Calculate the following commutators: (a) [P, ² ] (b) [â² , p²]
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- In spherical polar coordinates, the angular momentum operators, 1² and Îz, can be written, Ә β = -ħ² 1 ə sin Ꮎ ᎧᎾ 1 2² sin² 0 00² дф él sin)+ + and (a) îψ(θ, φ) = λιψ(θ, φ) = Î₂ = = -ih Apply these operators to the unnormalized eigenfunction, (0, 4) = sin²0 e-²ip, and determine the eigenvalues of that are associated with each operator. (b) Î₂¥(0, 0) = 1₂4(0,4)Calculate the standard uncertainty in z if z=Xsinθ using the angle (57.00 +- 0.85) degrees and the value X = (58.20 +- 0.77)Consider the real part of the spherical harmonic Y1, + 1. At which values of do angular nodes occur?
- Calculate Z for a single oscillator in an Einstein solid at a temperature T=4TE=4ϵ/kBT=4 TE=4 ϵ/kB . The value of Z isProblem 6: A uniform wooden meter stick has a mass of m = 763 g. A clamp can be attached to the measuring stick at any point P along the stick so that the stuck can rotate freely about point P, which is at a distance d from the zero-end of the stick as shown. Randomized Variables m = 763 g 1 2 3045 6 7 12 13 14 15 16 17 18 10 11 Otheexpertta.comSuppose that ak > 0 for all k e N and E ak < 0. For each of the following, prove that the given series converges. ak ( a ) ΣΕΙ 1+ k3ak (b) Lk=1 1+ ak Vak (c) Ek=1 k < 0o.
- Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.Let F = (z^2 cos y, −xz^2 sin y, 2xz cos y − cos z).a) Show that F is irrotational.b) Find a potential function f (x, yz) such that F = ∇f , and f (0, π, π/3) = 2compute the accepted value of g