Hydrogen-Like Atoms

Question 1

What do the energy eigenvalues depend on in the absence of an external field?

Answer 1

Only depend on n.

Question 2

What is meant by degeneracy?

Answer 2

Where several quantum states can share the same energy.

Question 3

What are the labels for the quantum number l?

Answer 3

l=0: s-state, l=1: p-state, l=2: d-state, l=3: f-state, l=4: g-state, and then in alphabetical order after that.

Question 4

How many different possible m(l) states are there for each (n,l) state? What is l less than or equal to?

Answer 4

2l+1 and l <= n-1

Question 5

How do we show the notation for increasing n and changing l?

Answer 5

The n goes first and then the letter for l. So say n=3, l=1, it would be 3p

Question 6

What is the shell called for n=1, 2, 3?

Answer 6

n=1: K shell, n=2: L shell, n=3: M shell

Question 7

What does a higher n mean for the electron?

Answer 7

Higher n means it is found further from the nucleus.

Question 8

What is the equation for the reduced mass μ? What is μ usually approximated to?

Answer 8

μ = mn*me/mn+me, where mn is the nuclear mass and me is the electron mass. mn&raquo_space; me, so μ ~ me

Question 9

What is the equation for the hydrogenic atom potential?

Answer 9

V(r) = -Z*e^2/4πε0r

Question 10

What do we do first to the radial part of the wavefunction: [-ћ^2/2m 1/r^2 * d/dr(r^2 d/dr) + l(l+1)ћ^2/(2mr^2) + V(r)]R(r) = E*R(r)?

Answer 10

Substitute R(r) = u(r)/r, find 1/r^2 * d/dr * r^2 *d/dr(u(r)/r) = 1/r^2 *d/dr *r^2(1/r du/dr - u/r^2), which can be simplified by putting r^2 inside brackets and find = 1/2 * d^2u/dr^2

Question 11

What do we do second after simplifying the radial part of the wavefunction?

Answer 11

Sub it in and rearrange, find final equation of -4πε0/2m *d^2u/dr^2+ [V(r) + l(l+1)ћ^2/(2mr^2)]u = Eu

Question 12

What does the part inside the square brackets represent?

Answer 12

The effective potential.

Question 13

What form does this new equation have solutions in?

Answer 13

R(r) = C(n,l) * f(ρ)exp(-ρ), ρ = r/a(B), a(B) is Bohr radius = 4πε0ћ^2/me^2

Question 14

What is meant by the probability function, PDF?

Answer 14

We want the chance of finding the electron in some volume dV.

Question 15

What is the equation for |Ψ|^2 dV?

Answer 15

|Ψ|^2 dV = |Ψ|^2 * r^2 sinθ dθdФdr

Question 16

What is |Ψ|^2 equal to?

Answer 16

|Ψ|^2 = |R(r)|^2 * |Θ(θ)|^2 * |ф(Ф)|^2

Question 17

What is |ф(Ф)|^2 equal to?

Answer 17

|ф(Ф)|^2 = AA* * exp(im(l)Ф) 8 exp(-im(l)Ф) = A^2

Question 18

What do we need to do to average over Θ and ф to get the probability that the electron is inside a shell of radius r and thickness dr?

Answer 18

P(r) dr = r^2 *|R(r)|^2 dr * integral from 0 to π of |Θ|^2 sinθ dθ * integral from 0 to 2π of |ф|^2 dФ -> both these integrals equal 1

Question 19

What are the energy eigenvalues?

Answer 19

En = -E1*z^2/n^2, n=1,2,3,…, E1 = Rydberg energy

Question 20

What is the main experimental evidence of all of this?

Answer 20

Main evidence is optical spectra -> transitions between different allowed quantum states by absorbing or emitting a photon.

Question 21

What is the Rydberg energy?

Answer 21

E1 = 13.6 eV

Question 22

How can we sketch the H atom energy levels?

Answer 22

Different lines for different energy level with E=0 at the top and goes more negative as you go down. Each line split into sections (like s, p, d etc)

Question 23

Which optical transitions are forbidden?

Answer 23

Ones where the quantum number l does not change (e.g. 2s -> 1s)

Question 24

Why are transitions where l = +/- 1 allowed?

Answer 24

Because photons have angular momentum, and to conserve this a photon must carry off or deliver some to/from the atom.