Transcript Chapter 17

Chapter 17
Commuting and Noncommuting Operators and
the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid
Objectives
• Introduction of Stern-Gerlach Experiment
• Understanding of Heisenberg Uncertainty
Principle
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
1. Commutation Relations
2. The Stern-Gerlach Experiment
3. The Heisenberg Uncertainty Principle
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.1 Commutation Relations
•
•
How can one know if two operators have a
common set of eigenfunctions?
We use the following

 

Aˆ Bˆ f x  Bˆ Aˆ f x  0
•
•
If two operators have a common set of
eigenfunctions, we say that they commute.
Square brackets is called the commutator of
the operators.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 17.1
Determine whether the momentum and (a) the
kinetic energy and (b) the total energy can be
known simultaneously.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
To solve these problems, we determine whether
two operators Aˆ and Bˆ commute by
evaluating the commutator Aˆ Bˆf ( x) Bˆ Aˆ f ( x) . If
the commutator is zero, the two observables
can be determined simultaneously and exactly.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
a. For momentum and kinetic energy, we evaluate
 h 2 d 2 
d  h2 d 2 
d 





 ih  
f
x



ih

 f x 
2 
2 

dx  2m dx 
dx 
 2m dx 
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.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
b. For momentum and total energy, we evaluate

 h2 d 2

d  h2 d 2
d 





 ih  

V
(
x
)
f
x



V
(
x
)

ih

 f x 
2
2



dx  2m dx
dx 

 2m dx

Because the kinetic energy and momentum
operators commute, per part (a), this expression is
equal to
d
d
 ih V x  f x   ihV x 
f x 
dx
dx
d
d
d
 ihV ( x)
f ( x)  ihf ( x) V ( x)  ihV ( x)
f ( x)
dx
dx
dx
d
 ihf ( x) V ( x)
dx
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
We conclude the following:
d
d

V ( x),ih dx   ih dx V ( x)  0
Therefore, the momentum and the total energy
cannot be known simultaneously and exactly.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.2 Wave Packets and the Uncertainty Principle
•
17.2 Wave Packets and the Uncertainty
Principle
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.2 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
c1
c2
2
2


 with c1  c2  1
2
2
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.2 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.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 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   b , and because diffraction requires
reasonably monochromatic radiation, we choose
p / p  0.01 . Show that with these parameters, the
uncertainty in the position of the electron is greater
than the slit spacing b.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.3 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.
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.3 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
1 ik0 x 1 nm i k0  nk x
 x   Ae  A  e
, with k  k
2
2 n m
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
17.3 The Heisenberg Uncertainty Principle
•
•
Both position and momentum cannot be known
exactly and simultaneously in quantum
mechanics.
Heisenberg famous uncertainty principle is
h
p x 
2
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Using the de Broglie relation, the mean momentum
is given by
6.6261034
 26
1
p  

6
.
626

10
kgm
s
 1001010
h
And
p  6.6261028 kgms1 .
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
The minimum uncertainty in position is given by
34
h
1.05510
8
x 


7
.
96

10
m
 28
2p 2 6.62610


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 explanation
for the observation that the electron appears to go
through both slits simultaneously!
Chapter 17: Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd