dx dx

we have

—

u, Lv ’ Lu, v = [p(u— v ’ u v)]b . (4.61)

a

Let us seek to impose boundary conditions separately at the two ends. Thus,

at x = a we want

—

(u— v ’ u v)|a = 0, (4.62)

4.2. THE ADJOINT OPERATOR 95

or

u — (a) v (a)

= , (4.63)

u—(a) v(a)

and similarly at b. If we want the boundary conditions imposed on v (which

de¬ne the domain of L) to coincide with those for u (which de¬ne the domain

of L† ) then we must have

v (a) u (a)

= = tan θa (4.64)

v(a) u(a)

for some real angle θa , and similar boundary conditions with a θb at b. We

can also write these boundary conditions as

±a y(a) + βa y (a) = 0,

±b y(b) + βb y (b) = 0. (4.65)

De¬ciency Indices

There is a general theory of self-adjoint boundary conditions, due to Hermann

Weyl and John von Neumann. We will not describe this theory in any detail,

but simply quote their recipe for counting the number of parameters in the

most general self-adjoint boundary condition: To ¬nd this number you should

¬rst impose the strictest possible boundary conditions by setting to zero the

boundary values of all the y (n) with n less than the order of the equation.

Next count the number of square-integrable eigenfunctions of the resulting

adjoint operator T † corresponding to eigenvalue ±i. The numbers, n+ and

n’ , of these eigenfunctions are called the de¬ciency indices. If they are not

equal then there is no possible way to make the operator self-adjoint. If

they are equal, n+ = n’ = n, then there is an n2 real-parameter family of

self-adjoint boundary conditions.

Example: The sad case of the “radial momentum operator.” We wish to

de¬ne the operator Pr = ’i‚r on the half-line 0 < r < ∞. We start with the

restrictive domain

D(T ) = {y, Pr y ∈ L2 [0, ∞] : y(0) = 0}.

Pr = ’i‚r , (4.66)

We then have

Pr† = ’i‚r , D(Pr† ) = {y, Pr†y ∈ L2 [0, ∞]} (4.67)

96 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

with no boundary conditions. The equation Pr† y = iy has a normalizable

solution y = e’r . The equation Pr† y = ’iy has no normalizable solution.

The de¬ciency indices are therefore n+ = 1, n’ = 0, and this operator

cannot be rescued and made self adjoint.

2

Example: The Schr¨dinger operator. We now consider ’‚x on the half-line.

o

Set

2

D(T ) = {y, T y ∈ L2 [0, ∞] : y(0) = y (0) = 0}.

T = ’‚x , (4.68)

We then have

T † = ’‚x ,

2

D(T † ) = {y, Tr†y ∈ L2 [0, ∞]}. (4.69)

Again T † comes with no boundary conditions. The√eigenvalue equation

T † y = iy has one normalizable solution y(x) = e(i’1)x/ 2 , and √the equation

T † y = ’iy also has one normalizable solution y(x) = e’(i+1)x/ 2 . The de¬-

ciency indices are therefore n+ = n’ = 1. The Weyl-von Neumann theory

now says that, by relaxing the restrictive conditions y(0) = y (0) = 0, we

can extend the domain of de¬nition of the operator to ¬nd a one-parameter

family of self-adjoint boundary conditions. These will be the conditions

y (0)/y(0) = tan θ that we found above.

2

If we consider the operator ’‚x on the ¬nite interval [a, b], then both

solutions of (T † ± i)y = 0 are normalizable, and the de¬ciency indices will

be n+ = n’ = 2. There should therefore be 22 = 4 real parameters in the

self-adjoint boundary conditions. This is a larger class than those we found

in (4.65), because it includes generalized boundary conditions of the form

B1 [y] = ±11 y(a) + ±12 y (a) + β11 y(b) + β12 y (b) = 0,

B2 [y] = ±21 y(a) + ±22 y (a) + β21 y(b) + β22 y (b) = 0

The next problem illustrates why we have spent so much time on identify-

ing self-adjoint boundary conditions: the technique is important in practical

physics problems.

Physics Application: Semiconductor Heterojunction. A heterojunction is

fabricated with two semiconductors, say GaAs and Alx Ga1’x As, having dif-

ferent band-masses. We wish to describe the conduction electrons in the

material by an e¬ective Schr¨dinger equation containing these band masses.

o

What matching condition should we impose on the wavefunction ψ(x) at

the interface between the two materials? A ¬rst guess is that the wavefunc-

tion must be continuous, but this is not correct because the “wavefunction”

4.2. THE ADJOINT OPERATOR 97

in an e¬ective-mass band-theory Hamiltonian is not the actual wavefunc-

tion (which is continuous) but instead a slowly varying envelope function

multiplying a Bloch wavefunction. The Bloch function is rapidly varying,

¬‚uctuating strongly on the scale of a single atom. Because the Bloch form

of the solution is no longer valid at a discontinuity, the envelope function is

not even de¬ned in the neighbourhood of the interface, and certainly has no

reason to be continuous. There must still be some linear relation beween the

ψ™s in the two materials, but ¬nding it will involve a detailed calculation on

the atomic scale. In the absence of these calculations, we must use general

principles to constrain the form of the relation. What are these principles?

ψ ψ

L R

? x

GaAs: m L AlGaAs:m R

Heterojunction wavefunctions.

We know that, were we to do the atomic-scale calculation, the resulting

connection between the right and left wavefunctions would:

• be linear,

• involve no more than ψ(x) and its ¬rst derivative ψ (x),

• make Hamiltonian into a self-adjoint operator.

We want to ¬nd the most general connection formula compatible with these

principles. The ¬rst two are easy to satisfy. We therefore investigate what

matching conditions are compatible with self-adjointness.

Suppose that the band masses are mL and mR , so that

1 d2

H=’ + VL (x), x < 0,

2mL dx2

1 d2

=’ + VR (x), x > 0. (4.70)

2mR dx2

Integrating by parts, and keeping the terms at the interface gives us

1 1

— —

— —

ψ1 , Hψ2 ’ Hψ1 , ψ2 = ψ1L ψ2L ’ ψ 1L ψ2L ’ ψ1R ψ2R ’ ψ 1R ψ2R .

2mL 2mR

(4.71)

98 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

Here, ψL,R refers to the boundary values of ψ immediately to the left or right

of the junction, respectively. Now we impose general linear homogeneous

boundary conditions on ψ2 :

ψ2L ab ψ2R

= . (4.72)

ψ2L cd ψ2R