QuantumPhysics.dvi

This equation is of a well-known form, namely that of a confluent hypergeometric function,

or Kummer function,

gℓ,ε(x) =M

(

ℓ+ 1−

iε

2

, 2 ℓ+ 2, 2 ix

)

(5.116)

To get a better idea of what these functions are, it is useful to givean integral representation,

M(a,b,z) =

Γ(b)

Γ(b−a)Γ(a)

∫ 1

0

dtetzta−^1 (1−t)b−a−^1 (5.117)

Most of the essential properties ofM, such as its asymptotics for large and smallz, may be

read off directly from this representation.

5.9 Self-adjoint operators and boundary conditions

The precise definition of self-adjointness is not just a matter of mathematical sophistication,

but instead has physical consequences. On the one hand, a given differential operator may

allow for inequivalent domains, resulting in inequivalent physical spectra. On the other hand,

an operator that “looks” self-adjoint, but is not actually self-adjoint, may not have a real

spectrum, and/or mutually orthogonal eigenspaces associated with distinct eigenvalues. We

illustrate these possibilities below with the help of some concrete examples.

5.9.1 Example 1: One-dimensional Schr ̈odinger operator on half-line

Consider a 1-dim quantum system given by the following Hamiltonian,

H 1 =−

̄h^2

2 m

d^2

dx^2

+V(x) (5.118)

As an operator on complex functions on the real line, and for a “reasonable” potentialV(x),

the HamiltonianH 1 is self-adjoint. The domainD(H 1 ) may be taken to be the sub-space of

L^2 (R) consisting of infinitely differentiable functions (denotedC∞) which vanish atx=±∞.

Self-adjointness then follows from the fact that all functions inD(H 1 ) vanish atx=±∞.

As an operator on functions onx∈[0,+∞], the HamiltonianH 1 may be self-adjoint for

certain choices of the domain, but not for others. The key relationis as follows,

(ψ,H 1 φ)−(H 1 ψ,φ) =

̄h^2

2 m

j(0) (5.119)

wherej(x) is the probability current density, defined by

j(x) =φ(x)

dψ∗

dx

(x)−ψ∗(x)

dφ

dx

(x) (5.120)