Quantum Physics Class
Section VI
Relativistic quantum mechanics, Dirac equation.
1) Show invariancy of the Klein–Gordon equation with respect to Lorentz transformation.
2) Let ˆH denote the Dirac Hamiltonian
H = c ˆˆ α ˆp + mc2β ,ˆ
where c stands for the light speed, ˆα = ( ˆαx, ˆαy, ˆαz), ˆβ are hermition operators which do not depend on time and space coordinates. We assume that
Hˆ2 = c2pˆ2+ m2c4.
a) Show that operator ˆαi, i = x, y, z, i ˆβ anticommutate in pairs.
b) Show that the eigenvalues of ˆαi and ˆβ are given by ±1.
c)Find a realization of ˆα and ˆβ.
3) Describe solutions of the Dirac equation for a free particle.
4 Consider a particle of the magnetic moment µ placed in the magnetic field B. Show that under the gauge transformation p → p −ecA, i¯h∂t → i¯h∂t+ecφ the Dirac equation takes the form
H − eφ c
!2
= |p − e
cA|2+ m2c2− e¯h c ˆσ · B , where ˆσ is the operator valued vector defined by means of ˆαi. a) Find ˆσ.
b) Derive non-relativistic approximation.
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