8. Hydrogenic Atoms

Question 1

State the Rydberg energy value

Answer 1

13.6 eV

Question 2

What does the Rydberg energy represent?

Answer 2

The ionisation energy of the hydrogen atom

Question 3

State the equation for the electron energy states in the hydrogen atom

Answer 3

E_n = -E_1 / n^2

where n is the principal quantum number

Question 4

What happens to the electron energy as n increases?

Answer 4

E_n becomes less negative - states are less tightly bound

Question 5

For what type of potential is the energy of the electron states true?

Answer 5

Coulomb potential (V(r))

Question 6

For what type of potential is the equations |L| = h_bar sqrt( l(l+1) ) and L_z = m_l h_bar true?

Answer 6

Any central potential

Question 7

What is a central potential?

Answer 7

When the potential energy of a system only depends on the distance from the origin

Question 8

Define degeneracy

Answer 8

Distinct quantum states with the same energy

Question 9

How many possible m_l states does a (n, l) state have?

Answer 9

2l + 1

Question 10

What happens to the position of an electron with respect to the nucleus as n increases?

Answer 10

It gets further away

Question 11

When modelling hydrogen like atoms, are we concerned with the size of the nucleus?

Answer 11

No as it is always much smaller than the Bohr radius

Question 12

Do we need to use the reduced mass when modelling hydrogen like atoms and why?

Answer 12

No as the mass of the nucleus is&raquo_space; the mass of the electrons for any amount of protons and neutrons

Question 13

Do the number of neutrons affect the Coulomb potential?

Answer 13

No

Question 14

State the factor to scale the lengths for an atom with atomic number Z

Answer 14

Z/a_B where a_B is the Bohr radius ~5.29e-11

Question 15

What happens to the positions of the electrons as Z increases?

Answer 15

They are confined closer to the nucleus

Question 16

Can electrons enter the nucleus and why?

Answer 16

No - Probability density funciton -> 0 and r -> 0

Question 17

What is the PDF mathematically?

Answer 17

The square of the modulus of the wave function

Question 18

What can be said about the radial probability distributions with respect to Φ and θ?

Answer 18

They are averaged over them

Question 19

What do the 3 quantum numbers (m_l, n, l) descruibe for electrons in a H atom?

Answer 19

The energy eigenvalues, wave funcitons of an electron and angular momentum

Question 20

For optical transitions between energy levels in an atom, what MUST be conserved?

Answer 20

Energy and angular momentum

Question 21

State the selection rule and an example of a forbidden transition

Answer 21

Δl = ± 1.

2s -> 1s is forbidden as for l: Δl = 0-0 = 0.

Question 22

For transitions, why must the angular momentum of the electron change?

Answer 22

Because photons carry away some angular momentum so the electron’s must change as it is conserved

Question 23

State the degenerate states for n = 1, 2, 3

Answer 23

n=1: 1s
n=2: 2s, 2p
n=3: 3s, 3p, 3d

Question 24

What does the Zeeman effect do to the degeneracy in m_l?

Answer 24

It lifts it (changes) so we need l > 0 to see them.

Question 25

What direction is the magnetic field applied in the Zeeman effect?

Answer 25

Along the z axis

Question 26

See page 4 on document for a Zeeman effect diagram

Answer 26

Do it lol

Question 27

What does w_l represent on the Zeeman effect diagram on page 4 of the document?

Answer 27

The Larmor frequency

Question 28

Give the equation for the Larmor frequency

Answer 28

w_l = e B_ext / 2m

Question 29

How does the Larmor frequency (w_L) vary with the applied frequency?

Answer 29

Linearly

Question 30

State the equation for the energy change in the Zeeman effect

Answer 30

ΔE = ± h_bar w_l = ± μ_B B_ext

where μ_B is the Bohr magneton = 9.274E-24 J/T

Question 31

What is the “normal Zeeman effect”?

Answer 31

When the spectral lines split into 3

Question 32

Why does the spectra always comprise of 3 lines?

Answer 32

Due to another selection rule:

Δm_l = 0 or ±1

Question 33

How does an electron in orbit interact with the magnetic moment along z?

Answer 33

It forms a current loop with the magnetic moment

Question 34

Give the equation for the magnetic moment for an electron forming a current loop along the z axis

Answer 34

μ_z = -μ_B*m_l