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Exercises For Advanced Quantum Mechanics: Hand in On 15th of November

This document provides two problems for an advanced quantum mechanics exercise involving time-dependent perturbations. Problem 1 involves calculating the electric dipole moment of a harmonic oscillator system placed in an oscillating external electric field. Problem 2 involves calculating the probability of a hydrogen atom initially in the ground state being in the 2p state after exposure to a changing electric field. Hints are provided for the wave functions needed to solve Problem 2.

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39 views1 page

Exercises For Advanced Quantum Mechanics: Hand in On 15th of November

This document provides two problems for an advanced quantum mechanics exercise involving time-dependent perturbations. Problem 1 involves calculating the electric dipole moment of a harmonic oscillator system placed in an oscillating external electric field. Problem 2 involves calculating the probability of a hydrogen atom initially in the ground state being in the 2p state after exposure to a changing electric field. Hints are provided for the wave functions needed to solve Problem 2.

Uploaded by

Anonymous oBNkcDk
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Exercises for advanced quantum mechanics

Lecturer: Bjorn Garbrecht


Sheets by Marco Drewes

WS 2013/14
Sheet 4

hand in on 15th of November

Problem 1: Harmonic oscillator in an external field


50%
Consider two charged particles in a bound state. For small displacements with respect to
each other the potential in the relative coordinate x can be approximated by a parabolic,
i.e. the Hamiltonian H0 is that of a harmonic oscillator with mass m and frequency 0 . Now
the system is placed in an oscillating external electric field, which we characterize by a time
dependent perturbation
V (t) = 0 x cos(t).
(1)
Calculate the electric dipole moment h|qx|i assuming that at initial time the system is in
an energy eigenstate of H0 and the perturbation is switched on at t = 0.
Problem 2: Hydrogen atom in a changing electric field
50%
Consider a hydrogen atom in the ground state in the infinte past (t ). It is exposed
~
to a homogeneous electric field E(t)
= (0, 0, E(t)) with
E(t) =

B
1
.
2
e + t2

(2)

Calculate the probability that the atom is in the 2p state at t .


Hint: The 1s wave function is given by
1s

1
=

1
a0

3/2

er/a0 ,

(3)

the wave functions in the 2p state are given by


2p,m=0
2p,m=1

 3/2
1
1
r r/2a0
=
e
cos ,
a0
4 2 a0
 3/2
1
r r/2a0
1
=
e
sin ei .
8 a0
a0

Here a0 is the Bohr radius and r, , are polar coordinates.

In both exercises it is sufficient to use lowest order perturbation theory.

(4)
(5)

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