QuantumPhysics.dvi

5.4 Propagation on the full line

In the problem of propagation on the circle, we may finally take the infinite volume limit

whereL→∞. This limit is identical to the limit where the lattice spacinga→0 is taken in

the problem of propagation on the infinite lattice. Both position and momentum operators

now have continuous spectra,

[X,P] =ihI ̄ X|x;X〉=x|x;X〉 x∈R

P|k;P〉= ̄hk|k;P〉 k∈R (5.41)

Since the operatorsXandPare self-adjoint, their eigenvalues are real, and the eigenspaces

corresponding to different eigenvalues are orthogonal to one another. The normalization of

each eigenstate may be chosen at will, since a quantum state corresponds to a vector in

Hilbert space, up to an overall complex multiplicative factor. The normalizations of the two

sets of basis vectors|x;X〉and|k;P〉may be chosen independently of one another. It will

be convenient to choose them as follows,

〈x′;X|x;X〉 = δ(x′−x)

〈k′;Pk;P〉 = 2πδ(k′−k) (5.42)

As a result, the completeness relations are completely determined (a complete derivation of

these results, for any Hamiltonian, will be given in section 5.5),

I=

∫

R

dx|x;X〉〈x;X|=

∫

R

dk

2 π

|k;P〉〈k;P| (5.43)

The overlap between a state in theX-basis, and a state in theP-basis, is given by,

〈x;X|k;P〉 = e+ikx

〈k;P|x;X〉 = e−ikx (5.44)

The energy relation is

Ep=

p^2

2 M

(5.45)

Notice that the completeness relation is just the formula for the Fourier transform.

〈x′;X|x;X〉=δ(x−x′) =

∫

R

dk

2 π

〈x′;X|k;P〉〈k;P|x;X〉=

∫

R

dk

2 π

eik(x

′−x)

(5.46)