1. Show that the relativistic relation between energy and momentum (Equation 15.2) reduces to...for the case when v ≪ c. Get solution
2. If ϕ is an eigenfunction of both energy and momentum, then another differential equation corresponding to Equation (15.2) is...Why is this a less desirable equation than the Klein–Gordon equation? Get solution
3. (a) If J is the Schrödinger probability current, show that...(b) What are the units of J? Get solution
4. Using the Klein–Gordon equation, the continuity equation, and the expression for J from Equation (15.10), derive the Klein–Gordon probability density:... Get solution
5. Write out explicitly the full 4 × 4 matrices corresponding to α1 and α2. Get solution
6. Multiply out the matrices in the Dirac equation to express the Dirac equation as four coupled differential equations for the four components of ψ: ψ1, ψ2, ψ3 and ψ4. Get solution
7. Write down the Dirac spinor corresponding to a spin-1/2 particle at rest with spin in the +x direction and positive energy. Get solution
8. (a) The general solution for the Dirac equation can be written in the form...where ϕ1, ϕ2, χ1, and χ2 are numbers independent of r and t. To take advantage of this form for the Dirac equation, use the shorthand...and...Using this form for the solution, show that ϕ and χ satisfy the coupled equations...and...(b) Use the results from part (a) to show that the general four-component solution of the Dirac equation may be written as... Get solution
2. If ϕ is an eigenfunction of both energy and momentum, then another differential equation corresponding to Equation (15.2) is...Why is this a less desirable equation than the Klein–Gordon equation? Get solution
3. (a) If J is the Schrödinger probability current, show that...(b) What are the units of J? Get solution
4. Using the Klein–Gordon equation, the continuity equation, and the expression for J from Equation (15.10), derive the Klein–Gordon probability density:... Get solution
5. Write out explicitly the full 4 × 4 matrices corresponding to α1 and α2. Get solution
6. Multiply out the matrices in the Dirac equation to express the Dirac equation as four coupled differential equations for the four components of ψ: ψ1, ψ2, ψ3 and ψ4. Get solution
7. Write down the Dirac spinor corresponding to a spin-1/2 particle at rest with spin in the +x direction and positive energy. Get solution
8. (a) The general solution for the Dirac equation can be written in the form...where ϕ1, ϕ2, χ1, and χ2 are numbers independent of r and t. To take advantage of this form for the Dirac equation, use the shorthand...and...Using this form for the solution, show that ϕ and χ satisfy the coupled equations...and...(b) Use the results from part (a) to show that the general four-component solution of the Dirac equation may be written as... Get solution
