QuantumPhysics.dvi

5.9.3 Example 3: One-dimensional Dirac-like operator in a box

A lesson we have learned from Example 2 is that the momentum operator corresponds to

a quantum system of left-movers only, (and no right-movers), a situation that vitiates the

possibility of imposing any reflecting boundary conditions. We may double the number of

degrees of freedom, however, and include one left-moving and oneright-moving degree of

freedom. Thus, we consider doublets of wave functionsψ 1 andψ 2 ,

ψ≡

(

ψ 1

ψ 2

)

φ≡

(

φ 1

φ 2

)

(5.132)

with Hermitean inner product,

(ψ,φ) =

∫ℓ

0

dx(ψ∗ 1 φ 1 +ψ∗ 2 φ 2 ) (x) (5.133)

and Hamiltonian (here we use the standard notation∂x≡∂/∂x),

H 3 = ̄hc

(

0 ∂x

−∂x 0

)

(5.134)

It is straightforward to evaluate the combinations,

(H 3 ψ,φ) = ̄hc

∫ℓ

0

dx

(

∂xψ 2 ∗φ 1 −∂xψ∗ 1 φ 2

)

(x)

(ψ,H 3 φ) = ̄hc

∫ℓ

0

dx

(

ψ 1 ∗∂xφ 2 −ψ∗ 2 ∂xφ 1

)

(x) (5.135)

and

(H 3 ψ,φ)−(ψ,H 3 φ) = ̄hc

(

ψ 2 φ 1 (ℓ)−ψ∗ 1 φ 2 (ℓ)−ψ 2 φ 1 (0) +ψ 1 ∗φ 2 (0)

)

(5.136)

Self-adjointness ofH 3 requires this combination to vanish. This may be achieved by imposing,

for example, theMIT bag boundary conditions (which were introduced to model quarks

confined to nucleons),

φ 2 (ℓ) =λℓφ 1 (ℓ) φ 2 (0) =λ 0 φ 1 (0)

ψ 2 (ℓ) =λℓψ 1 (ℓ) ψ 2 (0) =λ 0 ψ 1 (0) (5.137)

for two real independent constants λ 0 andλℓ. The spectrum ofH 3 is now real, but does

depend uponλ 0 andλℓ.