Download Quantum Physics 2005 Notes-3 Observables – (Chapter 5) Notes 3

Quantum Physics 2005
Notes-3
Observables – (Chapter 5)
Notes 3
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Introduction
• Observables are physical attributes of a system that
can be measured in the laboratory.
• In quantum physics, in the absence of a
measurement, a microscopic system does not
necessarily have values of its physical properties. (A
particle does not “have” a position until we measure
it. It has a set of possible positions.)
• We want to find out how to calculate observables
from wavefunctions. The mathematical approach is
through the introduction of operators.
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• In quantum scale measurements, the
measurement itself affects the state of the
particle, so we cannot use classical
definitions of observables.
• Example: position and momentum.
– The momentum is measured by measuring
velocity. This involves the measurement of
position at two times. The first measurement
affects the momentum and hence the later
position and consequent velocity.
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Two ways to compute momentum
1) Given ! ( x, t ), use the wave function to compute " p ( p).
Compute p from ∫ "* p"dp.
2) Find a way to express p as a function of x.
Compute p from ∫ ! * p( x)!dx.
Let’s compute each way for the lowest state
of the square well.
Notes 3
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Momentum in the square well
! ( x, t ) =
2
 $ x  i#t
cos 
e
L
L


1 %
1 L/2
 $ x  ipx /
& ipx /
"( p) =
!
=
(
x
,
0)
e
dx
cos
∫
∫

 e dx
L$ h & L / 2
2$ h &%
 L 
1 L/2
$x 
=
∫ cos 
 ( cos( px / h) + i sin( px / h) ) dx
L$ h & L / 2
 L 
=
L/2 
 $ p  
px
  $ p  
1 L/2
1
$x 
=
&
+
dx
x
cos
cos(
)
cos
cos
∫
∫  


 
  L + h  x   dx
L
L
h
h
&
L
/
2
L$ h & L / 2
L
$
h
2


 
 

 
L/2
  $ p  
 $ p  
sin
sin
x
&



 L + h  x
1    L h  
 

=
+
 $ p  
2 L$ h    $ p  
&
 L + h  
   L h  
  &L/ 2


 
=
Notes 3
  $ p  L 
 $ p  L 
sin
sin
&



 L + h  2 
1    L h  2 
 

+
 $ p  
L$ h    $ p  
&


 L + h  
  L h 
 


 
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Momentum in square well
L
"( p) =
2
  $ p  L 
 $ p  L 
 sin   L & h  2  sin   L + h  2  
1  
 
 

+
 $ p  L  
L$ h    $ p  L 
&
 L + h  2  
   L h  2 
  


 
  $ L 
$ L 

p
 sin   2 & 2h p   sin  +
L  

 2 2h  
"( p) =
+
4$ h   $ L 
$ L  
p
p 
&
 +
  2 2h 
 2 2h  

L
for plotting, let's set
= 2 (in some units).
2h
Notes 3
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Momentum in square well
One part of the momentum
distribution looks like this
The other looks like this
=
+
peaked at p = ±h$ / L
The sum
see phi_p for square_well.mws
Notes 3
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Momentum in square well
Now to calculate expectation value of momentum,
2
  $ L 
$ L 
sin
p
&
p
   2 2h   sin  +
%
%
L
2
2
h


+

  pdp
 
p = ∫ "* p"dp = ∫
4$ h   $ L 
$ L  
&%
&%
&
p
p 
 +
  2 2h 
2
2
h

 

(We don't really want to calculate this awful thing, so let's
think about symmetry. We note that " is symmetric about p = 0.
This means that p" is an odd function, and therefore the integral is zero.)
The situation would be much worse if I asked you to calculate p 2
because you'd really have to do the integral.
Notes 3
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The Momentum Operator: a plausibility
argument
We want to be able to do something like this:
%
px = ∫ ! * ( x, t ) px ! ( x, t ) dx
&%
but we don't yet know how to write p ( x).
Consider a pure momentum function, which
is used to make up a wavepacket:
1
! p ( x, t ) =
ei ( px & Et ) / h
2$ h
to extract p from this, we might take the derivative w.r.t. x.
'! p
ip
= !p
h
'x
Notes 3
'
⇒ & i h ! p ( x, t ) = p ! p
'x
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The Momentum Operator
• From a plausibility argument for a pure
momentum state, we found a “momentum
operator”
'
pˆ x ( &ih
'x
We generalize this idea and will apply this
operator to any wavefunction to extract
information on the momentum.
' 

*
ˆ
px = ∫ ! ( x, t ) px ! ( x, t ) dx = ∫ ! ( x, t )  &ih  ! ( x, t )dx
'x 
&%
&%

%
Notes 3
*
%
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Momentum in the square well
' 

px = ∫ ! ( x, t ) pˆ x ! ( x, t ) dx = ∫ ! * ( x, t )  &ih  ! ( x, t )dx
'x 
&%
&%

&ih 2 %
$x ' 
 $ x 
=
∫ cos 
  cos 
  dx
L &%
 L  'x 
 L 
%
*
%
&ih 2$ %
&ih 2$ %
$x  $x 
=
∫ cos 
∫ udu = 0
 sin 
 dx =
2
2
L &%
L &%
 L   L 
Notes 3
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Where are we?
(and where are we going?)
• We are in the middle of Chapter 5, learning the
concept of the operator for a quantum observable.
• We will review a few of the most frequently used
operators and some obvious ramifications of this
approach.
• By the end of this class, we will formally introduce the
Schrodinger equation and begin to address some
well-known solutions.
– stationary states and the time-independent Schrodinger
equation.
– the time dependent SE.
Notes 3
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A more general momentum operator derivation
•
following Morrison prob 5.1.
A reasonable starting point,
p =m
d x
dt
We will also use Schrodinger's equation
ih ' 2 ! * V *
'! ih ' 2 ! V
'! *
=
+ ! and
=&
& !
2
2
ih
ih
't 2m 'x
't
2m 'x
So:
   '! * 
 
' % *

*  '! 
p = m  ∫ ! x!dx  = m  ∫  
 dx 
 x! + ! x 
't  &%

 't   
   't 
2
   ih ' 2 ! * V * 
V  
*  ih ' !
& !  x! + ! x 
+ !  dx 
= m ∫   &
2
2
ih
ih   
   2m 'x

 2m 'x
Notes 3

ih  * ' 2 ! ' 2 ! *
x! dx
= ∫! x 2 &
2
2 
'x
'x
 F2005
Quantum
Physics
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A more general momentum operator derivation

ih  * ' 2 ! ' 2 ! *
p = ∫! x 2 &
x! dx
2
2 
'x
'x

  * '!  
ih  '  * '! '! *
*
x! + ! !  & 2  !
= ∫ ! x
&
 dx
2  'x 
'x
'x
'x  
 
*
%
%

ih  * '! '!
*
* '!
x! + ! !  & i h ∫ !
dx
= ! x
&
2
'x
'x
'x
&%
 &%
%
'!
p = ∫ ! ( &i h
)dx = ∫ ! * pˆ !dx
'x
&%
&%
'!
ˆp ( &ih
'x
%
Notes 3
*
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Momentum Operator
Whenever a wavefunction !(x,t) is used
to describe a state of a system, it is correct to
represent the momentum variable by means
of the differential operator:
'
pˆ ) &ih
'x
Notes 3
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The Operator Postulate
• In quantum mechanics, every observable is
represented by an operator that is used to
obtain physical information about the
observable from the state functions.
Notes 3
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Operator mathematics
• You already know about operators, you’ve used them
in Math, Theo Phys, Optics…
They are essentially a set of instructions on what to
do to a function (take the derivative, multiply it by…)
• Some rules:
– Operators must act on a function
– Operators act on every function to their right unless their
action is constrained by brackets
– The product of operators implies successive operation
(starting with the rightmost).
– The sequence of operations does not commute.
– addition or subtration of operators means to distribute the
operations
Notes 3
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How general operators differ from the specialized ones
you learned in elementary school Non-commuting operators
'
Let's look at operators Qˆ1 and Qˆ 2 . Take Qˆ1 = pˆ = &ih
and Qˆ 2 = xˆ = x.
'x
Is Qˆ Qˆ f ( x) = Qˆ Qˆ f ( x) ?
1
2
2
1
'
'f 

Qˆ1Qˆ 2 f ( x) = &ih [ xf ] = &ih  f + x 
'x
'x 

'f
Qˆ 2Qˆ1 f ( x) = &ihx
'x
A measure of whether operators commute is the difference due to order:
(Qˆ Qˆ & Qˆ Qˆ ) f = ( xpˆˆ & pxˆ ˆ ) f = ihf
2
1
1
2
Exercise 5 – Check this expression explicitly for f=e2x.
Notes 3
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Exercise 5 - solution
'
and Qˆ 2 = xˆ = x and f ( x) = e2 x .
Qˆ1 = pˆ = &ih
'x
Is Qˆ Qˆ f ( x) = Qˆ Qˆ f ( x) ?
1
2
2
1
'
Qˆ1Qˆ 2 f ( x) = &ih  xe 2 x  = &ih e 2 x + 2 xe 2 x 
'x
'
Qˆ 2Qˆ1 f ( x) = &ihx e 2 x  = &ih 2 xe2 x
'x
A measure of whether operators commute is the difference due to order:
(Qˆ Qˆ & Qˆ Qˆ ) e
2
1
Notes 3
1
2
2x
ˆˆ & px
ˆ ˆ ) e 2 x = &ih 2 xe 2 x & &ih e2 x + 2 xe2 x  = ihe2 x
= ( xp
Quantum Physics F2005
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Some other operators
To find the expectation value of a general observable: Q.
First construct an operator to extract Q, Qˆ .
(For wavefunctions, this can usually be accomplished
by rewriting the physical quantity in terms of x and p.)
Then compute the expectation integral as before
Q = ∫ ! *Qˆ ( x)!dx
Notes 3
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Some other operators
2
p
Classical kinetic energy T =
2m
(When an operator is squared, that means
to perform the operation twice)
1 2
1
1 
' 
' 
ˆ
ˆ
ˆ
ˆ
T* =
p* =
pp* =
 &ih  &ih 
'x 
'x 
2m
2m
2m 
&h2 ' 2
=
*
2
2m 'x
Position: xˆ = x
Constant: Cˆ = C
Notes 3
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Some other operators
Potential energy, V : Vˆ
Usually, the potential energy, V , is a function of x and t.
so we just contruct it from V(x,t).
Example- a spring: V = kx 2 / 2 ⇒ Vˆ = kxˆ 2 / 2
If it has time dependence we need to include that too.
The total energy operator in x: Hˆ = Tˆ + Vˆ
is called the Hamiltonian.
'
ˆ
The energy operator in terms of t : E = ih
't
Notes 3
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The Schrodinger Equation
• The Schrodinger equation is a relationship
between the space and time representations
of the energy operators.
• This equation allows you to calculate the
time-development of the state functions.
Hˆ * = Eˆ*
2
2
h '
'*
&
* + V ( x, t )* = ih
2
't
2m 'x
Notes 3
Quantum Physics F2005
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