Atomo idrogeno

Question 1

What is the importance of the hydrogen atom in the study of atomic structure?

Answer 1

The hydrogen atom serves as the basis for the study of the structure of all atoms.

Question 2

What are the relevant coordinates for the hydrogen atom?

Answer 2

The relevant coordinates for the hydrogen atom are the 3 cartesian coordinates for the nucleus and the 3 cartesian coordinates for the electron.

Question 3

What is the formula for the Hamiltonian in terms of cartesian coordinates?

Answer 3

The formula for the Hamiltonian in terms of cartesian coordinates is H = (1/2)M(·X^2_1 + ·Y^2_1 + ·Z^2_1) + 1/2m(·X^2_2 + ·Y^2_2 + ·Z^2_2) + V(r).

Question 4

Why is it impossible to separate the variables in the Schrödinger equation using cartesian coordinates?

Answer 4

It is impossible to separate the variables in the Schrödinger equation using cartesian coordinates because the potential energy of interaction between the nucleus and the electron mixes the coordinates of the nucleus and the electron.

Question 5

What is the motion of the hydrogen atom that can be separated from the internal motion?

Answer 5

The motion of the hydrogen atom that can be separated from the internal motion is the global translation of the atom in space.

Question 6

What is the form of the kinetic energy term in the Hamiltonian after separating the global translation?

Answer 6

The form of the kinetic energy term in the Hamiltonian after separating the global translation is 1/2m(·x^2_2 + ·y^2_2 + ·z^2_2) + 1/2M(·X^2_1 + ·Y^2_1 + ·Z^2_1).

Question 7

What is the formula for the kinetic energy term in the Hamiltonian after using the center of mass and polar coordinates?

Answer 7

The formula for the kinetic energy term in the Hamiltonian after using the center of mass and polar coordinates is 1/2(m + M)(·X^2 + ·Y^2 + ·Z^2) + 1/2M(·x^2_1 + ·y^2_1 + ·z^2_1).1/2m(·x^2_2 + ·y^2_2 + ·z^2_2).

Question 8

What is the formula for the Laplacian operator in spherical coordinates?

Answer 8

The formula for the Laplacian operator in spherical coordinates is ∇^2 = 1/r^2(∂/∂r(r^2∂/∂r) + 1/sinθ(∂/∂θ(sinθ∂/∂θ)) + 1/sin^2θ(∂^2/∂ϕ^2)).

Question 9

What is the relationship between the Hamiltonian and the angular momentum?

Answer 9

The Hamiltonian commutes with the angular momentum and its components.

Question 10

What are the common eigenfunctions of H, L^2, and Lz?

Answer 10

ψ(r, θ, ϕ) = R(r)Ylm(θ, ϕ) with l = 0, 1, 2, 3, … and m = −l, −l + 1, …, 0, …, l − 1, l

Question 11

What is the equation for the radial part of the wavefunction?

Answer 11

∂^2R/∂r^2 + 2/r ∂R/∂r + (2Za/r - l(l+1)/r^2 + 2Ea^2/(-2E))R = 0

Question 12

What is the general form of the radial part of the wavefunction for E > 0?

Answer 12

It’s equal to the free particle equation, R(r) = e^(±i√(2E/ae^2)·r)

Question 13

What is the general form of the radial part of the wavefunction for E < 0?

Answer 13

R(r) = K(r)e−cr, En = −μe^4Z^2/(2h^2n^2)

Question 14

What do the solutions for E ≥ 0 correspond to?

Answer 14

They correspond to unbound states where the electron dissociates from the nucleus and moves freely without feeling the Coulomb attraction.

Question 15

What is the general form of the hydrogen atom wavefunction?

Answer 15

ψnlm(r, θ, ϕ) = Rnl(r)Ylm(θ, ϕ) = Rnl(r)Slm(θ) 1/√2π e^imϕ with n = 1, 2, 3, 4, …, l = 0, 1, 2, 3, …, (n − 1) and m = −l, …, +l.

Question 16

What is the constant a used in the radial wavefunction?

Answer 16

a = h^2/μe^2

Question 17

What is the value of b0 in the ground state of hydrogen atom?

Answer 17

b0 = 2(Z/a)^(3/2) according to the normalization condition.

Question 18

What is the total wave function for the ground state of hydrogen atom?

Answer 18

Ψ100 = (1/√π)(Z/a)^(3/2)e^(-Zr/a)

Question 19

What is the value of the Bohr radius?

Answer 19

a = h^2/μe^2 = 0.5292 Å.

Question 20

What is the representation of a s-type function in a polar coordinate system?

Answer 20

A spherically symmetric function like s-type has a representation of a sphere.

Question 21

What is the representation of a p-type function in a polar coordinate system?

Answer 21

A p-type function with m=0 has a representation of two tangent circles in a plane or two tangent spheres in space.

Question 22

What is the radial distribution function?

Answer 22

|Ψ(r)|^2dr = |Rnl(r)|^2r^2dr.

Question 23

What is the Zeeman effect?

Answer 23

The Zeeman effect is the splitting of energy levels with different magnetic quantum numbers (m) due to the interaction between the magnetic moment associated with the electron’s rotation and an applied magnetic field.

Question 24

What causes the degeneracy of energy levels with different magnetic quantum numbers?

Answer 24

The degeneracy of energy levels with different magnetic quantum numbers is caused by the electron’s rotation, which is associated with a magnetic moment.

Question 25

What happens to the degeneracy of energy levels when an atom is immersed in a magnetic field?

Answer 25

The degeneracy of energy levels is lost when an atom is immersed in a magnetic field due to the interaction between the magnetic moment associated with the electron’s rotation and the applied magnetic field.

Question 26

What is the formula for the magnetic moment associated with an electron’s rotation?

Answer 26

μ = βe√l(l+1), where βe is the Bohr magneton and l is the orbital angular momentum quantum number.

Question 27

What is the Hamiltonian for an atom in the presence of an applied magnetic field?

Answer 27

HT = H + HB, where H is the unperturbed Hamiltonian and HB = LzBβe/h is the perturbation due to the applied magnetic field.

Question 28

What are Slater orbitals?

Answer 28

Slater orbitals are a set of orbitals that describe the behavior of electrons in a polyelectronic atom, taking into account the interaction with other electrons in addition to the attraction with the nucleus.

Question 29

What is the radial part of a Slater orbital?

Answer 29

The radial part of a Slater orbital is given by Rnl(r) = CnR^(n-1)e^(-ζr/ao), where cn is a normalization constant, n is the principal quantum number, l is the azimuthal quantum number, r is the distance from the nucleus, ζ is the orbital exponent, and ao is the Bohr radius.

Question 30

What is the effective potential for a Slater orbital?

Answer 30

The effective potential for a Slater orbital is given by Veff = -ζn/r + n(n-1)/2r^2 - l(l+1)/2r^2, where ζn is the effective nuclear charge, n is the principal quantum number and r is the distance from the nucleus.

Question 31

What is the role of the orbital exponent in Slater orbitals?

Answer 31

The orbital exponent in Slater orbitals represents an effective nuclear charge that takes into account the screening effect of other electrons. It is used as a parameter to adjust the shape of the orbital and can be estimated for different atoms.

Question 32

How are the orbital exponents in Slater orbitals related to the screening effect of other electrons?

Answer 32

The orbital exponents in Slater orbitals are related to the screening effect of other electrons in that they represent an effective nuclear charge that is reduced by the screening effect. The larger the screening effect, the smaller the effective nuclear charge and the larger the orbital exponent.