wang
(Wang)
#1
The operatorH 1 will be self-adjoint if a domainD(H 1 ) can be chosen for the functionsψ
andφsuch thatj(0) = 0 forany pairψ,φ∈ D(H 1 ). This will be the case if the domain is
defined to be the subspace ofC∞functions which obey
ψ(0) +λ
dψ
dx
(0) = 0 (5.121)
for a given real constantλ. Note thatλ= 0 and corresponds to Dirichlet boundary condi-
tions, whileλ=∞corresponds to Neumann boundary conditions. Thus,λparametrizes an
interpolation between Dirichlet and Neumann boundary conditions, and for each value ofλ,
H 1 is self-adjoint.
The spectrum ofH 1 depends onλeven though, at face value, the differential operatorH 1
of (5.118) does not involveλ. Consider, for example, the problem with a potentialV(x) = 0
for 0≤x < ℓ, andV(x) = +∞forx≥ℓ, and solve for the eigenvaluesEofH 1. To satisfy
the vanishing boundary condition atx=ℓ, we must have
ψ(x) = sink(x−ℓ) E=
̄h^2 k^2
2 m
(5.122)
wherekcan be any positive real number. Enforcing also theλ-dependent boundary condition
(5.121) atx= 0 renders the spectrum ofkdiscrete, and requires,
tg(kℓ) =λk (5.123)
We recover the special cases,
λ= 0 kn=
2 nπ
2 ℓ
n= 1, 2 , 3 ,···
λ=∞ kn=
(2n−1)π
2 ℓ
(5.124)
For intermediate values ofλ, the solutions are transcendental, and may be determined graph-
ically. They clearly interpolate between the above cases. Thus, thespectrum depends on
the precise domain, through the boundary conditions.
5.9.2 Example 2: One-dimensional momentum in a box
Next, consider the 1-dim quantum system given by the Hamiltonian,
H 2 =i ̄hc
d
dx
(5.125)
As an operator on functions on the real line,H 2 is self-adjoint. But what happens when we
attempt to put the system in a box? For example, canH 2 be self-adjoint when acting on
functions on the finite interval [0,ℓ] withℓ >0?
