Download Quantum Physics 2005 Notes-2 The State Function and its Interpretation

Quantum Physics 2005
Notes-2
The State Function
and its Interpretation
Notes 2
Quantum Physics F2005
1
In a particle-detection scheme, the wave-like
interference pattern builds up particle-by-particle
from Morrison
Notes 2
Quantum Physics F2005
• Note that the
arrival of each
particle appears
to be random,
until enough
particles have
arrived for a
statistically useful
sample.
• This suggests
that we should
use a probability
distribution
approach for
quantum physics
calculations
2
Discussion of probability
• The word event refers to obtaining a making
a measurement of an observable.
• To define the probability of an event result,
we consider the fraction of times that result
will occur for measurements on a very large
number of identical systems. Such a
collection of systems is called an ensemble.
• To perform an ensemble measurement of
an observable, we perform precisely the
same measurement on the members of an
ensemble of identical systems.
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Normalized Probability
The probability of a specific outcome is the number of times
that it occurs, divided by the total number of experiments:
n
P(a) = a
N
The probability of a set of outcomes is the sum of the
probabilities of each outcome:
c
1 c
P (a or b or ...) = ∑ P ( j ) = ∑ n j
j =a
N j =a
By this definition,
all possible outcomes
∑
P( j ) = 1
j =a
Notes 2
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Ensemble Averages
• Given the probabilities for a set of outcomes, we can
find useful quantities, like the average or expectation
value of a result q, through a simple sum:
M
∑ q j P(q j )
"mean of q"= q = q =
j =1
M
∑ P(q j )
j =1
• Another statistic that is useful in describing
distributions is the dispersion or variance:
variance=dispersion= ( q - q
Notes 2
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2
5
Continuous distributions
• In many physical cases, the result of a measurement
can be any value within a physical range, for
example, the position of a particle.
• We can define the probability of finding the particle
within a certain infinitesimal range as P(x)dx and we
call P(x) the probability density.
• Given a probability density, we can compute the
expectation value of any parameter q(x).
!
x = ∫ xP( x)dx
"!
!
q( x) = ∫ q ( x) P( x)dx
"!
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Matter waves
• Just adding probability distributions does not
give interference.
• In order to get an interference pattern, we
must be able to superpose two waves.
• A working hypothesis:
– Observations occur in proportion to the probability
that a particle will be in a state (r(t), p(t), E(t))
when it is measured.
– The probability is proportional to the square of a
wave amplitude.
Notes 2
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The State Function
• Every physically realizable state of a system
is described in quantum mechanics by a
state function # that contains all accessible
information about the system in that state.
– The phrase “wave function” is usually used to
mean a state function that is specified as a
function of position and time #(x,t).
Notes 2
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Born’s Postulate
If, at time t, a measurement is made to locate the particle
r
associated with the wave function # (r , t ), then the probability
r
P(r , t )dv that the particle will be found in the volume
r
dv around position r is equal to:
r
r
r
P (r , t ) dv = # *(r , t ) # (r , t ).
r
r
Where # *( r , t ) denotes the complex conjugate of # (r , t ).
Notes 2
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Principle of Superposition
If #1 and # 2 represent two physically realizable states of
a system, then the linear combination
# = c1#1 + c2 # 2
is also a physically realizable state of the system.
Notes 2
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Normalization
• Sometimes you will be given wave functions that lead
to an integrated probability that is not already 1. (In
fact, lazy professors do this a lot.)
• Before you try to do any calculation with a
wavefunction, you should
make sure it is properly
!
r
r
normalized, that is: ∫ # *( r , t )# (r , t )dv = 1
"!
!
r
r
Given a un-normalized wavefunction #' such that ∫ # '*(r , t )# '(r , t )dv = M
"!
just multiply #' by the constant
#(x,t)=
1
to create a new function:
M
#'(x,t)
that will be normalized.
M
Notes 2
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Physical constraints on wavefunctions
• A physical wavefunction must be
normalizable. (The particle must be
somewhere.)
• A physical wavefunction must be single
valued. (It can’t have two probabilities of
being at the same point.)
• A physical wavefunction and its spatial
derivatives must be continuous.
Notes 2
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Examples of allowed and disallowed
wavefunctions
(from Morrison)
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Expectation value
• In quantum mechanics, the ensemble average of an
observable for a particular state of the system is
called the expectation value of that observable.
!
*
#
∫ Q#dv
Q =
"!
!
*
#
#dv
∫
"!
• The uncertainty in the observable is the square root
of the dispersion.
$Q =  ( Q " Q

Notes 2
)
2
1/ 2


= Q

Quantum Physics F2005
2
2 1/ 2
" Q 

14
Einstein-de Broglie relations
As we continue to analyze behavior of wavefunctions
it is useful for us to keep in mind two basic relations for
matter waves, relating particle properites to wave properties:
E = hf = h%
and
h
p = = hk
&
Notes 2
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A quick peek at the Schrodinger Equation
Later in this course we will deduce and solve the wave
equation for matter waves (The Schrodinger Equation).
Here I will show it to you, make some simplifying
assumptions, and discuss the nature of the solutions.
Here it is, in one dimension,
 h2 ' 2

'# ( x, t )
 " 2m 'x 2 + V ( x)  # ( x, t ) = ih 't


Notes 2
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Free space solution for #
•
For V(x)=0, we can illustrate some physical points
by working through the wave function logic of this
physical situation without fully solving the differential
equation.
A. If # is to be normalizable, it must go to zero as
r(infinity.
B. # must be complex, because the right side of the
SE has an i in it.
C. # must not violate the homogeneity of free space.
(The probability must be the same after we translate
# by an arbitrary distance.)
Notes 2
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Three trial functions
1.) #( x, t ) = A cos(kx " %t + ) ) :a real harmonic wave
2.) #( x, t ) = Aei (kx"%t ) :a complex harmonic wave
3.) #( x, t ) = a wavepacket made up of complex harmonic waves
A. # normalizable?
B. # complex?
C. # must not violate the homogeneity of free space.
Notes 2
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Test the harmonic function
#( x, t ) = A cos(kx " %t + ) ) :a real harmonic wave
A. Normalizable? No, unless A goes to zero.
B.) Complex? No. Unlikely to satisfy the wave equation.
C.) Homogeneous P? No.
P( x, t ) = #( x, t )* #( x, t ) = A2 cos2 (kx " %t + ) )
P( x + a, t ) = #( x + a, t )* #( x + a, t )
#( x + a, t ) = A[ cos(kx " %t )cos ka + sin(kx " %t )sin ka]
P( x + a, t ) = P( x, t )cos2 ka +
+ A2 sin2 (kx " %t )sin2 ka + 2cos ka sin ka cos(kx " %t )sin(kx " %t )
Notes 2
Quantum Physics F2005
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Test the complex harmonic function
#( x, t ) = Aei ( kx"%t )
A. Normalizable? Only if A goes to zero.
"i ( kx "%t )
∫ P(x,t)dx = ∫ Ae
i ( kx"%t )
Ae
!
dx = ∫ Adx = A[ 2!]
"!
B.) Complex? Yes. It is indeed a solution to the SE.
C.) Homogeneous P? Yes.
P( x, t ) = #( x, t )* #( x, t ) = A2 (independent of position)
We seem to be on the right track.
Notes 2
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Notes on the complex harmonic function
# ( x, t ) = Aei ( kx "%t ) = A [ cos(kx " %t ) + i sin(kx " %t ) ]
= A cos ( ( px " Et ) / h ) + i sin ( ( px " Et ) / h ) 
Represents a state with a single value of momentum
and a single value of energy, therefore it is called a
pure momentum state.
•It extends over all space.
•The phase fronts move in the +x direction
%
E
at a speed v p of
(or ).
p
k
E p 2 / 2m
p
.)
=
(For a classical particle, v = =
2m
p
p
Notes 2
Quantum Physics F2005
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The complex wave packet
We can construct a localized/normalizable wavepacket out of complex harmonic waves
using the Fourier transform:
1 !
i ( kx "%t )
# ( x, t ) =
dk
∫ A(k )e
2* "!
⇒ Because this wavefunction can be localized, it is likely to be normalizable.
(We will need to test specific cases.)
⇒ Because each term in it is a solution to the wave equation,
the wave function must also be a solution.
⇒ Is the shape of the probability of the sum of complex harmonics translation invariant?
(
% ik2 x
% ik1 x + be
P( x, t ) = ae
*
) ( ae%
ik1 x
) (
% ik2 x = a% *e " ik1x + b%*e " ik2 x
+ be
)( ae%
ik1 x
% ik2 x
+ be
)
% " i ( k1 " k2 ) x + b%*ae
% " ik ' x + b%*ae
% i ( k1 " k2 ) x = a 2 + b 2 + a% *be
% ik ' x
= a 2 + b 2 + a% *be
% ik2 x +ik2+ + b%*e " ik2 x "ik2+ ae
% ik1x +ik1+
P( x + + , t ) = a 2 + b 2 + a% *e " ik1x "ik1+ be
% " ik '( x ++ ) + b%*ae
% ik '( x ++ )
= a 2 + b 2 + a% *be
This function is just the old one, moved over.
Notes 2
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In-class exercise 1
•Morrison 3.1 p. 89
This is a normalized state function
at t=0.
The magnitude of #(x,0) continues
to decay smoothly to zero as
x(±!.
1.
2.
3.
4.
Is this function physically admissible? Why or why not?
Where is the particle most likely to be found?
Estimate the expectation value of the position. Is it the same as for
part 2?
Is the uncertainty in momentum of this state equal to zero? Justify.
Notes 2
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In-class exercise 2
• Morrison 3.3 p. 90
Notes 2
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(In which I try to clarify what we are doing)
•
From evidence of interference in “particle” experiments, we introduced the idea
that matter must be described by some sort of wave.
–
•
•
The wave is not directly observable, but the resulting probability of various
outcomes is, hence we introduced a probability interpretation of quantum
physics experiments.
We then assumed that there existed a wave that could yield the correct
observables, with classical relations between time, energy, momentum, and
position. The ramifications were then explored.
–
–
–
–
•
De Broglie-Einstein relations: p=h/ &; E=hf
The probability of an outcome is proportional to the square of the wavefunction for that
outcome.
The probability interpretation leads to the ability to compute the expectation (average)
value for appropriate observables once the wave function is known.
The superposition of waves to form a packet leads to a mathematical statement of the
Heisenberg Uncertainty Principle.
We could deduce the form of a useful wave equation for matter waves.
We will now explore some examples of wave functions and their observable
ramifications.
–
–
Notes 2
Wavepackets, momentum-position transforms
More general state functions – the momentum state
Quantum Physics F2005
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Important results so far
Einstein- deBroglie relations: p = hk =
h
&
;
T = h% = hf
The probability interpretation: q = ∫ q ( x) P( x)dx;
P( x)dx = # *#dx
The general form for a wave packet using waves
1 !
i ( kx "%t )
with well-defined momentum: # ( x, t ) =
dk
∫ A( k )e
2* "!
Notes 2
Quantum Physics F2005
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In class exercise 3 – Developing the
matter-wave equation
(from Morrison Problem 2.7)
• Write down an expression in exponential form for a
harmonic traveling wave (because our matter wavepacket is composed of simple traveling waves.)
• Rewrite your wave function in terms of p(k) and E(%).
• Deduce the simplest differential equation that might
give rise to such a traveling wave, consistent with
E=p2/2m,.
',
' 2,
',
' 2%
+ a2 2 + ... = b1
+ b2 2 + ...
a0 + a1
'x
'x
't
't
Notes 2
Quantum Physics F2005
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In-class exercise notes
You should now have something like:
i
1 2
"i 2
p4
a0 + a1 p, " a2 2 p , = b1
p , + b2
h
h
2mh
4m 2
which can only be true if,
a
"i 2 m
a0 = 0; a1 = 0; b2 = 0; and 2 =
b1
h
so a plausible differential equation for the motion
of a free particle is:
h 2 ' 2,
',
"
= ih
2
2m 'x
't
Notes 2
Quantum Physics F2005
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A solution to the matter wave equation – the
infinite square well
h2 ' 2,
',
=
h
i
Starting with "
and specifying that the wavefunction must
2
't
2m 'x
go to zero at x= ± L/2 leads to some interesting results.
Only certain quantized values of wavelength are allowed.
 * x  " i%t
is one solution.
e
 L 
For further work, we want this function to be normalized:
, ( x, t ) = N cos 
*x
2 L
dx
N
1 = ∫ , *, dx = N ∫ cos 2 
=

L
2
-L/2
-L/2


L/2
, ( x, t ) =
2
L/2
⇒ N=
2
L
2
 * x  " i%t
cos 
e
L
 L 
This function looks and behaves like this
Notes 2
Quantum Physics F2005
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The square well continued
Now, find the expectation value of position.
2 L/2
2 *x 
x = ∫ xP( x, t )dx =
∫ x cos 
 dx = 0
L "L/ 2
"L/ 2
 L 
You can do this integral directly,
or argue its value based on symmetry.
L/2
Notes 2
Quantum Physics F2005
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The square well continued
Now, find the variance of position so we
can assess the range of positions over which
we might expect to find the particle.
x
x
2
2
2 L/2 2
2 *x 
x
cos
= ∫ x P ( x, t )dx =
∫

 dx
L "L/ 2
"L/2
 L 
L/2
2
2L
=  
L* 
3
* /2
2
2
∫ u cos ( u ) du
"* / 2
* /2
2
u cos 2u 
2L  u3  u 2 1 
= 3  +  "  sin 2u +

*  6  4 8
4  "* / 2
x
Notes 2
2
* 21
2 L2  * 3
1 
= 3  +0"  = L  " 2 
*  24
4
12 2* 
Quantum Physics F2005
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Particles and waves:
the Gaussian wavepacket
We will simplify calculations by assuming that the spatial
part of the wavefunction can be described at t = 0 by a
Normal Gaussian function centered at x0 and moving with
average wavenumber k0 ;
1/ 2
 1  ik0 x " ( x " x0 )2 / 4- x2
.
# ( x, 0) = 
 e e
 - x 2* 
(We have already solved a similar problem in Notes 1. )
1
So A(k ) =
2*
Notes 2
1/ 2
 1 


 - x 2* 
!
∫e
i ( k0 " k ) x " ( x " x0 )2 / 4- x2
e
dx
"!
Quantum Physics F2005
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Gaussian wavepacket continued
to simplify further let's take x0 = 0,
1/ 2
1  1  ! i ( k0 " k ) x " x2 / 2- x2
so A(k ) =
e
dx

 ∫e
2*  - x *  "!
Then letting k ' = k " k0 , we solve by completing the square:
2
1
 1

"ik ' x " 2 x 2 + - x2 k '2 " - x2 k '2 = 
x " i- x k '  + - x2 k '2
2- x
 2- x

A(k ) =
1
2*
1/ 2
 1 


*
 x

2
2
!
e "- x k ' ∫ e
"
2
1
2- x
x "i- x k '
dx
"!
1/ 2
1  1 
x
let x ' =
-i- x k ', then A(k ) =


2- x
2*  - x * 
1/ 2
1  1 
A(k ) =


2*  - x * 
Notes 2
2- x e
"- x2 k '2
1

2- x e
"- x2 k '2
!
2
" x'
∫ e dx '
"!
1
4
* =  - x2  e
*

Quantum Physics F2005
"
- x2 ( k " k0 )2
2
33
Gaussian wavepacket continued
1
4
- x2 ( k " k0 )2
1

2
A(k ) =  - x2  e
*

This is a Normal Gaussian k " distribution,
"
centered at k0 with width - k =
1
-x
.
• Although we specified a wave momentum k0 we find that the wavefunction
actually contains a spread of wave momenta,
1
-x
, which increases as the
$p
.
h
• Waves of differing k move at different speeds, so the original wavefunction
spatial Gaussian narrows. Remember that $k =
will spread with time.
Notes 2
Quantum Physics F2005
34
Inversely correlated widths
Note that if # is properly
normalized ( ∫ # * #dx = 1)
then the wavevector
amplitude function is
also normalized to 1:
2
∫ A(k ) dk = 1
Notes 2
Quantum Physics F2005
35
An aside on the definition of the
width of a Gaussian
The width of a Gaussian can
be defined in various ways.
Two conventional ways are
by the Full Width at Half
Maximum (FWHM) and twice
the standard deviation.
Morrison uses $L=standard
deviation of the position.
Notes 2
Quantum Physics F2005
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Time evolution of the wavefunction
The general form of the wavefunction is:
1 !
" i ( kx "% t )
# ( x, t ) =
A
(
k
)
e
dk .
∫
2* "!
This means that if you are given # ( x, 0) you can find # ( x, t ).
1 !
" i ( kx )
A(k ) =
#
(
x
,
0)
e
dx.
∫
2* "!
Note that to do the top integral, you must know % ( k ).
Time evolution of the wavefunction
Notes 2
Quantum Physics F2005
37
In class exercise 4 – Constructing a wave
packet from momentum amplitudes
Consider the amplitude function:

 0

 1
A(k ) = 
 $k

 0

1
k < k0 " $k
2
1
1
k0 " $k < k < k0 + $k
2
2
1
k > k0 + $k
2
a ) Find # ( x, 0).
b) Find P( x, 0).
c) Sketch P( x, 0) and find the positions of the first zeros.
d) Estimate $x$k .
Notes 2
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Formal derivation of group velocity
for a wave-packet
, ( x, t ) = ∫ A(k )e " i ( kx "% ( k ) t ) dk
Assuming that A(k ) has a fairly narrow range of important k 's.
we will allow ourselves to expand % (k ) :
 d% 
% ( k ) . % ( k0 ) + ( k " k 0 ) 
 + ...
 dk  k0
Letting k ' = k - k0
, ( x, t ) . ei ( k x "% ( k ) t ∫ dk 'e
0
Notes 2
0
  d%  
" ik '  x " 
t 
dk

 0

Quantum Physics F2005
39
Momentum Probability Amplitude
Note that A(k ) clearly contains information about momentum
because k = p / h.
It is useful (later) to be able to describe the state function
directly in terms of momentum states.
1  p
A   (the momentum probability amplitude)
h h
Then we can rewrite the wavefunction as;
1 !
ipx /
# ( x, 0) =
∫ / ( p )e dp and
2* h "!
1 !
" ipx /
/( p) =
dx.
∫ # ( x, 0)e
2* h "!
/( p) 0
Note that the factor of 1/ h is required for proper normailzation of # and /.
Notes 2
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Interpretation of the space and
momentum state functions
# ( x, t ) = the wavefunction and directly gives information about the probability
of finding a particle at a specific position, but it is difficult to directly
calculate properties like the expected momentum because momentum is not
written as a function of position.
/ ( p ) = the momentum probability amplitude and directly gives information
about the probability of finding a particle with a specific momentum.
They both describe the same quantum state.
If the members of an ensemble are in the quantum state #(x,t)
with Fourier transform / ( p) = F [ # ( x, 0)] then
P ( p) dp = / *( p )/ ( p )dp
is the probability that in a momentum measurement at time t a
particle's momentum will be found to be between p and p + dp.
Notes 2
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41
Expectation values
Given the momentum amplitude function and the associated probability
distribution, we can compute expectation values:
!
p = ∫ / ( p ) * p/ ( p )dp = expectation value (average) of p.
"!
$p = momentum uncertainty= ( p " p

p
2
)
2 1/ 2


= p

2
2 1/ 2
" p 

!
= ∫ / ( p ) * p 2 / ( p ) dp
"!
Notes 2
Quantum Physics F2005
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