MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

HERMITIAN OPERATORS IN QUANTUM MECHANICS 115 where 8.42 8.43 ries of a square wave can now be written as 8.7 HERMITIAN OPERATORS IN QUANTUM MECHANICS In quantum mechanics the state o

GENERALIZED FOURIER SERIES 113

pointwise convergence but not vice versa We conclude this section by stating

a theorem from Courant and Hilbert (p 427, vol I)

The expansion theorem: Any piecewise continuous function defined in the

fundamental interval [a, b] with a square integrable first derivative (i.e., sufficiently smooth) could be expanded in an eigenfunction series:

m=O which converges absolutely and uniformly in all subintervals free of points of discontinuity At the points of discontinuity this series r e p resents (as in the Fourier series) the arithmetic mean of the right- and the left-hand limits

In this theorem the function F (x) does not have to satisfy the boundary conditions This theorem also implies convergence in the mean and point- wise convergence That the derivative is square integrable means that the integral of the square of the derivative is finite for all the subintervals of the fundamental domain [a, b] in which the function is continuous

8.5 GENERALIZED FOURIER SERIES

Series expansion of a sufficiently smooth F (z) in terms of the eigenfunction set {urn ( 3 ) ) can now be written as

Using the basic definition of the Dirac-delta function, that is,

It is needless to say that this is not a proof of completeness

8.6 TRIGONOMETRIC FOURIER SERIES

Trigonometric Fourier series are defined with respect to the eigenvalue pro& lem

sin mx sin nxdx = A,&,,

cos mx cos nxdx = B,S,,,

(8.39)

(8.41)

sin mx cos nxdx = 0,

HERMITIAN OPERATORS IN QUANTUM MECHANICS 115

where

(8.42) (8.43)

ries of a square wave

can now be written as

8.7 HERMITIAN OPERATORS IN QUANTUM MECHANICS

In quantum mechanics the state of a system is completely described by a complex valued function, @(z), in terms of the real variable z Observable

quantities are represented by differential operators (not necessarily second order) acting on the wave functions These operators are usually obtained from their classical expressions by replacing position, momentum, and energy with their operator counterparts as

H = v2 + V ( Z ) The observable value of a physical property is given by the expectation value

of the corresponding operator L as

( L ) = /@*L@dx (8.51)

Because ( L ) corresponds to a measurable quantity it has to be real; hence observable properties in quantum mechanics are represented by Hermitian operators For the real Sturm-Liouville operators Hermitian property [Eq

(8.20)] was defined with respect to the eigenfunctions u and v, which sat- isfy the boundary conditions (8.13) and (8.15) To accommodate complex operators in quantum mechanics we modify this definition as

/ 9;L@adz = (L@1)*92dx, J (8.52) where 9land 9 2 do not have to be the eigenfunctions of the operator L The fact that Hermitian operators have real expectation values can be seen from

= /(L@)*@dx

HERMITIAN OPERATORS /N QUANTUM MECHANICS 117

A Hermitian Sturm-Liouville operator must be second order However,

in quantum mechanics order of the Hermitian operators is not restricted Remember that the momentum operator is first order, but it is Hermitian because of the presence of a in its definition:

a

ax

rm ( p ) = / 9*(-itz-)9dZ

A general boundary condition that all wave functions must satisfy is that they have to be square integrable, and thus normalizable Space of all square integrable functions actually forms an infinite dimensional vector space called

L 2 or the Hilbert space Functions in this space can be expanded as general-

ized Fourier series in terms of the complete and orthonormal set of eigenfunc- tions, {urn (z)}, of a Hermitian operator Eigenfunctions satisfy the eigenvalue equation

Lum(z> = Amum(z), (8.58) where A, represents the eigenvalues In other words, {urn(.)} spans the infinite dimensional vector space of square integrable functions The inner product (analog of dot product) in Hilbert space is defined as

and the

Schwartz inequality:

1911 I%l L I ( ~ l , * Z ) l (8.62)

An important consequence of the Schwartz inequality is that convergence of

( @ I , 9 2 ) follows from the convergence of (@I, 91) and ( 9 2 , @2)

can be brought into the self-adjoint form by multiplying it with e-"

8.2 Write the Chebyshev equation

(1 - X2)Tl(X) - XTL(X) +n2Tn(x) = 0

in the self-adjoint form

8.3 Find the weight function for the associated Laguerre equation

8.5 Show that the Legendre equation can be written as

d

dx -[(l - x">4] + l(Z+ 1)9 = 0

8.6 For the Sturm-Liouville equation

with the boundary conditions

Y(0) = 0

Y( ) - Y ' ( 4 = 0,

Hint: 73-y the substitution x = tant

8.8 Show that the Hermite equation can be written as

8.9 Given the Sturm-Liouville equation

If yn(x) and y,(x) are two orthogonal solutions and satisfy the appropriate boundary conditions, then show that &(x) and yA(x) are orthogonal with the weight function p(x)

8.13

show that they have the same eigenfunctions

a) What are their eigenvalues?

b) Write the L, and L, operators in spherical polar coordinates

Write the operators t2, and L, in spherical polar coordinates and

8.14 For a Sturm-Liouville operator

let u(z) be a nontrivial solution satisfying Xu = 0 with the boundary condition

a t z = a, and let V ( X ) be another nontrivial solution satisfying Lu = 0 with the boundary condition a t x = b Show that the Wronskian

is equal to A / p ( z ) , where A is a constant

8.15 For the inner product defined as

(*I, * 2 ) = J * ; ( x ) * 2 ( X ) d X 7 prove the following properties, where a is a complex number:

8.16 Prove the triangle inequality:

and the Schwartz inequality:

1*i1 I*2\ 2 l ( * i 7 * 2 ) l

8.17 Show that the differential equation

Y” +Pl(X)Y’ + [132(x) + Wx)ly(x) = 0 can be put into self-adjoint form as

9

FACTORIZATION

METHOD

The factorization method allows us to replace a Sturm-Liouville equation, which is a second-order differential equation, with a pair of first-order differ- ential equations For a large class of problems satisfying certain boundary conditions the method immediately yields the eigenvalues and allows us to write the ladder or the step-up/-down operators for the problem These o p

erators are then used to construct the eigenfunctions from a base function Once the base function is normalized, the manufactured eigenfunctions are also normalized and satisfy the same boundary conditions as the base func- tion First we introduce the method of factorization and its features in terms

of five basic theorems Next, we show how eigenvalues and eigenfunctions are obtained and introduce six basic types of factorization In fact, factor- ization of a given second-order differential equation is reduced to identifying the type it belongs to To demonstrate the usage of the method we discuss the associated Legendre equation and spherical harmonics in detail We also discuss the radial part of Schradinger’s equation for the hydrogen-like atoms, Gegenbauer polynomials, the problem of the symmetric top, Bessel functions, and the harmonic oscillator problem via the factorization method Further details and an extensive table of differential equations that can be solved by this technique can be found in Infeld and Hull (1951), where this method was introduced for the first time

121

9.1 ANOTHER FORM FOR T H E STURM-LIOUVILLE EQUATION

The Sturm-Liouville equation is usually written in the first canonical form

as

d

-& [ P ( X ) F ] + q(z)*(z) + AW(X)*(Z) = 0, 2 E [alp], (9.1)

where p(z) is different from zero in the open interval (a, p); however, it could have zeroes a t the end points of the interval We also impose the boundary conditions

and

where

eigenvalue Solutions also satisfy the orthogonality relation

and Ik are any two solutions corresponding to the same or different

If p(z) and w(.) are never negative and w ( ~ ) / p ( ~ ) exists everywhere in (a, p),

using the transformations

METHOD OF FACTORlZATlON 123

and

m = mo,mo + 1,mo + 2, (9.10) However, we could take mo = 0 without any loss of generality The orthogw nality relation is now given as

We now define two operators O*(z,m) as

dk(z,m) + k 2 ( z , m ) = -r(z,m)-p(m), (9.18)

dz

+ k 2 ( z , m + 1 ) = T(z,m)-p(m+l) (9.19) dk(z, m + 1)

dz

9.3 THEORY OF FACTORIZATION AND THE LADDER

OPERATORS

We now summarize the fundamental ideas of the factorization method in terms

of five basic theorems The first theorem basically tells us how to generate solutions with different m given yc(z)

Theorem I: If yc(z) is a solution of Equation (9.12) corresponding to the eigenvalues X and m, then

O+(z,m + I)YE(Z) = Y:+'(z) (9.20) and

o - ( z , m ) y g ( z ) = y:-W (9.21) are also solutions corresponding to the same X but different m as indi- cated

ProoE Multiply Equation (9.17) by O+(m + 1) :

O+(z,m+ 1) [ O - ( Z , ~ + l)O+(z,m+ l)yg(z)] (9.22)

THEORY OF FACTORIZATION AND THE LADDER OPERATORS 125

Similarly, 0- ( z , m) can be used to generate the solutions with the eigenvalues

".) (m - 3), (m - 2), (m - 1) (9.27) O*(z, m) are also called the step-up/-down or ladder operators

Theorem 11: If yl(z) and y2(z) are two solutions satisfying the boundary condition

then

We say that 0- and O+ are Hermitian, that is 0- = 0; with respect to yl(z) and y2(z) Note that the boundary condition [Eq (9.28)] needed for the factorization method is more restrictive than the boundary con- ditions [Eqs (9.2) and (9.3)] used for the Sturm-Liouville problem

Condition (9.28) includes the periodic boundary conditions as well as

the cases where the solutions vanish at the end points

Proof: Proof can easily be accomplished by using the definition of the ladder operators and integration by parts:

(9.30)

Finally, using the boundary condition [Eq (9.28)]) we write this as

(9.31)

Solution 0- ( z , m ) y E ( z ) in Equation (9.21) is equal toyE-'(z) only up

to a constant factor Similarly, o+(z, m ) y E - l ( z ) is only equal to yg(z)

up to another constant factor Thus we can write

where C(1, m) is a constant independent of z but dependent on 1 and m

We are interested in differential equations the coefficients of which may have singularities only at the end points of our interval Square inte grability of a solution actually depends on the behavior of the solution near the end points Thus it is itself a boundary condition Hence, for

THEORY OF FACTORIZATION AND THE LADDER OPERATORS 127

a given square integrable eigenfunction yT ( z ) , the manufactured eigen-

function yc-'(z) is also square integrable as long as C(I, m ) is different from zero Because we have used Theorem 11, yz-'(z) also satisfies the same boundary condition as yc(z) A parallel argument is given for yz+'(z) In conclusion, if yc(z) is a square integrable function satisfy- ing the boundary condition [Eq (9.28)], then all other eigenfunctions manufactured from it by the ladder operators O+(z,m) are square inte- grable and satisfy the same boundary condition For a complete proof

C(I, m) must be studied separately for each factorization type For our purposes it is sufficient to say that C(1, m) is different from zero for all physically meaningful cases

Theorem IV If p(m) is an increasing function and m > 0, then there exists

a maximum value form, say mmax = 1, and X is given as X = p(1+ 1) If

p ( m ) is a decreasing function and m > 0, then there exists a minimum value for m, say mmin = l', and X is X = p(l')

Proof: Assume that we have some function yx";(z), where m > 0 , which satisfies the boundary condition [Eq (9.28)] We can then write

(9.40)

If p(m) is an increasing function of m, eventually we are going to reach

a value of m, say mmax = 1, that leads us to the contradiction

unless

128 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD

0- ( Z , l ) & ( 2 ) = 0 (9.46)

A in this case is determined as

Cases for m < 0 are also shown in Figure 9.1

We have mentioned that the square integrability of the solutions is itself

a boundary condition, which is usually related to the symmetries of the

THEORY OF FACTORIZATION AND THE LADDER OPERATORS 1%

problem For example, in the case of the associated Legendre equation the end points of our interval correspond to the north and south poles

of a sphere For a spherically symmetric problem, location of the poles

is arbitrary Hence useful solutions should be finite everywhere on a sphere In the Frobenius method this forces us to restrict X to certain integer values (Chapter 2) In the factorization method we also have

to restrict A, this time through equation (9.40) to ensure the square

integrability of the solutions for a given p(m)

Theorem V: When Theorem I11 holds, we can arrange the ladder operators

to preserve not just the square integrability but also the normalization

of the eigenfunctions When p(m) is an increasing function of m, we can define new normalized ladder operators

which ensures us the normalization of the manufactured solutions When p(m) is a decreasing function, normalized ladder operators are defined as

Proof: Using the last equation in Equation (9.40) we write

Thus, if y c ( z ) is normalized, then the eigenfunction manufactured from yc(z) by the operator -E+ is also normalized Similarly, one could show that

are also normalized Depending on the functional forms of p ( m ) , L*( z, I , m)

are given in Equations (9.48) and (9.49)

9.4 SOLUTIONS VIA THE FACTORIZATION METHOD

We can now manufacture the eigenvalues and the eigenfunctions of an equa- tion once it is factored, that is, once the k ( z , m ) and the p ( m ) functions cor- responding to a given r(z, m) are known For m > 0, depending on whether

p ( m ) is an increasing or a decreasing function, there are two cases

9.4.1

In this case, from Theorem IV there is a maximum value for m,

Case I ( m > 0 and p ( m ) is an increasing function)

m = 0 , 1 , 2 , , 1, (9.54) and the eigenvalues are given as

x = x1 = p(2 + 1) (9.55) Since there is no eigenstate with m > 1, we can write

0 + ( z , 1 + l)yl'(z) = 0 (9.56)

SOLUTIONS VIA THE FACTORIZATION METHOD 131

Thus we obtain the differential equation

For a given 1, once y;“”(z) is found, all the other normalized eigenfunctions with

m = 1,l- 1,l- 2, , 2,1,0, can be constructed by repeated applications of the step down operator L- ( z , l , m) as

9.4.2

In this case, from Theorem IV there is a minimum value for m, where

Case I I ( m > 0 and p ( m ) is a decreasing function)

C is again determined from the normalization condition

9.5 TECHNIQUE AND THE CATEGORIES OF FACTORIZATION

In Section 9.2 we saw that in order to accomplish factorization we need to determine the two functions k ( z , m ) and p(m), which satisfy the two equations