
Therefore,
in
the
interval
(c,
b) the
approximate
solution, asgivenby
Equation
(13.49), is
t(x)
'"
A
1/4
cos(l
X
t~
[E - Vex)] dx -
::'4)
,
(E - V) a
I<
where
A is some
arbitrary
constant.
Thesolution thatis
bounded
to therightof b mustalso
beexponentially
decaying.
Hence,thesolution fora < x < b is
8The
interested
reader
is
referred
tothebookby
Mathews
and
Walker,
pp.27-37.
(13.50)
382 13.
SECOND-ORDER
LINEAR
DIFFERENTIAL
EQUATIONS
Since
these
two
expressions
givethesame
function
inthesame
region,
theymustbe
equal.
Thus,
A = B,
and,
more
importantly,
cos
(f
J~~
[E -
V(x)]
dx
-
~)
= cos
({
J~~
[E -
V(x)]
dx
-
~)
,
or
lab
JZm[E
-
V(x)]
dx
= (n +
!)rrn.
Thisis essentiallytheBohr-Sommerfeld
quantization
condition ofpre-1925
quantum
me-
chanics.
IIllII
13.8.1 Classical Limit of the Schrodlnger Equation
As long aswe are approximating solutions of second-orderDEs that arise naturally
from the Schrodinger eqnation, it is instructive to look at another approximation
to the Schr6dinger equation, its classical limit in which the Planck constant goes
to
zero.
The idea is to note that since
1/r(r,
t) is a complex function, one can write it as
1/r(r,
t)
=
A(r,
t)
exp
[~s(r,
t)]
,
where
A(r,
t) and
S(r,
t) are real-valued functions. Snbstituting (13.50) in the
Schrodinger equation and separating the real and the imaginary parts yields
(13.51)
These two equations are completely equivalent to the Schrodinger equation.
The second equation has a direct physical interpretation. Define
p(r,
t)
==
A
2
(r ,
t)
=
11/r(r,
t)1
2
and
VS
J(r, t) sa A
2
(r, t) - = pv,
m
___
(13.52)
=V
multiply the second equation in (13.51) by
2A/m,
and note that it then can be
written
as
ap
-+V·J=O,
at
(13.53)
wltich is the continuity equation for probability. The fact that
J is indeed the
probability current density is left for Problem 13.30.