Exercises in "Atomfysik FK"
2- Elements of Quantum Mechanics
2-1 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 formulas):
a
0 sin(bx)sin(cx)dx = -1/2[sin{(b+c)a}/(b+c) - sin{(b-c)a}/(b-c)]
sin((n+1/2))=(-1)(n+1)/2 n even
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 
4 2  1 2
, an 
(n odd)
 (n  2)( n  2)
2
1
2-2 A stream of particles of mass m and energy E encounters a potential step of heigth W.
1) Write the wavefunction (Distinguish between two cases W< E and W>E).
2) Calculate the probablility current and give an expression for the reflection and
transmission coefficients in the two cases.
Answer: see Kvantmekanik AK
2.3. 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 What does it mean when =0?
4) Now assume =0. Find the new eigenvectors (wavefunctions) +, -. It is helpful to
use the angle defined as W12=W ei.
The general solution is given by
+= cos/2 ei/2 1+ sin/2 e-i/2 2
-= sin/2 ei/2 1- cos/2 e-i/2 2
With tg=W/.
5) If the system is in the state 1 at t=0, what is the probability that the system is in level 2
at time t? (some mathematics here)
2 2
2
  W 2 t)
Answer: 2) E  Em  2  W12 5) sin 2 ( ) sin 2 (

1
2.4 We consider the (electronic) spin of a hydrogen atom, described by the spin operator S
=  /2 , where  are the Pauli matrices, and the eigenvectors  and , corresponding to
ms = +1/2 and ms = -1/2.
1) Calculate Sz, Sz, Sx, Sx, S2, and S2
2) Assume that the system is first in state . What is the expectation value of Sz? Sx?
1
3) The system is in the superposition state
( + ). Is this wavefunction an
2
eigenfunction of Sz? Of any other spin component? What is now < Sz>? and < Sx>?
How could you describe the state of this system?
2.5 Consider a particle with angular momentum  =1, wavefunctions Y11, Y10, Y1-1. Use the
raising and lowering operators L to find how Lx acts on Y11, Y10, Y1-1. Hence find the
eigenfunctions and eigenvalues of Lx .
A beam of particles with  =1 is travelling along the y axis and passes through a SternGerlach magnet with its magnetic field along the x axis. The emerging beam with m=1 is
separated from the other two beams and passes through a second Stern-Gerlach magnet
with the magnetic field along the z axis. Into how many beams is this beam further split
and what are the relative numbers of atom in them?
Same question for the other two beams (see Fig. below).
m
z
x
z
z
2.6 Rigid rotator in a magnetic field
The Hamilton operator of a rigid rotator with axis in the z direction is given by
H 0  AL2  BL z ,where A and B are constant factors.
a) Give the energy eigenvalues and eigenvectors.
The rotator is placed in a magnetic field perpendicular to the z axis. The interaction
between the rigid rotator and the magnetic field is described by H1  CLy .
b) What is the expectation value of H1 in a stationary state of H0?
c) Use perturbation theory up to lowest nonvanishing order to approximate the enegy
eigenvalues of H0+H1. (Hint: use the operators L+, L-).
2
Answer: c)
2mC 2 
4B
3- One-electron atoms
Useful formula:


0
r n e r / a dr  n!a n1
3-1 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 an eigenfunction of the parity operator?
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, , ,9.84eV ,0,  2 ,
3 6 6
3
6
6
3-2 Calculate the expectation values of r, <r>, and of p2 for the 1s state.
2
Answer: 1.5 a 0 , 2
a0
3-3 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.
  
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?
3
11) Describe qualitatively the effect of the hyperfine structure? What is the Lamb shift?
Answer: 4)  =0,1, j=1/2, 3/2 8) 3 energy levels: 0, -  2 p ,  2 p /2
3. 5 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
4- Interaction of atoms with electromagnetic radiation
4-1 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=+1/2 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
4
Transition rate: 4.2 102 s-1
4-2 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) – using time-dependent perturbation theory.
4
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.
L1
v
c) Plot (approx.) the variation of F1 and F2 as functions of (0-).
L

sin 2  ( 0   ) 
L  L
2v 


Answer: F1  F0
; F2  4 F1 cos 2  ( 0   ) 1

2
2

( 0   )
2v 

4.3 Average oscillator strengths
The average oscillator strength for the transition ni  i  n f  f is defined as
 
E.D fi

f n f  f ni  i 
f
1

2 i  1 m f    f
2
i
f
mi    i
n f  f m f ni  i mi
.
Integration over magnetic quantum numbers leads to:
2
2m fi Max( i ,  f )   
3
f n f  f ni  i 
 0 Rn f  f (r ) Rni  i (r )r dr  .

3
2 i  1 
a) Calculate the average oscillator strength for the transitions 2 p  3s and 2 p  3d in

Hydrogen. (Hint: use  r k e r / a dr  k!a k 1 )
0
The average oscillator strengths obey a sum rule
f
n f  f ni  i
 1 , which can be
nf  f
decomposed into two sums:
(  1)( 2 i  3)
f n f  i1ni  i  i
,

3(2 i  1)
nf
f
n f  i 1 ni  i

nf
 i (2 i  1)
.
3(2 i  1)
b) Use these sum rules to show that the electromagnetic dipole transitions in which the
principal quantum number n and the angular quantum number  change in the same
sense (i.e. both decrease or both increase) tend to be more probable than transitions in
which n and  change in the opposite sense. (Hint: give a simple argument based on
the sign of the average oscillator strength. Use n f  ni   fi  0 ).
Answer: a) 0.014, 0.7
5- Interaction of atoms with magnetic and electric fields
5-1 Now we consider the n=2 level of hydrogen. In the absence of fine structure this level
is 4 times degenerate. Now we switch on a perturbation in the form of a constant electric
5
field in the z-direction. This lifts (some of) the degeneracy – we are interested in finding
out how, and how much. The Hamiltonian for this can be written H’ = eEz = eErcos,
where E is the field strength. H’ is large compared to the fine structure splitting.
1) Which of the four n=2 levels are coupled by H’? Call these levels 1 and 2
2) Write down H’ in the basis of 1 and 2. The overlap integrals will give you that
H’12 = -3eEa0.
3) Find the eigenvalues and eigenstates of the new Hamiltonian H0 + H’.
What you have calculated is called the linear Stark effect.
Answer: see book
6- Many-electron atoms
6-1 In this problem we look at the energies for the 1snl (nl≠1s) configurations in the
helium atom. The Hamiltonian is written as:
H = H1 + H2 + e2/|r1-r2|, where Hj = pj2/2m – 2e2/rj
a) If we first neglect the electron repulsion, what is the degeneracy of the 1snl
configuration? (also neglect spin-orbit coupling)
b) Which are the possible values of the total spin?
c) Write down the possible spatial wavefunctions for the 1snl configuration in terms of the
wavefunctions 100(rj) and nlm(rj) for Hj.
d) Which spin states do these wavefunctions correspond to?
e) What is the energy for the 1snl configuration?
f) We now include the electron-electron repulsion as a perturbation. Write down the
expression for the first order correction to the 1snl energy.
Answer: see book
6.2- Positronium in a time-dependent magnetic field
An electron-positron system (positronium) is described by the Hamilton operator
 
 
H 0  AS1  S2 where A is a constant factor and S1 , S 2 the spins of the electron and of the
  
positron. The total spin is S = S1 + S 2 .
a) Show that H0, S 2 och Sz commute with each other.
 

2  2
(Hint: use S1  S 2 = 12 ( S 2  S1  S2 ) ). Write down the wavefunctions common to
H0, S 2 och Sz as well as the energy levels (Hint: use the results obtained for the spin
functions in helium). What is the degeneracy of the different energy levels?
The positronium is placed at time t=0 in a magnetic field B (t )  B0 e  t / in the z direction.
eB (t )
( S1 z  S 2 z ) .
The interaction with the magnetic field is described by the operator V 
mc
The total Hamiltonian is H=H0+V. The system is assumed to be initially in the singlet
level.
b) Use time-dependent perturbation theory to calculate the probability P(t) for the system
to be in a triplet level after a time t.
6
3
1
e2 B2
Svar: a)   2 A,  2 A b) 2 2
4
4
m c
1  2 cos( At )e t /   e 2t / 
1
A2  2  2

7