The Hamiltonian (Equation 4.158) is e (11.97) = ehBo À (t) = CB-S= [sin na cos (wt)ox + sin a sin (wt)oy + cos ao;] 2m h
Posted: Wed May 18, 2022 4:41 pm
- The Hamiltonian Equation 4 158 Is E 11 97 Ehbo A T Cb S Sin Na Cos Wt Ox Sin A Sin Wt Oy Cos Ao 2m H 1 (61.21 KiB) Viewed 47 times
YB = Y(T) - y0). (11.95) Example 11.4 Imagine an electron (charge e-e, mass m) at rest at the origin, in the presence of a magnetic field
The exact solution to the time-dependent Schrödinger equation is (Problem 11.20): (61-6) cos (at/2) - i (11.103) x(t) = sin (at/2) cos (a/2)e-ior/2 [cos (21/2) – ; (@y+m) sin (11/2)] sin (a/2)e+ion/2 where (11.104) X = w2 +w; – 200 cosa, or, expressing it as a linear combination of X+ and X-: at (11.105) x(t) = cos at 2 (01 – cosa) - i 2 sin -i01/2 e X+(t) 2 ()] -)... [li ( λι i sin a sin +iot/2 e *x-(t).
Problem 11.20 Check that Equation 11.103 satisfies the time-dependent Schrödinger equation for the Hamiltonian in Equation 11.97. Also confirm Equation 11.105, and show that the sum of the squares of the coefficients is 1, as required for normalization.