Exercises 2006
1. We consider a particle in an infinitely deep box, which extends from 0 to L.
1) Write down the wavefunctions and energies
We assume that the particle is in the lowest level of the box. At time t=0, the box is
suddenly expanded to twice its original width, and now extends from 0 to 2L. (At t=0,
the system can be described by the wavefunction from 1).)
2) Write the wavefunction as a superposition of the eigenstates of the new box (Hint:
use the following formula):
a
0 sin(bx)sin(cx)dx = -1/2[sin{(b+c)a}/(b+c) - sin{(b-c)a}/(b-c)]
3) Describe what will happen to this system as a function of time – will the particle
start moving? (Will the wavefunction be time-dependent?).
This is a good example of a wavepacket. Calculate at what times the wavepacket will
come back to its original state. This is called revival of the wavepacket.
n 1
Answer: 2) a 2 
Useful formula:


0
4 2  1 2
, an 
(n odd)
 (n  2)( n  2)
2
1
r n e r / a dr  n!a n1
2. Consider a hydrogen atom. At time t=0, the wavefunction is the following superposition
of energy eigenfunctions nlm(r):
(r,t=0) = N[2100(r) - 210(r) + 311(r)]
a) Normalize the wavefunction
b) Is the wavefunction of even or odd parity?
c) What are the probabilities of finding the system in the ground state?
The state (200)? Or (311)? Another eigenstate?
d) What is the expectation value of the total energy?
Of the operators L2, and Lz?
1
2 1 1
2

Answer:
, No, ,0, , ,9.89eV ,  2 ,
3 6 6
3
6
6
3. Classically-forbidden region in hydrogen
Classically, any region of space where the kinetic energy of a particle is negative is
forbidden.
a) Show that the classically forbidden region for the ground state of hydrogen is r >
2a0, where a0 is the Bohr radius.
b) Calculate the probablity of finding the electron in this region.
hint:


b
r 2 e r / a dr  (ab 2  2a 2 b  2a 3 )e b / a
Answer: 0.238
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4. A tritium atom includes a 3H nucleus (a proton and two neutrons) and an electron. 3H
decays rapidly to 3He (two protons and a neutron) via -emission. (We assume that
this happens very rapidly compared to the characteristic atomic times and that the
tritium atom is its ground state when the transition occurs).
Calculate the probability that immediately after the transition, the He+ ion is
a) in its ground state.
b) in another state (incl. continuum states).
c) in the 2s state.
d) in a state with   0 .
Answer: 0.7, 0.3, 0.25, 0
5. To describe the intensity of a transition between two levels 0,1, one can use the concept
2m( E1  E0 )  2
of oscillator strength: f10 
| r10 |
3 2
We want to calculate the oscillator strength for the transition 1s  2 p in Hydrogen.
We call 1s level 0 and 2p level 1.
a) Calculate the radial part


0
1* ( r )r 3 ( r )dr as a function of the Bohr radius a0.
0
b) Calculate the integral z10 using the properties of the spherical harmonics.
c) Calculate f 10z .
d) Show, using a simple argument that f 10x = f 10y = f 10z for 1s  2 p .
e) Calculate f 10 . What percentage of the total oscillator strength is in the transition
1s  2 p ? (The oscillator strengths obey a sum rule:
f
ka
 1 ).
k
Answer: 1.29 a0, 1.29a0 / 3 ,0.14, 0.42
6. Fine structure in Hydrogen
We want to calculate the fine structure for the n=2 level in H. We consider the energies
and wavefunctions corresponding to the unperturbed Hamilton operator H0, which is the
sum of the kinetic energy operator and the Coulomb potential. The electron has a ½.spin.
1) What is the degeneracy of the n=2 level?
2) Which orthonormal basis can be used to describe level 2? Which quantum numbers are
necessary to characterize the wavefunctions?
3) The fine structure is described with the Hamilton operator Wf=Wmv+Wso+WD, where
Wmv corresponds to the relativistic mass increase, WD is the Darwin term, and Wso

describes the spin-orbit interaction. Wmv is proportional to p 4 , WD depends only on r,
 
and Wso= VLS(r) L  S . The hyperfine interaction is neglected. Show that Wso does not
commute with Lz. What does it imply for the matrix that describes Wso in the basis
nm ms for n=2?

4) Show that Wso commutes with L2 and that the matrix consists of a 2x2 matrix for 2s
and a 6x6 matrix for 2p.
2
  
5) To simplify the problem, consider the total angular momentum operator J  L  S .
Use the triangle rule to determine which j,  and s are possible for n=2. These states
are labelled 2 s 1  j .

 
 
6) Show that L  S can be written as 12 ( J 2  L2  S 2 ) .

7) Show that J 2 and Jz commute with H0 and Wso.
 

8) In which basis are J 2 , Jz , H0, Wso , L2 , S 2 diagonal?
9) Write down Wso in the above matrix using the integrals  2   2 VLS (r)  2 where
 2  is the radial wavefunction for 2  . How many energy levels are there?

10) It can be shown that Wmv, WD commute with L . What is the effect of Wmv, WD on
the n=2 shell?
Answer: 4)  =0,1, j=1/2, 3/2 8) 3 energy levels: 0, -  2 p ,  2 p /2
7. Stark effect for the 2s1/2 and 2p1/2 levels in Hydrogen (difficult).
An hydrogen atom is exposed to an electric field with strength E. eEa0 is small compared
to the fine structure but not relative to the Lamb shift  . The aim of the exercise is to
calculate the Stark effect for the 2s1/2 och 2p1/2 levels using perturbation theory.
a) Which states need to be considered in the problem?
b) Express the wavefunctions |  sjmj> for these states as a function of Ym and  m s . Use
the web page http://www.gleet.org.uk/cleb/ to find these coefficients (called Clebsch
Gordan).
c) Use the selection rules to decide which states are coupled by the interaction with the
electric field.
d) Write the matrix that represents the Hamilton operator in the basis |  sjmj>. Show that
 s 0 0 


 s 0 0 0
it can be written as 
. Give an expression for s, s' as function of eEa0.
0 0  s' 


 0 0 s' 0 


(The radial integral is <2s|r|2p>=3 3 a0)
e) Calculate the first order energies. Show that the energy shifts vary quadratically with
the field when eEa0<<  and linearly when eEa0>>  . Could one have guessed the
result without doing the above calculation?
Answer: b) The Clebsch Gordan coefficients are  1/ 3 , 
2 / 3 . c) s= 3eEa0
8. Zeeman effect in very strong fields
a)
b)
A hydrogen atom is exposed to a very strong magnetic field, so that the interaction
with the field is larger than the spin-orbit interaction. Show that this problem can
be solved exactly. Express the energies and wavefunctions.
Consider the spin-orbit interaction as a small perturbation. Calculate approximately
the energy levels. (Paschen Back effect)
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c)
We now assume that the interaction with the magnetic field is a small perturbation
of the spin-orbit Hamiltonian. What are the energies and wavefunctions?
Answer: b)  B B( m  2ms )  m ms
9. Calculation of a transition rate.
Normally, atoms are found in their ground state. What could you do to prepare a gas of
hydrogen atoms in a) the 2pm=0 state? b) the 2pm=1 state? Can you think of a way to
excite the 2s state?
In this problem, we want to calculate the transition rate for the 1s to 2pm=0 state, using a
linearly polarized laser field (resonant with the transition) with an intensity of I = 1
mW/cm2. The width of the transition is 1 GHz.
a) Start by calculating the radial part of the integral. Express your answer in units of the
Bohr radius.
b) Now calculate the angular part. It is easy to do this integral if you write it using the
spherical harmonics.
c) Calculate the transition rate.
Answer: Radial part 1.29 a0, Angular part
1
3
Transition rate: 4.2 104 s-1
10. Ramsey fringes.
A beam of atoms with velocity v is sent through a chamber of length L, in which they
interact with a radio-frequency field Ecos(t) of frequency . The atoms are initially
in state i, and the frequency  is close to resonant with an atomic transition i->f
(frequency 0). The perturbation from the rf-field is described by H’ = -E*Dcos(t),
where D is the dipole operator.
L
v
a) Express the flux F1 of atoms in the excited state at the exit of the chamber (the
incoming flux is F0) – making the rotating wave approximation and using timedependent perturbation theory.
b) A second chamber of length L is added to the path of the atoms. The two chambers
are separated by L1 > L. Find the flux F2 of atoms in the excited state after the second
chamber. Express F2 as a function of F1.
v
L1
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c) Plot (approx.) the variation of F1 and F2 as functions of (0-).
Answer: F1  F0
 
E.D fi
2
2
L

sin 2  ( 0   ) 
L  L
2v 


; F2  4 F1 cos 2  ( 0   ) 1

2
( 0   )
2v 

11. Two level atom perturbed by a time-independent and time-dependent
perturbation
Consider a physical system (an atom) described with two levels 1 and 2. The energies are
E1 and E2, the wavefunctions 1, 2 and the Hamiltonian H0. The system is exposed to a
time-independent perturbation W. W couples 1 to 2, but not 1 to 1 and 2 to 2.
1) Write matrix expressions for H0, W and H0+W in the two-dimensionel vector space
generated by 1, 2. 
2) Calculate the new eigenvalues E+, E- for the perturbed system. Use the notations
Em=(E1+E2)/2, =(E1-E2)/2. 
3) Draw approximately E+, E- as a function of |W|What does it mean when =0?
4) Compare with the light shifts due to a perturbing electromagnetic field.
Answer: 2) E  Em  2  W12
2
12. Hydrogen atoms in a time-dependent electric field
An ensemble of hydrogen atoms in the ground state is placed in a time-dependent electric
field E=0, t<0, E(t)=E0 exp(-t/) t>0.
a) Use time-dependent perturbation theory to calculate what is the fraction of atoms in
the 2p state after a long time? Express this fraction as a function of a0 (Bohr radius), ,
(1s-2p transition frequency), , e (electron charge) and E0.
Hint: consider only the 1s, 2p states and express the variation of the coefficients of the
wave function in time using the Schrödinger equation. Solve the equation for c2p
assuming that the probability to go from 1s to 2p is very small.
b) Assume that = 1 ps. Which strength should the electric field have to move 0.1% of
the atom population in the 2p state? (Such fields can be achieved in laser-plasma
interactions or in atom-ion collisions).
c) Which strength should the electric field have to move 0.1% of the atom population in
the 2s state?
0.55a 02 e 2 E 02
Answer: a)
b) 8.2 109 V/m c) impossible
1

2
 2   0  
13. Laser cooling of atoms with hyperfine structure
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a) The treatment of Doppler cooling in the course assumes a two-level atom. In real
experiments with a MOT, the hyperfine structure of the ground configuration causes
complications. We consider Sodium atoms with I=3/2. Draw approximately an energy
level diagram of the hyperfine structure of the 3s 2S1/2 and 3p 2P3/2 levels and indicate the
allowed electric dipole transitions.
b) In a laser cooling experiment, the transition 3s 2S1/2, F=2, to 3p 2P3/2, F'=3 is excited by
light that has a frequency detuning of = -/2= -5 MHz (to the red of the transition).
Selection rules dictate that the excited state decays back to the initial state so that there is a
nearly closed cycle of absorption and spontaneous emission. However, there is some offresonant excitation to the F'=2 state which can decay to the F=1 state and be "lost" from
the cycle. The F'=2 level lies 60 MHz below the F'=3 level. Estimate the average number
of photons scattered by an atom before it falls into the lower hyperfine level of the ground
configuration. (Assume that the transitions have similar strengths, and that I<<Isat)). To
counteract the leakage out of the laser cooling cycle, experiments use an additional laser
beam that excite atoms out of the F=1 level (so that they come back to F=2). Suggest a
suitable transition for this repumping process.
Answer: b)  122
14. Magnetic trapping of sodium atoms
a) Sketch the energy of the hyperfine levels of the 3s 2S1/2 ground state of sodium as a
function of the applied magnetic field strength (assumed to be much weaker than the
hyperfine structure). Give the energy difference between the hyperfine levels at zero
magnetic field. (The hyperfine structure constant is A= 886 MHz; sodium has a nuclear
spin equal to 3/2). Show that for both hyperfine levels the magnetic levels have a splitting
of 7 GHz/T.
b) The sodium atoms are placed in a magnetic trap trapping the atoms radially, (see Fig.
10.2) with a radius 10 mm, and a field gradient b'=1.5 Tm-1. Which magnetic sub-levels
will be trapped? (These should be "low-field-seeking" states, i.e. such that the magnetic
(Zeeman) energy decreases as the atom moves into a lower magnetic field). Estimate the
maximum temperature of the atoms that can be trapped for the relevant magnetic sublevels.
c) A radio frequency radiation field with frequency 70 MHz is applied. At which distance
of the center of the trap will atoms move to other magnetic states and cease to be trapped?
What is the new maximum temperature of the trap?
Answer: a) 7 GHz/T b) 5 and 10 mK c) 6.7 mm; Tmax=0.67 Tmax b)
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