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IISER-Bhopal, PHYS 402
Spring 2017, mid-term solutions
1. Hyperfine structure:
(a) (2 points) What are the physical reasons (as discussed in the lecture) that give rise to the
Hyperfine structure of atomic energies? [list three, max 4 lines]
(b) (2 points) Consider the coupling of a nuclear spin (quantum number I), with a total
electron angular momentum (quantum number J). What are the allowed values for the
coupled total angular momentum quantum number F ?
(c) (4 points) The atomic isotope 87 Rb (Rubidium with 87 nucleons) has nuclear spin I = 3/2.
Consider also the valence electron. Finestructure effects already lift the orbital angular
momentum degeneracy and isolate the | 5 2 P3/2 i state. Into how many non-degenerate
energy levels does the energy level of this state split when you include hyperfine-structure?
Which quantum number does this splitting/ shift depend on?
(d) (2 points) Consider the ratio of hyperfine-shift ∆E to the basic Hydrogenic energy En =
−Ry/n2 . How does the ratio |∆E/En | scale with n?
Solution:
(a) Nuclear spin, finite nuclear radius, nuclear electric quadrupole moment
(b) F = I + J, · · · , |I − J|, in integer steps.
(c) From Eq. (2.81) of the lecture HFS depends on F , I, j, l. According to (b), we can
have F = 0, 1, 2, 3. Since the levels were already split according to j and l, HFS splits
it into an additional 4 levels.
(d) From the same equation, we know ∆E ∼ n−3 , hence ∆E/En ∼ 1/n.
2. Stark effect: Consider the effect of a small static electric field E = E0 k̂ (i.e. along the z-axis)
on a hydrogen atom. The total Hamiltonian is
Ĥ = Ĥ0 + eE · r = eE0 r cos θ,
(1)
where all bold-face symbols are 3D vectors.
(a) (1 point) Explain the mathematical reasons and symmetry reasons, why
h nlm |eE0 r cos θ| n0 l0 m0 i = 0 unless m = m0 [max 4 lines]. Note: It is also zero unless
l 6= l0 , but you do not have to show or explain this.
(b) (2 points) Calculate the shift of the state | φ100 i (n = 1, l = m = 0) to first order perturbation theory, using (a).
(c) (5 points) Calculate the perturbed eigenstates and eigenenergies belonging to the n = 2
set of states to first order perturbation theory. You may use h 210 |eE · r| 200 i = −3ea0 E0 .
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(d) (2 points) In (c) you should have found two of the unperturbed eigenstates that remain
eigenstates of the perturbed system without any energy shift. Write which corresponding
states | nlm i are unshifted within the n = 42 set of states and explain why [max 4 lines].
Solution:
R 2π
(a) The azimuthal integration is 0 exp [i(m0 − m)ϕ]dϕ, which is 0 unless integers m = m0 .
This is due to the underlying cylindrical symmetry of the Hamiltonian.
(b) First order shift is ∆E = h φ100 |eE0 r cos θ| φ100 i, which vanishes due to (a) [l=l’=0].
(c) Following the lecture we have to use degenerate perturbation theory. Using (a), among
all states | φ2lm i, the only non-vanishing matrix element of the perturbation is h 210 |eE·
r| 200 i = V , given in the question. We just want to diagonalize the 4 by 4 matrix


0 0 V 0
 0 0 0 0 

H=
(2)
 V 0 0 0 ,
0 0 0 0
here written with basis ordering | φ200 i, | φ21(−1) i, | φ210 i, | φ211 i. We can read-off /
guess/ calculate the 4 eigenvalues Ek and eigenvectors | ϕk i which answer the question:
√
| ϕ1 i = | φ21(−1) i, E1 = 0, | ϕ2 i = | φ21(1) i, E2 = 0, | ϕ3 i = (| φ200 i + | φ210 i)/ 2,
√
E3 = V , | ϕ4 i = (| φ200 i − | φ210 i)/ 2, E4 = −V . Note: This is verbatim from the
lecture contents and actually an easier version of assignment 2, where you did this for
n = 3 states.
(d) The states with maximal angular momentum z-component modulus | φn=42(l=41)m=41 i,
| φn=42(l=41)m=−41 i are unshifted, since m = m0 from (a) does not allow them to couple
to any other states.
3. Helium atom:
(a) (2 points) Write down the Hamiltonian for the two electrons in the helium atom in atomic
units. Ignore spin. Describe your variables and discuss essential symmetries of this Hamiltonian [max 6 lines].
(b) (2 points) Ignoring spin of the two electrons and ignoring electron-electron (e-e) interactions, how can you write appropriate eigenfunctions and eigenenergies of the Hamiltonian
found in (a), based on your knowledge of the hydrogen atom? Note (1): Your solutions
should respect the symmetries found in (a). Note (2): Please write your solutions in terms
of abstract Hydrogen wavefunctions φnlm (r), without explicitly writing their full form.
(c) (2 points) Consider now the spin of the electrons, but still ignore e-e interactions. Let
Ŝ = ŝ1 + ŝ2 be the total spin operator of the electron pair, and ŝ1,2 the individual spins of
each electron. Each electron has spin quantum number s = 1/2 described by the two basis
states | ↑ ik = | s = 12 , ms = 12 ik , | ↓ ik = | s = 21 , ms = − 12 ik , where k = 1, 2 numbers the
electron. In terms of these states, write all four allowed total spin states (eigenstates of Ŝ)
for these electrons in terms of the states | ↑ ik , | ↓ ik and indicate the total spin quantum
number S and total-spin z-component quantum number mS for each total spin state.
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(d) (2 points) What does the fact that electrons are Fermions imply for a two electron state
in Helium, considering positions and spin? What famous ”principle” follows from this?
[max 6 lines].
(e) (2 points) Now combine your results from (b), (c), (d) to infer the correct spatial and spin
state of the two electrons in their total ground-state (overall lowest energy), still neglecting
e-e interactions.
Solution:
(a) Ĥ = −∇2r1 /2 − ∇2r2 /2 − Z/r1 − Z/r2 + 1/|r1 − r2 |. rk is position of electron k, Z is
nuclear charge. Ĥ is symmetric (unchanged) under r1 ↔ r2 .
(b) Due to the symmetry above, solution can be chosen as symmetric or anti-symmetric
under r1 ↔ r2 . Since we now ignore the interactions 1/|r1 − r2 |, the Hamiltonian is
a sum of hydrogenic Hamiltonians for each electron, hence we can write the
√ overall
solution as Ψnlm,n0 l0 m0 (r1 , r2 ) = (φnlm (r1 )φn0 l0 m0 (r2 ) ± φn0 l0 m0 (r1 )φnlm (r2 ))/ 2. Their
energies are Enlm,n0 l0 m0 = En + En0 , where En are the hydrogen energies.
(c) As in the lecture:
√
| S = 0, mS = 0 i = (| ↑ i1 | ↓ i2 − | ↓ i1 | ↑ i2 )/ 2,
| S = 1, mS = 1 i = | ↑ i1 | ↑ i2 ,
√
| S = 1, mS = 0 i = (| ↑ i1 | ↓ i2 + | ↓ i1 | ↑ i2 )/ 2,
| S = 1, mS = −1 i = | ↓ i1 | ↓ i2 .
(d) Grouping position rk and spin sk into the multi-variable qk , the wavefunction must
satisfy Ψ(q1 , q2 ) = −Ψ(q2 , q1 ), i.e. be antisymmetric under swap of both electrons.
This gives rise to the Pauli-exclusion principle, that no two electrons can occupy the
same single-electron quantum state.
(e) From (b) we see that the lowest energy spatial eigenfunction is Ψ100,100 (r1 , r2 ) =
φ100 (r1 )φ100 (r2 ), symmetric under r1 ↔ r2 . To satisfy (d), we have to combine
this with the anti-symmetric spin state, hence in total Ψ(q1 , q2 ) = φ100 (r1 )φ100 (r2 ) ⊗
| S = 0, mS = 0 i.
4. Atom-light coupling
(a) (2 points) Write the complete atom-radiation Hamiltonian in the Coulomb Gauge, discuss
each term [max 4 lines].
(b) (2 points) For a Hydrogen atom in some initial electronic state | φa i = | φna la ma i, how
would you proceed to determine the probability for the atom to make a transition to
another state | φb i = | φnb lb mb i within a short time T due to a weak electro-magnetic field?
(Just describe the required steps in words, you do NOT need to provide any calculation)
[max 6 lines].
(c) (2 points) Suppose the energies fulfill Ea < Eb . Which physical quantities mainly govern
the transition rate a → b for a single atom? Ignore fundamental constants.
(d) (2 points) How is this transition rate for absorption related to the rate for stimulated
emission b → a? What is meant by spontaneous emission? [max 6 lines].
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(e) (2 points) What is meant by an ”electric-dipole allowed transition”? Explain how higher
order multipole transitions (magnetic dipole, electric quadrupole, · · · ) arise mathematically, and in which cases they have to be considered? [max 6 lines].
Solution:
(a) See Eq. (3.7) of the lecture.
(b) For weak electro-magnetic fields we can use first order time-dependent perturbation
theory. This essentially requires the quantity Hba (t) = h φb |Ĥ 0 (t)| φa i.
(c) The main dependence of the rate Wba is on the transition matrix element Mba see Eq.
(3.12), the light intensity I(ωba ) and ωba .
(d) The emission rate Wab = Wba . Spontaneous emission is the photon emitted from a
decay b → a taking place without there already being an electromagnetic field.
(e) Transitions are electric dipole allowed if h φb |er| φa i =
6 0. Higher order multipoles arise
from the expansion of exp [ikr] in the general transition matrix element. Normally they
only play a role if the lower order multi-poles vanish.
5. (4 points) What is the electron configuration of Silicon (Z = 14) in the ground-state. How
many and which electrons are not in a closed (complete) sub-shell?
Solution: (1s)2 , (2s)2 , (2p)6 , (3s)2 , (3p)2 . The last two, (3p)2 are not in a closed sub-shell.
6. (4 points) Order the following neutral elements according to their ionisation energy Eion in the
ground-state, from Eion high to low: Potassium (Z=19), Argon (Z=18), Boron (Z=5). Explain
why/ how you can deduce this order without any calculation or reference to a data bank.
Note: Eion =−Ebind , where Ebind < 0 is the energy of the most weakly bound outer electron.
Solution: Potassium, Boron, Argon. We know this, since noble-gases (Argon) have the
highest ionisation energies, Alkalis (Potassium) the lowest. Boron is sth. in between.
7. (4 points) For each of the following four transitions, explain if they are allowed by electric
dipole selection-rules or why not:
(i) | nlm i = | 1s0 i → | n0 l0 m0 i = | 2s0 i, (ii) | 1s0 i → | 2p1 i,
(iii) | 2p1 i → | 5f 1 i, (iv) | 2p1 i → | 5d3 i.
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Solution: (i) no, same l. (ii) yes, (iii) no, l-change by 2, (iv) no, m-change by 2.
8. (4 points) Into how many non-degenerate energy levels does the | 3d5/2 i state of hydrogen split
in the presence of weak magnetic field (splitting small compared to the fine-structure)? Why
Solution: In the weak field limit, Zeeman splitting is according to mj . Here j = 5/2, so
we have 6 possibilites for mj .
9. (4 points) If an atom is initially in state | φa i, and we then apply a resonant Rabi π/2 pulse
of laser light on the transition a → b, what is the state of the atom after the pulse? If directly
after this, we apply an additional π-pulse, what are the probabilities to find the atom in state
| φa i or | φb i? Note: α × π-pulse means its duration is t = απ/Ω, where Ω is the Rabi frequency
√
Solution: After the π/2-pulse the atom is in | Ψ i = (| φa i + | φb i)/ 2. Subsequently
applying a π-pulse sends | φa i → | φb i and vice versa (with a sign not discussed in the
lecture). Probabilities are unaffected by the sign and are still 50% for both.