Download Questions for learning Quantum Mechanics of FYSA21

Questions for learning Quantum Mechanics of
FYSA21
Ulf von Barth, Tomas Brage and Peter Samuelsson
October 27, 2014
Experimental background
1. At the beginning of last century, four important effects/experiments/models
that led to problems when analysed within the framework of classical
physics were
• black body radiation,
• Rutherfords model of the atom,
• the photoelectric effect,
• Compton scattering.
Chose two of the four examples, describe the problems with a classical
physics analysis and sketch the quantum mechanical concepts introduced to solve the problems. (5p)
2. Describe Youngs’s double slit experiment with electrons. Also, describe what you can learn about physical properties of quantum particles from the double slit experiment. (5p)
Waves, wave functions and wavepackets
1. What is meant by the statement “the wave function describes the
probability amplitude of a quantum particle”? (1p)
2. Write down the time dependent wave function for a free quantum
particle moving to the right on the x-axis (traveling, plane wave).
What changes when you consider a wave traveling to the left? (2p)
3. What is meant by normalization of a wave function Ψ(x, t)? Why does
one, on physical grounds, require wave functions to be normalized?
(2p)
4. A particle on the x-axis is described by a normalized wave function
(at time t) Ψ(x, t). What is the probability to detect the particle in
the interval 0 < x < a? (1p)
5. What is meant by the collapse of the wave function when measuring
the position of a particle? Give an example. (2p)
6. What is a wavepacket and how does it differ from a traveling, plane
wave? (2p)
7. What is the difference between the phase and the group velocity of a
wave? (2p)
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Schrödinger equation
1. How does the the Schrödinger equation for free particles look like and
how can you ”derive/motivate” it? (3p)
2. How can the Schrödinger equation be generalized in order to describe
a particle moving in a one-dimensional potential V (x)? (1p)
3. How does the time-independent Schrödinger equation for a particle at
energy E, moving in a one-dimensional potential V (x) look like? How
can one derive the time independent Schrödinger equation from the
time dependent one? (3p)
4. From a mathematical point of view, what is the order of the Schrödinger
equation with respect to time? How does this compare with Newton’s
second law for classical particles? (1p)
One-dimensional bound states
1. What is the wave function in a region where the potential becomes
infinitely repulsive? (1p)
2. Solve the time independent Schrödinger equation in an infinitely deep
one dimensional potential well located at 0 < x < a. Sketch the
wavefunctions of the lowest-in-energy bound states. How do the bound
state energies change when the width a is changed? (4p)
3. When there is a change in the potential from one region to the next,
which conditions must the wave function obey at the boundary? (2p)
4. Describe the key steps in the solution of the time independent Schrödinger
equation in a finite one dimensional potential well located at 0 < x < a.
How do the wavefunctions and energies of the bound state differ from
the ones of the corresponding infinite well? Motivate why? (3p)
5. Does every negative one-dimensional potential possess a bound state?
Does it always possess excited states? (1p)
One-dimensional scattering problems
1. Starting from the definition of the probability density and using the
time dependent Schrödinger equation, derive the expression for the
probability current. (3p)
2. What is probability current of a left- (right-) traveling, plane wave?
What can one say about the position and time dependence of the
current? (2p)
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3. How are the transmission and reflection probabilities for traveling,
plane waves incident on a scattering potential in one dimension defined
in Quantum Mechanics? (2p)
4. Consider a potential step defined by V (x) = 0 for x < 0 and V (x) =
−V0 for x > 0. Sketch the derivation of the transmission probability
for a plane wave incident from the left, at energy E > 0. Describe
qualitatively the transmission probability as a function of energy and
relate it to the corresponding case for a particle described by classical
mechanics. (4p)
5. Consider a plane wave at energy E > 0 incident on a potential well
defined by V (x) = 0 for x < 0 and x > a and V (x) = −V0 for
0 < x < a. Describe qualitatively the transmission probability as a
function of energy. What is a resonance in the transmission? (3p)
6. Consider a plane wave at energy E > 0 incident on a potential barrier
defined by V (x) = 0 for x < 0 and x > a and V (x) = V0 for 0 < x < a.
Describe qualitatively the transmission probability as a function of
energy. (3p)
7. For the potential barrier in the previous question, how would an incident particle with kinetic energy E > 0, described by classical mechanics, behave? Based on your answer, discuss the concept of quantum
tunneling. (2p)
Physical observables
1. How does the operator corresponding to momentum look like in Quantum Mechanics? Motivate it. (2p)
2. How does the operator corresponding to the kinetic energy look like
in Quantum Mechanics? Motivate it. (2p)
3. How does the operator corresponding to the total energy look like in
Quantum Mechanics? Motivate it. (2p)
4. What is meant by an Hermitian operator? (1p)
5. What is meant by the Hermitian conjugate to an operator? (1p)
6. Why are physical observables in Quantum Mechanics represented by
Hermitian operators? (1p)
7. What is meant by an eigenvalue and an eigenfunction of an operator?
(1p)
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8. If you measure an physical quantity represented by the operator Â
what values can you get? (1p)
9. If a Quantum Mechanical system has the wave function Ψ(x) and
you measure the observable represented by the operator  with an
eigenvalue a and the corresponding normalised eigenfunction Ψa (x),
what is the probability of obtaining the value a? (1p)
10. If you have just measured the obeservable  of a Quantum Mechanical
system and obtained the value a and then immediately remeasure the
same observable, which value will you obtain? Why? (2p)
11. Define the expectation value of an operator  in the state represented
by the normalised wave function Ψ(x). (1p)
12. Formulate the expansion/superposition theorem of Quantum Mechanics. (2p)
13. Use the expansion theorem to show why the term ”expectation value”
in Quantum Mechanics is a direct analogy to the same concept in
statistics. (3p)
Simultaneous measurements
1. How do you define the commutator between two operators? (1p)
2. What is meant by commuting and non-commuting observables? (1p)
3. What does Heisenbergs’s uncertainty relation have to say about the
uncertainties in the simultaneous measurement of two non-commuting
observables? (2p)
4. Show that two commuting observables with non-degenerate spectra
have simultaneous eigenfunctions. (2p)
5. What is the commutator between the position- and the momentum
operators? (1p)
6. What is meant by a minimum uncertainty wave packet? Give an
example of such a wave packet. (2p)
The Harmonic oscillator
1. How does the Hamiltonian Ĥ of the Harmonic oscillator look like and
what is the physical background to its classical analogue? (2p)
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2. How do you define ”ladder” operators â and ↠in terms of the operators for position and momentum. (1p)
3. How can you write the Hamiltonian Ĥ in terms of ”ladder” operators?
(1p)
4. What is the commutation relation between â and ↠? (1p)
5. Making use of the commutation relation between Ĥ and â, show that
if ϕn is an eigenstate of Ĥ, so is âϕn . (2p)
6. What is the distance in energy between the eigenvalues of the Harmonic oscillator? What is the ground state energy? (1p)
7. Describe/sketch qualitatively the lowest-in-energy eigenfunctions of
the Harmonic oscillator? (2p)
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