QuantumPhysics.dvi

Before launching into any math, let’s solve the Schr ̈odinger equation forH 2 ,

i ̄h

∂

∂t

ψ(t,x) =i ̄hc

∂

∂x

ψ(t,x) (5.126)

or equivalently

(

∂

∂t

+c

∂

∂x

)

ψ(t,x) = 0 (5.127)

The general solution of this equation isψ(t,x) =f(x−ct) for any functionfof one variable.

The wave function corresponds to a purely right-moving particle. The Schr ̈odinger equation

turns out to be so restrictive that no left-moving particles can be allowed in the spectrum.

We now immediately see why imposing a boundary condition on this Hamiltonian could

be problematic. Both Dirichlet and Neumann boundary conditions would correspond to

a reflection of waves, which cannot happen. The only allowed boundary condition on an

interval would be periodic boundary conditions, since the right-moving wave could then

freely continue to travel without having to be reflected.

This may be seen concretely on the interval [0,ℓ], as follows,

(ψ,H 2 φ)−(H 2 ψ,φ) =i ̄hc

(

ψ(x)∗φ(x)

)∣∣

∣∣

x=ℓ

x=0

(5.128)

Self-adjointness ofH 2 requires the vanishing of the right hand side. If we requireψ(ℓ) =

ψ(0) = 0, thenφ(ℓ) andφ(0) can take any values, so this choice of domain does not lead to

a self-adjointH 2. Assuming now thatψ(0) 6 = 0, andφ(0) 6 = 0, the vanishing of (5.128) is

equivalent to,

ψ∗(ℓ)

ψ∗(0)

φ(ℓ)

φ(0)

= 1 (5.129)

whose general solution is given by the Bloch wave periodicity condition,

ψ(ℓ) = eiθψ(0)

φ(ℓ) = eiθφ(0) (5.130)

for a real parameterθ. The spectrum ofH 2 again depends onθ, since the eigenstate wave

functions are given by

ψn(x) =eiknx kn= (θ+ 2πn)/ℓ n∈Z (5.131)

The eigenvalues ofH 2 are then given by− ̄hckn, and are real. Notice that the eigenfunctions

are also mutually orthogonal for distinctn, as we indeed expect from a self-adjoint operator.