By Professor Dr. Walter Greiner (auth.)
ISBN-10: 3540616217
ISBN-13: 9783540616214
ISBN-10: 3662034255
ISBN-13: 9783662034255
Relativistic Quantum Mechanics - Wave Equations concentrates typically at the wave equations for spin-0 and spin-1/2 debris. bankruptcy 1 offers with the Klein-Gordon equation and its houses and purposes. The chapters that stick to introduce the Dirac equation, examine its covariance homes and current a number of techniques to acquiring options. a variety of purposes are mentioned intimately, together with the two-center Dirac equation, gap thought, CPT symmetry, Klein's paradox, and relativistic symmetry rules. bankruptcy 15 provides the relativistic wave equations for larger spin (Proca, Rarita-Schwinger, and Bargmann-Wigner). The vast presentation of the mathematical instruments and the sixty two labored examples and difficulties make this a distinct textual content for a sophisticated quantum mechanics path.
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Additional info for Relativistic Quantum Mechanics: Wave Equations
Sample text
L + 1 = c (13) so that results. Now we try to solve the differential equation for/((}) with the help of a power series expansion, is not true in the special case I = 0. Here the solution with v_ can also be normalized, and one must set another criterion in order to exclude this case. For example, one can demand that an expectation value of the kinetic energy should exist. 9 in which the analogue problem for the Coulomb solutions of the Dirac equation is carefully discussed. n'-1 -a ~ ~an'(! n'-1 =0 (15) n'=O n'=1 A comparison of coefficients of equal powers in e gives aao a1=- a2 = c a1(a + 1) 2(c + 1) (16) or, generally am-1 a +m -1 am=-----m c+m-1 (17) Therefore, f(e) can be written as 2 00 I ~ (a)n' en ) a a+ 1 (!
E. lia = O ' (a = 1' 2) 2 A ' A >) A) = - 2fimo v2 (;r;< ~ 73 + 172 a+ moe 2 (;r; ~73 a If we had defined the Lagrange density with l]it instead of~. the same equation of motion (1) would have resulted from the variation with respect to lli!. 7. ;:2 ( f 3+ if2) - moc 2f3) lli] d3x dt 1 must be real. 110)] to be true. The integral I would not be real if, instead of W, the spinor y;t had been used. e. 108). 108) we can now guess the generalization of the expectation values of arbitrary operators Land define the mean value (expectation value) (L) by (L) = j wtf3Lllid3x , where (L) must be real.
1 2m0 + V(x) is the Hamiltonian. c r. _V'l/J*·V'l/J+'l/J*V'l/J=H 2mo , (11) ° that is, in accordance with (9). T0 gives the energy density of a given system. e. the expectation value of the Hamiltonian. 19 3 20 1. 4. Calculation of the energy flux S: In analogy to the Poynting vector in electrodynamics, S = E x B, the canonical formulation generally follows as (12) where e; are the Cartesian unit vectors. In the case of the Schrodinger field we get (13) Calculation of the momentum density p: = e1T1° + e2T2° + e3T3° = (V'I/J) 8~ + (V'IjJ*) 8~ p 8'1/J =- 8'1/J* ~('ljJ*V'ljJ- '1/JV'I/J*) (14) This is the expression known to us from elementary quantum mechanics.