Mathematical Methods for Physics and Engineering - Matematica.NET

QUANTUM OPERATORSFor a particle of mass m moving in a one-dimensional potential V (x), proveEhrenfest’s theorem:〈 〉d〈p x 〉 dVd〈x〉= −and = 〈p x〉dt dxdt m .19.4 Show that the Pauli matricesS x = 1 2 (0 11 0), S y = 1 2 (0 −ii 0)( )1 0, S z = 1 ,2 0 −1which are used as the operators corresponding to intrinsic spin of 1 in nonrelativisticquantum mechanics, satisfy S 2 x = S 2 y = S 2 z = 1 4 2 I, and have the2same commutation properties as the components of orbital angular momentum.Deduce that any state |ψ〉 represented by the column vector (a, b) T is an eigenstateof S 2 with eigenvalue 3 2 /4.19.5 Find closed-form expressions for cos C and sin C, whereC is the matrixC =(1 11 −1Demonstrate that the ‘expected’ relationshipscos 2 C +sin 2 C = I and sin 2C =2sinC cos Care valid.19.6 Operators A and B anticommute. Evaluate (A + B) 2n for a few values of n andhence propose an expression for c nr in the expansionn∑(A + B) 2n = c nr A 2n−2r B 2r .r=0Prove your proposed formula for general values of n, using the method ofinduction.Show that∞∑ n∑cos(A + B) = d nr A 2n−2r B 2r ,n=0 r=0where the d nr are constants whose values ( you should ) determine.0 1By taking as A the matrix A =, confirm that your answer is1 0consistent with that obtained in exercise 19.5.19.7 Expressed in terms of the annihilation and creation operators A and A † discussedin the text, a system has an unperturbed Hamiltonian H 0 = ωA † A. The systemis disturbed by the addition of a perturbing Hamiltonian H 1 = gω(A + A † ),where g is real. Show that the effect of the perturbation is to move the wholeenergy spectrum of the system down by g 2 ω.19.8 For a system of N electrons in their ground state |0〉, the Hamiltonian isN∑ p 2 xH =n+ p 2 y n+ p 2 N∑z n+ V (x n ,y n ,z n ).2mn=1n=1Show that [ ]p 2 x n,x n = −2ipxn , and hence that the expectation value of thedouble commutator [[x, H ] ,x], wherex = ∑ Nn=1 x n,isgivenby〈0 | [[x, H ] ,x] | 0〉 = N2m .672).