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
ò0 sin(bx)sin(cx)dx = -1/2[sin{(b+c)a}/(b+c) - sin{(b-c)a}/(b-c)]
sin((n+1/2)p)=(-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.
Answer: 2) (n odd)
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 f1, f2 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 f1, f2.
2) Calculate the new eigenvalues E+, E- for the perturbed system. Use the notations Em=(E1+E2)/2, D=(E1-E2)/2.
3) Draw approximately E+, E- as a function of D. What does it mean when D=0?
4) Now assume D=0. Find the new eigenvectors (wavefunctions) f+, f-. It is helpful to use the angle a, defined as W12=W eia.
The general solution is given by
f+= cosq/2 eia/2 f1+ sinq/2 e-ia/2 f2
f-= sinq/2 eia/2 f1- cosq/2 e-ia/2 f2
With tgq=W/D.
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)
Answer: 2) 5)
2.4 We consider the (electronic) spin of a hydrogen atom, described by the spin operator S = /2 s, where s are the Pauli matrices, and the eigenvectors a and b, corresponding to ms = +1/2 and ms = -1/2.
1) Calculate Sza, Szb, Sxa, Sxb, S2a, and S2b
2) Assume that the system is first in state a. What is the expectation value of Sz? Sx?
3) The system is in the superposition state ( a + b). Is this wavefunction an 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 to find how acts on Y11, Y10, Y1-1. Hence find the eigenfunctions and eigenvalues of .
A beam of particles with =1 is travelling along the y axis and passes through a Stern-Gerlach 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).
2.6 Rigid rotator in a magnetic field
The Hamilton operator of a rigid rotator with axis in the z direction is given by ,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 .
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-).
Answer: c)
3- One-electron atoms Useful formula:
3-1 Consider a hydrogen atom. At time t=0, the wavefunction is the following superposition of energy eigenfunctions Ynlm(r):
Y(r,t=0) = N[2Y100(r) - Y210(r) + Y311(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?
Answer:
3-2 Calculate the expectation values of r, <r>, and of p2 for the 1s state.
Answer: 1.5 ,
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 , WD depends only on r, and Wso= VLS(r). 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 for n=2?
4) Show that Wso commutes with 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 . Use the triangle rule to determine which j, and s are possible for n=2. These states are labelled .
6) Show that can be written as .
7) Show that and Jz commute with H0 and Wso.
8) In which basis are , Jz , H0, Wso , , diagonal?
9) Write down Wso in the above matrix using the integrals where is the radial wavefunction for 2. How many energy levels are there?
10) It can be shown that Wmv, WD commute with . What is the effect of Wmv, WD on the n=2 shell?
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
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:
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 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(wt) of frequency w. The atoms are initially in state i, and the frequency w is close to resonant with an atomic transition i->f (frequency w0). The perturbation from the rf-field is described by H’ = -E*Dcos(wt), where D is the dipole operator.
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.
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.
c) Plot (approx.) the variation of F1 and F2 as functions of (w0-w).
Answer: ;
4.3 Average oscillator strengths
The average oscillator strength for the transition is defined as
.
Integration over magnetic quantum numbers leads to:
.
a) Calculate the average oscillator strength for the transitions and in Hydrogen. (Hint: use)
The average oscillator strengths obey a sum rule , which can be decomposed into two sums:
, .
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 ).
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 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 = eErcosq, 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 f1 and f2
2) Write down H’ in the basis of f1 and f2. 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 f100(rj) and fnlm(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
where A is a constant factor and ,the spins of the electron and of the positron. The total spin is =+.
a) Show that H0,och Sz commute with each other.
(Hint: use =). Write down the wavefunctions common to H0,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 in the z direction. The interaction with the magnetic field is described by the operator . 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.
Svar: a) b)
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