Transcript General Structure of Wave Mechanics (Ch. 5) • • • • Sections 5-1 to 5-3 review items covered previously use Hermitian operators to represent observables (H,p,x) eigenvalues.

General Structure of Wave Mechanics (Ch. 5)
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•
•
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Sections 5-1 to 5-3 review items covered previously
use Hermitian operators to represent observables (H,p,x)
eigenvalues of Hermitian operators are real and give the expectation values
eigenvectors for different eigenvalues are orthogonal and form a complete set of
states
• any function in the space can be formed from a linear series of the eigenfunctions
• some variables are conjugate (position, momentum) and one can transform from one
to the other and solve the problem in either’s “space”
eigenfunctions : ui ( x )
 i | j  ui | u j 
 ( x, t ) 
C
N
*
u
i
 u j dx   ij  dot product
uN ( x )e iEt / 
 p   * pop dx   * ( i
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
) d x
x
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Notation
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•
•
•
there is a very compact format (Dirac notation) that is often used
|i> = |ui> = eigenfunction
<c|f is a dot product between 2 function
|i><j| is an “outer” product (a matrix). For example a rotation between two different
basis
• if an index is repeated there is an implied sum
eigenfunctions : ui ( x )
 i | j  ui | u j 
 ( x) 
C
N
 u u dx  
*
i
uN ( x )  
j
ij
 dot product
 CN n  n n 
n n  projectionoperator
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Degeneracy (Ch. 5-4)
• If two different eigenfunctions have the same eigenvalue they are degenerate
(related to density of states)
• any linear combination will have the same eigenvalue
Ou i  aui
Ou j  au j
 O (ui   u j )  a (ui   u j )
• usually pick two linear combinations which are orthogonal
• can be other operators which have only some specific linear combinations being
eigenfunctions. Choice may depend on this (or on what may break the degeneracy)
• example from V=0
sin kx, coskx, e
ikx
,e
ikx
 2k 2
E 
2m
sin kx, coskx orthogonal:  sin kx coskxdx  0
eikx , e ikx
orthogonal:  ( e ikx )*eikxdx  0
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Degeneracy (Ch. 5-4)
• Parity and momentum operators do not commute
 ( x )
 (  x )
)  i
x
x
 (  x )
pop ( P ( x ))  i
x

[ P, pop ]  Ppop  pop P  2i
P
x
P ( pop ( x ))  P ( i
• and so can’t have the same eigenfunction
• two different choices then depend on whether you want an eigenfunction of Parity or
of momentum
sin kx, coskx
eikx , eikx
Parity
m om entum
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Uncertainty Relations
(Supplement 5-A)
• If two operators do not commute then their uncertainty product is greater then 0
• if they do commute  0
• start from definition of rms and allow shift so the functions have <U>=0
( A) 2  A2    A  2
 A2   * A2 dx
U  A  A  U  0, ( U ) 2  ( A) 2
• define a function with 2 Hermitian operators A and B U and V and l real
f ( x )  (U  ilV ) ( x )
U  A  A 
V  B  B 
• because it is positive definite
I (l )   f *f d x  0
• can calculate I in terms of U and V and [U,V]
I 
 ((U  ilV ) )
*
(U  ilV ) dx
  * (U dag  ilV dag )(U  ilV ) dx
U dag  U T *  U
Herm itian
  * (U  ilV )(U  ilV ) dx
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Uncertainty Relations
• rearrange
I 



*
*
(Supplement 5-A)
(U  ilV )(U  ilV ) dx
(U 2  il (UV  VU )  l2V 2 ) dx
• But just the expectation values
I  U 2  il ( UV    VU )  l2  V 2 
 U 2  il  [U ,V ]  l2  V 2 
• can ask what is the minimum of this quantity
dI
i  [U ,V ] 
 0  lmin  
dl
2 V2 
• use this  “uncertainty” relationship from operators alone
I (lmin )  U 2  ilmin  [U ,V ]   l2min  V 2   0
1
 i[U ,V ]  2
4
1

 i[ A, B ]  2
4
  U 2  V 2  
 ( A) 2 ( B ) 2
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Uncertainty Relations -- Example
• take momentum and position operators
• in position space
xop  x
pop  i

x



px  i
x  i  ix
x
x
x
 px  xp  [ p, x ]  i
xp  ix
• that x and p don’t commute, and the value of the commutator, tells us directly the
uncertainty on their expectation values
( A) 2 ( B ) 2 
 px 
1
 i[ A, B ]  2
4
1
1
 i[ p, x ]  2 
4
2
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 i ( i)  2 

2
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Time Dependence of Operators
• the Hamiltonian tells us how the expectation value for an operator changes with time
 A t 

*
( x, t ) A ( x, t )dx
d
A
 A  t   * ( x , t )
 ( x, t )dx 
dt
t


*
*
[

(
x
,
t
)]
A

(
x
,
t
)
dx


(
x
,
t
)
A
[
 ( x, t )]dx
 t

t
• but know Scrod. Eq.

1
 *
1
i

H 
(
H )*  H * *
t
i
t
i

H *T  H  ( H )* A   *HA
• and the H is Hermitian
• and so can rewrite the expectation value
d
A
 A t 
dt
t
d
A
 A t 
dt
t

i
i
*
*

HA

dx


AHdx





i
H , A

t
t
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Time Dependence of Operators
II
• so in some sense just by looking at the operators (and not necessarily solving S.Eq.)
we can see how the expectation values changes.
d
A
i
H , A
 A t 

t
dt
t

• if A doesn’t depend on t and [H,A]=0  <A> doesn’t change and its observable is a
constant of the motion
• homework has H(t); let’s first look at H without t-dependence
• and look at the t-dependence of the x expectation value
d
i
x 
dt

H , x 
V ( x ), x   0

d
x 
dt
p
2
i


p2
H 
 V ( x)
2m
 p2

 2m  V ( x ), x 



, x  p p , x    p , x  p 
2
p
i
p
m
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Time Dependence of Operators
III
• and look at the t-dependence of the p expectation value
d
i
p 
dt

H , p 
i


 p2

 2m  V ( x ), p 



i
 p ,V ( x ) 

( pV  Vp )  p (V )  V ( p )
 d
 d
 dV

(V )  V
 

i dx
i dx
i dx

• rearrange giving
 p ,V ( x )  
and
 dV
d

i dx
dt
d2
m
dt2
x 
p

dV ( x )
dt
dV ( x )
dt
• like you would see in classical physics
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