130_notes.dvi

To make a narrow beam of light, one must adjust the phases of various components of the beam
carefully. Another version of the uncertainty relation is that ∆N∆φ≥1, wherephiis the phase of
a Fourier component andNis the number of photons.

Of course the Electromagneticwaves of classical physics usually have very large numbers
of photonsand the quantum effects are not apparent. A good condition to identify the boundary
between classical and quantum behavior is that for the classical E&M to be correct thenumber of
photons per cubic wavelength should be much greater than 1.

33.11Emission and Absorption of Photons by Atoms

The interaction of an electron (See section 29.1) with the quantizedfield is already in thestandard
Hamiltonian.

H =

1

2 m

(

~p+

e

c

A~

) 2

+V(r)

Hint = −

e

2 mc

(~p·A~+A~·~p) +

e^2

2 mc^2

A~·A~

= −

e

mc

A~·~p+ e

2

2 mc^2

A~·A~

For completeness we should add theinteraction with the spin of the electronH=−~μ·B~.

Hint=−

e

mc

A~·~p+ e

2

2 mc^2

A~·A~− e ̄h

2 mc

~σ·∇×~ A~

For an atom with many electrons, we mustsum over all the electrons. The field is evaluated at
the coordinatexwhich should be that of the electron.

This interaction Hamiltonian contains operators to create and annihilate photons with transitions
between atomic states. From our previous study of time dependent perturbation theory (See section
28.1), we know that transitions between initial and final states areproportional to thematrix
element of the perturbing Hamiltonianbetween the states,〈n|Hint|i〉. The initial state|i〉
should include adirect product of the atomic state and the photon state. Lets concentrate
on one type of photon for now. We then could write

|i〉=|ψi;n~k,α〉

with a similar expression for the final state.

We will first consider theabsorption of one photon from the field. Assume there aren~k,α
photons of this type in the initial state and that one photon is absorbed. We therefore will need
a term in the interaction Hamiltonian that contains on annihilation operator (only). This will just
come from the linear term in A.

〈n|Hint|i〉 = 〈ψn;n~k,α− 1 |−

e

mc

A~·~p|ψi;n~

k,α〉

= −

e

mc

〈ψn;n~k,α− 1 |

1

√

V

√

̄hc^2

2 ω

ǫˆ(α)

(

ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ

)

·~p|ψi;n~k,α〉