(4.2) The relativistic equations studied in Chapter 3 gen- erally predict the the corresponding particles have...
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(4.2) The relativistic equations studied in Chapter 3 gen- erally predict the the corresponding particles have Landé g factor equal to 2. We can explore this for particles of spin using the Dirac equation. (a) A field obeying the Dirac equation in the pres- ence of a background electromagnetic field also obeys the second-order equation (iy Du + m)(iy Dv m) = 0, (4.45) where Du = (Ou + ie Au). Simplify this equa- tion by using the identity y²y² = = √ {y²², ²} + { [v²v²]. (4.46) and show that it reduces to the Klein-Gordon equation plus one extra term. 8² Ət² ² + m²)o(t, 2) = 0. (b) Simplify the new term by proving the identity [Du, Dv] = +ieFuv. (4.47) Using the explicit form of the y matrices, evaluate this term in a background magnetic field for which Fij = −¤ijkBh and Foi = 0. V = (c) Act the resulting equation on the Dirac equa- tion solution (3.50). (3.2) = ($) ₁- e Show that, to first order in B, the energy of the state is shifted by a term of the form of AE = - B. In the expression for , identify g = 2. -imt (3.50) (4.2) The relativistic equations studied in Chapter 3 gen- erally predict the the corresponding particles have Landé g factor equal to 2. We can explore this for particles of spin using the Dirac equation. (a) A field obeying the Dirac equation in the pres- ence of a background electromagnetic field also obeys the second-order equation (iy Du + m)(iy Dv m) = 0, (4.45) where Du = (Ou + ie Au). Simplify this equa- tion by using the identity y²y² = = √ {y²², ²} + { [v²v²]. (4.46) and show that it reduces to the Klein-Gordon equation plus one extra term. 8² Ət² ² + m²)o(t, 2) = 0. (b) Simplify the new term by proving the identity [Du, Dv] = +ieFuv. (4.47) Using the explicit form of the y matrices, evaluate this term in a background magnetic field for which Fij = −¤ijkBh and Foi = 0. V = (c) Act the resulting equation on the Dirac equa- tion solution (3.50). (3.2) = ($) ₁- e Show that, to first order in B, the energy of the state is shifted by a term of the form of AE = - B. In the expression for , identify g = 2. -imt (3.50)
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a By using the identity in Equation 446 we can simplify Equation 445 to iy Dum iy Dvm DDv DvD 2m2 Mu... View the full answer
Related Book For
Applied Regression Analysis and Other Multivariable Methods
ISBN: 978-1285051086
5th edition
Authors: David G. Kleinbaum, Lawrence L. Kupper, Azhar Nizam, Eli S. Rosenberg
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