Hydrogen-Like Atoms Flashcards
What do the energy eigenvalues depend on in the absence of an external field?
Only depend on n.
What is meant by degeneracy?
Where several quantum states can share the same energy.
What are the labels for the quantum number l?
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.
How many different possible m(l) states are there for each (n,l) state? What is l less than or equal to?
2l+1 and l <= n-1
How do we show the notation for increasing n and changing l?
The n goes first and then the letter for l. So say n=3, l=1, it would be 3p
What is the shell called for n=1, 2, 3?
n=1: K shell, n=2: L shell, n=3: M shell
What does a higher n mean for the electron?
Higher n means it is found further from the nucleus.
What is the equation for the reduced mass μ? What is μ usually approximated to?
μ = mn*me/mn+me, where mn is the nuclear mass and me is the electron mass. mn»_space; me, so μ ~ me
What is the equation for the hydrogenic atom potential?
V(r) = -Z*e^2/4πε0r
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)?
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
What do we do second after simplifying the radial part of the wavefunction?
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
What does the part inside the square brackets represent?
The effective potential.
What form does this new equation have solutions in?
R(r) = C(n,l) * f(ρ)exp(-ρ), ρ = r/a(B), a(B) is Bohr radius = 4πε0ћ^2/me^2
What is meant by the probability function, PDF?
We want the chance of finding the electron in some volume dV.
What is the equation for |Ψ|^2 dV?
|Ψ|^2 dV = |Ψ|^2 * r^2 sinθ dθdФdr
What is |Ψ|^2 equal to?
|Ψ|^2 = |R(r)|^2 * |Θ(θ)|^2 * |ф(Ф)|^2
What is |ф(Ф)|^2 equal to?
|ф(Ф)|^2 = AA* * exp(im(l)Ф) 8 exp(-im(l)Ф) = A^2
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?
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
What are the energy eigenvalues?
En = -E1*z^2/n^2, n=1,2,3,…, E1 = Rydberg energy
What is the main experimental evidence of all of this?
Main evidence is optical spectra -> transitions between different allowed quantum states by absorbing or emitting a photon.
What is the Rydberg energy?
E1 = 13.6 eV
How can we sketch the H atom energy levels?
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)
Which optical transitions are forbidden?
Ones where the quantum number l does not change (e.g. 2s -> 1s)
Why are transitions where l = +/- 1 allowed?
Because photons have angular momentum, and to conserve this a photon must carry off or deliver some to/from the atom.
