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Transcript
The Measuring Process:
a Superposition
or
a Mixture?
1
The Rotational Motion of a Diatomic Molecule
Upon first approximation let us regard a molecular
rotating system as a free particle on a ring (a twodimensional hard rotor.)
The working assumptions are:
1. The molecule is moving on a plane (approximation)
2. The molecular radius is rigid (approximation)
3. The central atom is stationary (approximation)
Or
4. The system is composed of a single particle with
reduced mass, moving around the center of gravity
(precise.)
2
The Postulates of Quantum Mechanics for a Free
Particle on a Ring
1. (The game tools) The state of the system can be described by a
wavepacket, (the game board) belonging to the space of
continuous functions in angle  :
( , t )   g (m)  m ( , t ) ;  m ( ,0) 
1
2
m
e
im
2. (The game rules) For each component in the wavepacket, the
following is true:
 m ( , t )  e
E
i t

 m ( ,0) ; E  (m) / 2 I
2
3. (The interface) the measuring operation has the following
probability of finding the particle in the angle element d  :
( , t )  d
2
3
The Representation of a Free Particle on a Ring
y
The system state is defined for a classical particle
by the particle’s angle and its angular momentum.
In quantum mechanics the system state is defined
by a wave function that has a complex value for
each angle. For example:
y

y
A particle with a preferred
axis in space. There is a
complete uncertainty of the
x angular momentum
( ) 
x
1
2
px
[ 1 ( )   1 ( )]
y
(+ )H
x

(- )F
A particle with a definite
angular momentum.
x There is a complete
uncertainty of the angle
y
 2 ( ) 
x
1
2
ei 2
m2
Lz  2
4
The Stern-Gerlach Spectroscopy
The Phenomenon
A beam of oxygen molecules passing
through a non-homogeneous magnetic
field splits into a number of beams
z
5
The Stern-Gerlach Spectroscopy
A Classical Model (for a planetary HF)
The motion of the positively charged hydrogen is equivalent to a ring
current. The current inspires a magnetic moment µB relative to the
angular momentum, the external field Bz exerts force Bz on the
magnetic moment.
B  Lz
B  Lz
Bz
N
 Fz  Bz   B
S
6
The Stern-Gerlach Spectroscopy
A Quantum Model
The angular momentum is a singular measurement. Each
particle is moving along one of the orbits according to its angular
momentum.
Lz  m
N
2

0
A mixture (each particle in a
different quantum state)
S

 2
7
Filtering a Superposition
The measuring system is a filter that separates basis states. While
measuring, each particle appears in one point only. The
probability for this is:
P(m)  g ( m)
2
1 ( )
px
1
2
N
[ 1 ( )   1 ( )]
Superposition (one particle in
two states)
S
 1 ( )
m=1
50%
m=-1
50%
8
Measuring as a Destructive Process
The Uncertainty Principle
Measuring results in a certainty of one property in
exchange to an uncertainty of another one, which had been
known prior to the measuring process.
Before measurement: The particle is in the x axis direction,
the angular momentum is unknown.
After measurement: The angular momentum is known, the
particle’s direction is unknown.
  Lz  h
9
Separating to Basis States
Basis State: a quantum state with a well-defined particle property
(position, momentum, angle, angular momentum, polarization,
energy, etc.) A set of basis states is measured for each
dimension. A basis state for one measurement is not necessarily
a basis for another.
A basis state for measuring
polarization (direction in
space)
A basis state for measuring
the angular momentum
y
y
x
x
10
The Completeness of the Basis
(Fourier Theorem)
Each periodic function defined on [ , -] can be decomposed
linearly:
( )   g (m)  m ( )
m
 m ( ) 
1
2
e
im

; g (m)   m ( )  ( ) d
*

In this way it is possible to
calculate the probability of
finding the particle in a given
angular momentum for each
continuous function on the ring
-
0

11
Superposition Versus Mixture
The wave function in a superposition is simultaneously in
various quantum states. The measuring process causes
each particle to choose only one basis state and this results
in a mixture.
After measuring

1
2
2
1   2
2

Before measuring

1
  1  2
2

2
12
How Can We Distinguish Between the Two?
By measuring another property: the probability of a reaction
in different angles. The angular dependency of the steric
factor in a nucleofilic charge reaction is examined. The basis
set of the direction measurements differentiates between
superposition and a mixture of states of angular momentum.
Mixture: a homogeneous
distribution in all directions
Superposition: a
preference to an
aignement in the x axis
direction
1
2
1

2

2



2

2

2
13
An Experiment with Crossing Beams
The reaction Li + HF  LiF + H is examined by crossing
beams of reactants and measuring the amount of the output
in different angles. The HF molecule is aligned in the Px
state.
H-F /F-H
Li
H-F /F-H
Li
14