130_notes.dvi

= − 2 α ̄h 3 πm^2 c^2 ∑ j ωcut∫−off 0 ωnj ω−ωnj |〈n|~p|j〉|^2 dω = − 2 α ̄h 3 πm^2 c^2 ∑ j ωnj[log(ω−ωnj)]ω 0 cut−off|〈n|~p|j〉|^2 ...

= − ̄h^2 2 〈n|e^2 δ^3 (~x)|n〉 = − e^2 ̄h^2 2 |ψn(0)|^2 Only thes states will have a non-vanishing probability to be at the origi ...

theorists to develop Quantum ElectroDynamics. The correction increases the energy of s states. One may think of the physical ori ...

36 Dirac Equation 36.1 Dirac’s Motivation TheSchr ̈odinger equation is simply the non-relativistic energy equationoperating on a ...

36.2 The Schr ̈odinger-Pauli Hamiltonian In the homework on electrons in an electromagnetic field, we showedthatthe Schr ̈odinge ...

36.3 The Dirac Equation We canextend this concept to use the relativistic energy equation(for now with no EM field). The idea is ...

Now take the sum and the difference of the two equations. i ̄h~σ·∇~(φ(L)−φ(R))−i ̄h ∂ ∂x 0 (φ(R)+φ(L)) +mc(φ(R)+φ(L)) = 0 i ̄h~σ ...

̄hγμ ∂ ∂xμ ψ+mcψ= 0 TheDirac equation in the absence of EM fieldsis ( γμ ∂ ∂xμ + mc ̄h ) ψ= 0. ψis a4-component Dirac spinorand, ...

that gave us the correct interaction with the electron’s spin. We have applied this same momentum operator relativistically, not ...

Our problem to find a similarprobability and flux for Dirac theory is similarbut a little more difficult. Start with the Dirac e ...

We have found aconserved current. Some interpretation will be required as we learn more about the solutions to the Dirac equatio ...

This is the two component equation which is equivalent to the Dirac equation for energy eigenstates. The one difference from our ...

We will be attempting to get the correct Schr ̈odinger equation to orderα^4 , like the one we used to calculate the fine structu ...

We have only kept terms to orderα^4. Now we mustsimplify two of the terms in the equation which contain the momentum operator ac ...

This was an important test of the Dirac equation and it passed with flying colors. TheDirac equa- tion naturally incorporatesrel ...

ψ~p=u~pei(~p·~x−Et)/ ̄h Plugging this into the equation, we get arelation between the momentum and the energy. −p^2 ̄h^2 + E^2 ̄ ...

   1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 − 1       A 1 A 2 B 1 B 2   = +    A 1 A 2 B 1 B 2       A 1 A 2 −B 1 −B ...

With this matrix defining the spin, thethird component is the one with spin upalong the z direction for the “negative energy sol ...

We should find four solutions. Lets start with one that gives aspin up electronin the first two components and plug it into the ...

We again see that for a non-relativistic electron, the last two components are small compared to the first. This solution is tha ...