Momento

Question 1

What is the formula for the angular momentum of an object in rotational motion?

Answer 1

L = r x p = rmωr = mr^2ω = Iω

Question 2

What is the formula for the three Cartesian components of the angular momentum?

Answer 2

Lx = ypz - zpy, Ly = zpx - xpz, Lz = xpy - ypx

Question 3

What is the formula for the modulus squared of the angular momentum?

Answer 3

L^2 = Lx^2 + Ly^2 + Lz^2

Question 4

What is the importance of the angular momentum in the study of molecular properties?

Answer 4

It is an important dynamic variable that is conserved in isolated systems and commutes with the Hamiltonian in problems with a central field, such as in an atom.

Question 5

What is the commutation relation between the position and momentum operators?

Answer 5

[x, px] = ih

Question 6

What is the commutation relation between the angular momentum operators?

Answer 6

[L^2, Lx] = [L^2, Ly] = [L^2, Lz] = 0, [Lx, Ly] = ihLz, [Ly, Lz] = ihLx, [Lz, Lx] = ihLy

Question 7

What is the formula for the operator associated with the z-component of the angular momentum?

Answer 7

Lz = h/i(x * d/dy - y * d/dx)

Question 8

What is the purpose of changing variables from cartesian to spherical coordinates when finding eigenvalues and eigenfunctions of L2 and Lz?

Answer 8

The is to express the operators of interest in the new coordinate system and obtain a simpler form of the equations.

Question 9

What are the relationships between cartesian and spherical coordinates?

Answer 9

x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ.
e
r2 = x2 + y2 + z2

Question 10

What is the form of the separation of variables used to solve for the eigenvalues of L2 and Lz?

Answer 10

Y(θ,ϕ) = S(θ)T(ϕ).

Question 11

What is the form of the normalized eigenfunctions of Lz?

Answer 11

T(ϕ) = 1/√(2π)e^(imϕ), where m = 0, ±1, ±2, ±3, …

Question 12

What is the solution for the eigenvalues of L2?

Answer 12

a = l(l+1)/h^2, where l = 0, 1, 2, 3, …

Question 13

What is the range of values for the quantum number m in terms of the quantum number l?

Answer 13

-l ≤ m ≤ l.

Question 14

What is the maximum value of Lz in terms of the quantum number l?

Answer 14

Lz(max) = l*h.

Question 15

What are the associated Legendre polynomials?

Answer 15

They are the solutions to the differential equation (1-x^2)d^2P/dx^2 - 2xdP/dx + [l(l+1)-m^2/x^2]P = 0, and are used to express the normalized eigenfunctions of L2 in spherical coordinates.

Question 16

What is the normalization factor for the eigenfunctions of L2 in spherical coordinates?

Answer 16

Sl,m(θ) = N*P^|m|
l(cos θ)

Question 17

What are the eigenvalues and eigenfunctions of L2 and Lz used for?

Answer 17

They are used to describe the quantization of angular momentum in quantum mechanics.

Question 18

What is the physical significance of the quantum numbers l and m?

Answer 18

The quantum number l determines the magnitude of the angular momentum, while the quantum number m determines the projection of the angular momentum onto a chosen axis.

Question 19

What is the difference between the eigenvalues obtained using the new operators and the classical differantion approach?

Answer 19

The eigenvalues obtained using the new operators can also be semi-integer values, whereas the classical approach only yields integer values.

Question 20

What is the expression for L^2 obtained using the operators properties?

Answer 20

The expression for L^2 obtained using the new operators is j(j+1)ħ^2.