Q. 1 (a) Apply the lowering operator L- to Y22 (θ,φ) to find Y21 (θ,φ). (b)...
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).
Homework Answers
Add Answer to:
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)...
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,...
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 ∇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 [...
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)...
(f) (1 point) Using values v0 = 10 m/s, θ = 45◦ , φ = −12◦...
(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
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....
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
(2.) Consider the orbital angular momentum operator defined in terms of the position and momentum operators...
(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,...
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 (...
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...
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...
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)...
(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
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
(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...
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