Q. 1 (a) Apply the lowering operator
L- to Y22 (θ,φ) to find
Y21 (θ,φ).
(b) Apply the raising operator L+ to
Y30 (θ,φ) to find Y31
(θ,φ).
Q. 1 (a) Apply the lowering operator L- to Y22 (θ,φ) to find Y21 (θ,φ). (b)...
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other. 2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
2. Find b 5.0 35° 3. Find a & θ 9.5 84° 1.0 4. Find φ, θ, & c 4.0 3.0
Find the solution to ∇2 Ψ(r, θ, φ) = 0 inside a sphere with the following boundary conditions: ∂Ψ (1,θ,φ)=sin2θcosφ.∂r Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
3. (a) Given that e2i4 sin20 32m show by direct differentiation using the raising operator L+ that 1,12,-2(9,0) 0 (b) Also for e 32T sin 2,-2(0,9) show using the raising operator L. that 3. (a) Given that e2i4 sin20 32m show by direct differentiation using the raising operator L+ that 1,12,-2(9,0) 0 (b) Also for e 32T sin 2,-2(0,9) show using the raising operator L. that
(f) (1 point) Using values v0 = 10 m/s, θ = 45◦ , φ = −12◦ , find r. (g) (1 point (bonus)) For the case where φ = 0◦ (flat ground), give the simplified expressions for horizontal displacement and time of flight. Disregard the numerical values from previous part. 2" (6 points) A projectile is launched with initial velocity vo at angle θ above horizontal The ground is sloped at angle φ w.rt. horizontal, where φ < θ 0
(2.) Consider the orbital angular momentum operator defined in terms of the position and momentum operators as p. Define the angular momentum raising and lowering operators as L± = LztiLy. Use the commutation relations for the position and m omentum operators and find the commutators for: (a.) Lx, Lz and Ly, Lz; (b.) L2, Lz; (c.) L+,L
3.58 (a) If U(x, y, z) = xy72, find ▽U and V2U. (b) If V(p, φ, z)- P (c) If W"(r, θ, φ.)-z? sin θ cos φ, find W and VzW. sın, find wandV2V.
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...