physical chemistry 2 nd edition
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Chapter 17 Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement. Physical Chemistry 2 nd Edition. Thomas Engel, Philip Reid. Objectives. Introduction of Stern-Gerlach Experiment Understanding of Heisenberg Uncertainty Principle. Outline. - PowerPoint PPT PresentationTRANSCRIPT
Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid
Chapter 17 Chapter 17 Commuting and Noncommuting Operators and Commuting and Noncommuting Operators and the Surprising Consequences of Entanglementthe Surprising Consequences of Entanglement
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
ObjectivesObjectives
• Introduction of Stern-Gerlach Experiment
• Understanding of Heisenberg Uncertainty Principle
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
OutlineOutline
1. Commutation Relations2. The Stern-Gerlach Experiment3. The Heisenberg Uncertainty Principle
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.1 Commutation Relations17.1 Commutation Relations
• How can one know if two operators have a common set of eigenfunctions?
• We use the following
• If two operators have a common set of eigenfunctions, we say that they commute.
• Square brackets is called the commutator of the operators.
0ˆˆˆˆ xfABxfBA
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Example 17.1Example 17.1
Determine whether the momentum and (a) the kinetic energy and (b) the total energy can be known simultaneously.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
To solve these problems, we determine whether two operators commute by evaluating the commutator . If the commutator is zero, the two observables can be determined simultaneously and exactly.
BA ˆ and ˆ )(ˆˆ)(ˆˆ xfABxfBA
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
a. For momentum and kinetic energy, we evaluate
In calculating the third derivative, it does not matter if
the function is first differentiated twice and then once
or the other way around. Therefore, the momentum
and the kinetic energy can be determined
simultaneously and exactly.
xfdx
dih
dx
d
m
hxf
dx
d
m
h
dx
dih
2
22
2
22
22
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
b. For momentum and total energy, we evaluate
Because the kinetic energy and momentum operators commute, per part (a), this expression is equal to
)()(
)()()()()()(
xVdx
dxihf
xfdx
dxihVxV
dx
dxihfxf
dx
dxihV
xfdx
dxihVxfxV
dx
dih
xfdx
dihxV
dx
d
m
hxfxV
dx
d
m
h
dx
dih
)(
2)(
2 2
22
2
22
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
We conclude the following:
Therefore, the momentum and the total energy cannot be known simultaneously and exactly.
0)(),(
xV
dx
dih
dx
dihxV
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.2 Wave Packets and the Uncertainty 17.2 Wave Packets and the Uncertainty PrinciplePrinciple
• 17.2 Wave Packets and the Uncertainty Principle
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.2 The Stern-Gerlach Experiment17.2 The Stern-Gerlach Experiment
• In Stern-Gerlach experiment, the inhomogeneous magnetic field separates the beam into two, and only two, components.
• The initial normalized wave function that describes a single silver atom is
1 with 22
2
2
2
121 cccc
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.2 The Stern-Gerlach Experiment17.2 The Stern-Gerlach Experiment
• The conclusion is that the operators A, “measure the z component of the magnetic moment,” and B, “measure the x component of the magnetic moment,” do not commute.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Example 17.2Example 17.2
Assume that the double-slit experiment could be carried out with electrons using a slit spacing of b=10.0 nm. To be able to observe diffraction, we choose , and because diffraction requires reasonably monochromatic radiation, we choose . Show that with these parameters, the uncertainty in the position of the electron is greater than the slit spacing b.
b
01.0/ pp
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.3 The Heisenberg Uncertainty Principle17.3 The Heisenberg Uncertainty Principle
• 17.3 The Heisenberg Uncertainty Principle
• As a result of the superposition of many plane waves, the position of the particle is no longer completely unknown, and the momentum of the particle is no longer exactly known.
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.3 The Heisenberg Uncertainty Principle17.3 The Heisenberg Uncertainty Principle
• Heisenberg uncertainty principle quantifies the uncertainty in the position and momentum of a quantum mechanical particle.
• It is concluded that if a particle is prepared in a state in which the momentum is exactly known, then its position is completely unknown.
• Superposition of plane waves of very similar wave vectors given by kkeAAex
mn
mn
xknkixik
with ,2
1
2
100
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
17.3 The Heisenberg Uncertainty Principle17.3 The Heisenberg Uncertainty Principle
• Both position and momentum cannot be known exactly and simultaneously in quantum mechanics.
• Heisenberg famous uncertainty principle is
2
hxp
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
Using the de Broglie relation, the mean momentum is given by
And . 12810626.6 kgmsp
12610
34
10626.610100
10626.6
kgmsh
p
© 2010 Pearson Education South Asia Pte Ltd
Physical Chemistry 2nd EditionChapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
SolutionSolution
The minimum uncertainty in position is given by
which is greater than the slit spacing. Note that the concept of an electron trajectory is not well defined under these conditions. This offers an explanationfor the observation that the electron appears to go through both slits simultaneously!
mp
hx 8
28
34
1096.710626.62
10055.1
2