Momentum
Operators and Observables
Recall the time-independent Schrodinger equation:
# ! 2 "2
&
% ! 2m "x 2 + U(x) ( ) (x) = E) (x)
$
'
E is the total energy of the particle:
p2
E = KE + U =
+U
2m
Comparing the two equations we see that:
! 2 "2
p2
"
!
= KE =
# p = !i!
2
2m "x
2m
"x
Momentum is described by a derivative operation in the
position representation.
Momentum
Operators and Observables
Every observable quantity is described by an operator which can
operate on wave functions.
For position wave functions ψ(x),
Position : x (operation : multiply by x)
Momentum : p = !i! " "x (operation :partial derivative)
For momentum wave functions ζ(p),
Position : i! ! !p
Momentum : p
Observables that are functions of x and p are described by operators that
are corresponding functions of the x and p operators
Momentum
Observables and corresponding operators (position wave functions)
Momentum
Expectation values
"
f (x) =
#
!"
"
f (x)P(x)dx = # f (x) $ (x,t) dx
2
!"
What about the mean value of p ? Recall that
p = mv = m
d x
dx
! p =m
dt
dt
Using
"
x =
#
"
!"
xP(x)dx = # x $ (x,t) dx
2
!"
And taking the time derivative, after some manipulation using
the Schrodinger equation we get,
$
$
! "!
p = %!*
dx = % ! * p! dx
i "x
#$
#$
Momentum
Expectation values
$
$
* ! "!
p = %!
dx = % ! * p! dx
i "x
#$
#$
In general the average value of any observable O is
#
O ==
*
!
$ Ô! dx
"#
Where Ô is the associated operator.
Momentum Distribution
Any function can be written as a sum of cosine and sine waves:
Periodic function: Fourier Series:
Spatial function at a fixed time:
f (x) =
"
#ae
n= !"
ikn x
n
"
=
# a (cos(k x) + isin(k x))
n
n= !"
L
n
L
n
1
1
! ikn x
an =
f
(x)e
dx
=
f (x) cos(kn x) ! isin(kn x) dx
"
"
2L ! L
2L ! L
(
)
Temporal function at a fixed position:
V (t) =
#
$ae
n= "#
1
an =
2T
T
# V (t)e
!T
ik! n t
n
! i" n t
=
#
$ a (cos(! t) + isin(! t))
n= "#
1
dt =
2T
n
n
n
T
# V (t) (cos(" t) ! isin(" t)) dt
n
!T
n
Momentum Distribution
Non-periodic function: Fourier Integral
Spatial function at a fixed time:
f (x) =
a(k) =
#
1
$ a(k)e
2!
dk =
"#
#
1
2!
ikx
$
f (x)e" ikx dx =
"#
1
2!
1
2!
#
$ a(k) (cos(kx) + isin(kx)) dk
"#
#
$
"#
(
)
f (x) cos(kx) " isin(kx) dx
Temporal function at a fixed position:
V (t) =
g(! ) =
$
1
2!
%
g(" )ei" t d" =
#$
1
2"
$
# i! t
V
(t)e
dt =
%
#$
1
2!
1
2"
$
% g(" ) (cos(" t) + isin(" t)) d"
#$
$
% V (t) (cos(! t) # isin(! t)) dt
#$
Momentum Distribution
Examples:
See http://www.falstad.com/fourier/ for a demonstration of Fourier
series
Momentum Distribution
Fourier decomposition of some functions:
f(x)
a(k)
A cos ( k1 x ) + A cos ( k2 x )
f(x)
a(k)
Momentum Distribution
Fourier decomposition of some functions:
f(x)
a(k)
# !(x ! x )2 &
o (
f x = C exp %
%$ 2" 2 ('
x
# !(k ! k )2 &
o (
a k = N exp %
%$ 2" 2 ('
k
()
()
Gaussian
Gaussian
x
k
Recall the reciprocity relations:
!x!k " 1, !#!t " 1
For Gaussian functions, the uncertainties are minimized
!x!k =
1
1
, !"!t =
2
2
Heisenberg Uncertainty Principle
Minimum uncertainty:
!x!k =
1
1
, !"!t =
2
2
p = !k, E = !!
!p!x "
•
•
!
2
!E!t "
!
2
Our knowledge of conjugate quantities is inherently uncertain.
No matter how good our measuring instruments are we cannot
simultaneously know x and p or E and t with complete precision.
Gaussian
For Gaussian functions:
!
!p!x =
2
x
!
!E!t =
2
Gaussian
k
Momentum Distribution
Momentum space wave functions
Since p=hk/2π, the functions a(k) can be easily written as functions ζ(p).
Given a wave function in coordinate space representing the position of a
particle, the corresponding momentum wave function is the Fourier
integral of the the position wave function:
! ( p,t) =
1
e
%
( 2" ! )
1/2
#i
( xp )
!
$ (x,t)dx
If the position wave function is normalized, the momentum wave
function is also normalized to unity:
" ! ( p,t)
2
dp = 1
Momentum Distribution
Momentum space wave functions
We can interpret the square of the quantity
P( p,t) = ! ( p,t) dp
2
of the normalized wave function as the probability at time t of the
momentum of a particle being in the volume dp around p
Hence the normalization requirement is a statement that the
probability of finding the particle with some value of momentum is 1.
" ! ( p,t)
2
dx = " P( p,t)dp =1
Momentum Distribution
Momentum space wave functions
• The probability of finding a particle with momentum a < p < b is
then
b
" ! ( p,t) dp
2
a
This is the area under the momentum probability density curve
between p=a and p=b:
! ( p,t)
2
p
Momentum Distribution
Given a wave function representing the momentum of a particle, the
corresponding position space wave function is the inverse Fourier
integral of the momentum wave function:
! (x,t) =
1
e
#
( 2" ! )
1/2
i
( px )
!
$ ( p,t)dp
If the momentum wave function is normalized, the position wave
function is also normalized to unity.
Note that the position and momentum wave functions represent the
same state of the system. All information about the state can be
obtained from either wave function.