Download LECTURE 8

HEINSENBERG’S UNCERTAINTY PRINCIPLE
“It is impossible to determine both position and momentum of a particle
simultaneously and accurately. The product of uncertainty involved in the
determination of position and momentum simultaneously is greater or equal
to h/2Π ”
h
x  p x 
2
h
E  t 
2
Significance: “Probalility” replaces “Exactness”
Heisenberg - 1927
An event which is impossible to occur according to classical physics has a finite probability of
occurrence according to Quantum Mechanics
1
The Uncertainty Principle
Since we deal with probabilities we have to •
ask ourselves:
“How precise is our knowledge?”
Specifically, we want to know Coordinate •
and Momentum of a particle at time t = 0
If we know the forces acting upon the particle –
than, according to classical physics, we know
everything about a particle at any moment in the
future
The Uncertainty Principle
But it is impossible to give the precise position
of a wave
A wave is naturally spread out
Consider the case of diffraction
Most of the energy arriving at a distant screen
falls within the first maximum
•
•
•
•
The Uncertainty Principle
Can we know Coordinate and Momentum •
(velocity) at some time t = 0 exactly, if we
deal here with probabilities?
The answer in Quantum Mechanics is •
different from that in Classical Physics, and
is presented by the Heisenberg’s Uncertainty
Principle
Classical Uncertainty
d
Consider classical diffraction•
Most of light falls within first maximum•
The angular limit of the first maximum is at the •
first zero of intensity which occurs at an angle 
set by the condition, d sin = , so we can say that
the angle of light is between + and -
Consider the following:
d sin  ~ 
d
sin 

~ 1, or d
d k sin  ~ 2
2

sin  ~ 2
Now, d  y and k y ~ 2k y
So have, y
k y
2
~ 2 , yk y ~ 4
As the uncertainty in y increases the uncertainty in the y-component of the k-vector
decreases
Classical Uncertainty
Multiply yk y ~ 4 by 
y (k y ) ~ 4
yp y ~ 4 ~ 
The classical uncertainty relation
The Uncertainty Principle
An experiment cannot simultaneously
determine a component of the
momentum of a particle (e.g., px) and
the exact value of the corresponding
coordinate, x.
The best one can do is

(p x )( x) 
2
The Uncertainty Principle
The limitations imposed by the uncertainty .1
principle have nothing to do with quality of the
experimental equipment
The uncertainty principle does imply that one .2
cannot determine the position or the momentum
with arbitrary accuracy
It refers to the impossibility of precise knowledge about
–
both: e.g. if Δx = 0, then Δ px is infinity, and vice versa
The uncertainty principle is confirmed by .3
experiment, and is a direct consequence of the de
Broglie’s hypothesis
HOWEVER
Since the wavefunction, Ψ(x,t), describes a •
particle, its evolution in time under the
action of the wave equation describes the
future history of the particle
Ψ(x,t) is determined by Ψ(x, t = 0) –
Thus, instead of the coordinate and velocity •
at t = 0 we want to know the wavefunction at
t=0
Thus uncertainty is built in from the beginning –
and the wavefunction at all times is related to
the evolution of probability
Examples:
Bullet
p = mv = 0.1 kg × 1000 m/s = 100 kg·m/s •
If Δp = 0.01% p = 0.01 kg·m/s –
34
 1.05  10 J  s
x 

 1.05  10 32 m
p
0.01 kg  m/s
Which is much more smaller than size of the –
atoms the bullet made of!
So for practical purposes we can know the –
position of the bullet precisely
Examples:
p
Electron (m = 9.11×10-31 kg) with energy •
4.9eV
Assume Δp = 0.01% p •
2 mE  2  9.1  10 31 kg  4.9  1.6  10 19 J  1.2  10 24 kg  m / s
p  0.01% p  1.2  10 - 28 kg·m/s

 1.05  10  34
6
4
x 

 10 m  10 A
- 28
p 1.2  10
Which is much larger than the size of the atom! –
So on atomic scale uncertainty plays a key role –
Quantum Mechanics
The methods of Quantum Mechanics •
consist in finding the wavefunction
associated with a particle or a system
Once we know this wavefunction we •
know “everything” about the system!
The Uncertainty Principle
Between energy and time •

(E )( t ) 
2
Quantum Mechanical Operators
Physical Quantity  Operators: “formal form” 
“actual operation”

Momentum
p x  pˆ x  i
x

ˆ
Total Energy
E  E  i
t
x  xˆ  x
Coordinate
U ( x)  Uˆ ( x)  Potential
U ( x) Function
Wave Function of Free Particle
Since the de Broglie expression is true for •
any particle, we must assume that any free
particle can be described by a traveling
wave, i.e. the wavefunction of a free particle
is a traveling wave
For classical waves: •
A cos[ kx  t ], A sin[ kx  t ],
Wave Function of Free Particle
However, these functions are not eigenfunctions of the •
momentum operator, with them we do not find,
 
pˆ x  
 p  k
i x But see what happens if we try, •
“manipulations”
( x, t )  Acos[kx After
 t ] some
i sin[ kx
 t ]  A exp( ikx)we get •
 ( x, t )  Acos[ kx  t ]  i sin[ kx  t ]  Aei{kx t }
 ( x, t )  Aeikxe it
 ( x, t )   ( x)e it
with  ( x)  Aeikx
Wave Function of Free Particle
2
p
E that •
Recall
k
 and   2 
 

Thus, the wavefunction of a Free Particle •
 x, t   Ae
p E
i  x 


 Ae
p
E
i x i t


e
or
 ( x, t )   ( x )e
E
i t

with, ( x)  Ae
p
i x

This wave function is an eigenfunction of momentum and energy!
Expectation Values
Only average values of physical quantities can be •
determined (can’t determine value of a quantity at a point)
These average values are called Expectation Values •
These are values of physical quantities that quantum mechanics –
predicts and which, from experimental point of view, are averages
of multiple measurements
Example, [expected] position of the particle •

x 

xP( x)dx , with



P( x)dx  1

Expectation Values
Since P(r,t)dV=|Ψ(r,t)|2dV, we have a way to •
calculate expectation values if the wavefunction for the
system (or particle) is known

x 



2
2
xP( x , t )dx  x ( x , t ) dx , since ( x , t )   * ( x , t )( x , t )




x 
 * ( x , t )x( x , t )dx

In General for a Physical Quantity W •
Below Ŵ is an operator (discussed later) acting on –
wavefunction Ψ(r,t)

W 



ˆ ( x , t ) dx
 * ( x, t) W

Expectation Value
for Momentum of a Free Particle





p   ( x) * pˆ  ( x)dx 
Generally •




 ( x) *   i
 


(
x
)
dx


x 


 ( x) 
*
*  ( x)
p   ( x )   i
dx
dx  i  ( x) 
x 
x




 ( x) 

p
i x
Ae 
Free •
Particle

 ipx
2
with  ( x) dx   Ae  



 


 ipx
p   Ae  


 




*
*
 ipx
 Ae   dx  1


p

i x

Ae   dx
  i
x


*

 ipx p  ipx
 ipx
p
p  i  Ae   i  Ae   dx  ii
 Ae  
 

  


 

h
p  i 2 p  1  p  k 



*
 ipx
 Ae   dx


Properties of the Wavefunction and its First
Derivative
must be finite for all x .1
must be single-valued for all x .2
must be continuous for all x .3


U ( x)   * ( x)U ( x) ( x)dx


px
 

  ( x)  i  ( x)dx
x 



*
 
E   ( x, t ) i  ( x, t )dx
 t 

*