8. Hydrogenic Atoms Flashcards
State the Rydberg energy value
13.6 eV
What does the Rydberg energy represent?
The ionisation energy of the hydrogen atom
State the equation for the electron energy states in the hydrogen atom
E_n = -E_1 / n^2
where n is the principal quantum number
What happens to the electron energy as n increases?
E_n becomes less negative - states are less tightly bound
For what type of potential is the energy of the electron states true?
Coulomb potential (V(r))
For what type of potential is the equations |L| = h_bar sqrt( l(l+1) ) and L_z = m_l h_bar true?
Any central potential
What is a central potential?
When the potential energy of a system only depends on the distance from the origin
Define degeneracy
Distinct quantum states with the same energy
How many possible m_l states does a (n, l) state have?
2l + 1
What happens to the position of an electron with respect to the nucleus as n increases?
It gets further away
When modelling hydrogen like atoms, are we concerned with the size of the nucleus?
No as it is always much smaller than the Bohr radius
Do we need to use the reduced mass when modelling hydrogen like atoms and why?
No as the mass of the nucleus is»_space; the mass of the electrons for any amount of protons and neutrons
Do the number of neutrons affect the Coulomb potential?
No
State the factor to scale the lengths for an atom with atomic number Z
Z/a_B where a_B is the Bohr radius ~5.29e-11
What happens to the positions of the electrons as Z increases?
They are confined closer to the nucleus
Can electrons enter the nucleus and why?
No - Probability density funciton -> 0 and r -> 0
What is the PDF mathematically?
The square of the modulus of the wave function
What can be said about the radial probability distributions with respect to Φ and θ?
They are averaged over them
What do the 3 quantum numbers (m_l, n, l) descruibe for electrons in a H atom?
The energy eigenvalues, wave funcitons of an electron and angular momentum
For optical transitions between energy levels in an atom, what MUST be conserved?
Energy and angular momentum
State the selection rule and an example of a forbidden transition
Δl = ± 1.
2s -> 1s is forbidden as for l: Δl = 0-0 = 0.
For transitions, why must the angular momentum of the electron change?
Because photons carry away some angular momentum so the electron’s must change as it is conserved
State the degenerate states for n = 1, 2, 3
n=1: 1s
n=2: 2s, 2p
n=3: 3s, 3p, 3d
What does the Zeeman effect do to the degeneracy in m_l?
It lifts it (changes) so we need l > 0 to see them.
What direction is the magnetic field applied in the Zeeman effect?
Along the z axis
See page 4 on document for a Zeeman effect diagram
Do it lol
What does w_l represent on the Zeeman effect diagram on page 4 of the document?
The Larmor frequency
Give the equation for the Larmor frequency
w_l = e B_ext / 2m
How does the Larmor frequency (w_L) vary with the applied frequency?
Linearly
State the equation for the energy change in the Zeeman effect
ΔE = ± h_bar w_l = ± μ_B B_ext
where μ_B is the Bohr magneton = 9.274E-24 J/T
What is the “normal Zeeman effect”?
When the spectral lines split into 3
Why does the spectra always comprise of 3 lines?
Due to another selection rule:
Δm_l = 0 or ±1
How does an electron in orbit interact with the magnetic moment along z?
It forms a current loop with the magnetic moment
Give the equation for the magnetic moment for an electron forming a current loop along the z axis
μ_z = -μ_B*m_l
