130_notes.dvi

If we integrate the last term by parts, (and the fields fall to zero at infinity), then that term contains a∇ ·~ E~ which is zer ...

✷χ=− [ ∂Aν ∂xν ] old TheMaxwell equation with the Lorentz conditionnow reads ✷Aμ=− jμ c . There isstill substantial gauge freedo ...

33 Quantum Theory of Radiation 33.1 Transverse and Longitudinal Fields In non-relativistic Quantum Mechanics, the static Electri ...

We will now study the radiation field in a region with no sources so that∇·~ A~= 0. We will use the equations B~ = ∇×~ A~ E~ = − ...

so that kρxρ=k·x=~k·~x−ωt. We can then write the radiation field in a more covariant way. A~(~x,t) =√^1 V ∑ k ∑^2 α=1 ˆǫ(α) ( ck ...

33.3 The Hamiltonian for the Radiation Field We now wish tocompute the Hamiltonian in terms of the coefficientsck,α(t). This is ...

The total Hamiltonian we are aiming at, is the integral of the Hamiltonian density. H= ∫ d^3 xH When we integrate over the volum ...

It waswith the Hamiltonian that we first quantized the non-relativistic motion of parti- cles. Theposition and momentum became o ...

This one checks out OK. Theother equation of Hamiltonis ∂H ∂Pk,α = Q ̇k,α Pk,α = 1 c ( ̇ck,α+ ̇c∗k,α) − iω c (ck,α−c∗k,α) = 1 c ...

Since we are dealing with harmonic oscillators, we want to find the analog of theraising and lowering operators. We developed th ...

constants. ak,α|nk,α〉 = √ nk,α|nk,α− 1 〉 a†k,α|nk,α〉 = √ nk,α+ 1|nk,α+ 1〉 Thenk,αcan only take oninteger valuesas with the harmo ...

33.6 Photon States It is now obvious that the integernk,αis thenumber of photons in the volumewith wave number~kand polarization ...

33.8 Quantized Radiation Field The Fourier coefficients of the expansion of the classical radiation field should now bereplaced ...

The total momentum in the (transverse) radiation field can also be computed (from the classical formula for the Poynting vector) ...

33.9 The Time Development of Field Operators The creation and annihilation operators are related to thetime dependent coefficien ...

Of course the average value of the Electric or Magnetic field vectoris zero by symmetry. To get an idea about the size of field ...

To make a narrow beam of light, one must adjust the phases of various components of the beam carefully. Another version of the u ...

〈n|Hint(abs)|i〉 = − e mc 1 √ V √ ̄hc^2 2 ω 〈ψn;n~k,α− 1 |ǫˆ(α)·~p ( ak,α(0)eikρxρ ) |ψi;n~k,α〉 = − e m 1 √ V √ ̄h 2 ω 〈ψn;n~k,α− ...

The approximation thate−i~k·~r≈1 is a very good one and is called theelectric dipole or E1 approximation. We previously derived ...

Putting these two vectors together is like adding toℓ= 1 states. We can get total angular momentum quantum numbers 2, 1, and 0. ...