Download Group and phase velocity

Schrödinger
We already know how to find the momentum eigenvalues of a system.
How about the energy and the evolution of a system?
Schrödinger Representation:
  qi , t  ˆ
i
 H   qi , t 
t
The evolution of a state of a system is given by the application of the Hamiltonian
operator to the state wavefunction.
In the Schrodinger representation, the evoultion is given by the
wavefunction   r , t  , and not by the Hˆ
the Hˆ is the operator corresponding to the Energy of a system...
if the question is What is the En erg y of a stat e? the operator re presenting
the obs ervable i s Hˆ
For a time-independent Hˆ , although   qi , t  evolves in time,
the Energy remains a co nsant
What is the Hamiltonian?
Hˆ 
Tˆ
Kinetic
operator
1 2 pˆ 2
T  mv =
2
2m
Vˆ

potential

and considering p  pˆ  i
 qi
2

 Tˆ  
2m  qi2
2
2
2
2
2
2






2
for a 3D system, Tˆ 







2m   x 2  y 2  z 2 
2m
V  Vˆ  qi  or Vˆ  x, y , z  this is the charactertistic potential defining
the boundary conditions in each system,
examples: V  qi   0  free particle
V  qi )  
1
2
m  qi  qo   Harmonic Oscillator
2
Separation of variables
i
  qi , t  ˆ
 H   qi , t 
t
If V(qi) is t-independent, then the Hamiltonian is also t-independent, and we can
proposed a separation of variables



 separation of variables:

f  x, t 

g x 
x
t
 f  x, t   x  t 
 f  x, t  =g  x   h  t 

f  x, t 
h
t

t



x

It is not possible to separate

 instead, for f  x, t   x  t  x  t  f  x , t  into 2 functions of  f  x, t   g  x   h  t 


independent variables
















Time and coordinate separation
Starting with i
  qi , t  ˆ
 H   qi , t  , and proposing   qi , t     qi     t 
t
i
   qi     t 
t
 Hˆ   qi     t 

   qi 
  t  
ˆ ˆ


i  t
   qi 
  T  V  qi    qi     t 
t
t 

2
 2  qi     t  ˆ
  t 
i   qi 

 V  qi   qi     t 
2
t
2m
 qi
  2   2  qi 
  t 
ˆ  q   q 
i   qi 
  t 


t
V


i
i
t
2m
 qi2
  2   2  qi 
 t 
i   qi    t 
 t 


Vˆ  qi   qi 
2
  qi     t   t
  qi     t  2m
  qi     t 
 qi
Cont.
2
i   t 
1   2     qi 
1 ˆ


V  qi   qi   constan t  W
2
  t  t
  qi  2m
  qi 
 qi
depends only
on  t 
depend s only on
  qi 
2
1   2     qi 
1 ˆ

V  qi   qi   W
2
  qi  2m
  qi 
 qi
i   t 
W
 t  t
  t 
iW

 t 
t
 t   e

iW
  2   2  qi 
2m
 qi2
 Vˆ  qi   qi   W  qi 
   2  2

ˆ
 V  qi    qi   W  qi 

2
 2m  qi

t
Hˆ   qi   W  qi 
  qi , t     qi   e

iW
t
where W is the t -independent en energy of the system!
The phase function
  qi , t     qi   e

iW
t
When Hˆ does not depend on time V  V  t   , then the system is in a
stationary state of energy W. Its time dependence is simply a phase
  iW t
W
W 
W
function  e
 cos t  i sin t  , with an oscillatory frequency


Probability density:
  qi , t 
2
   qi     t     qi   e
2

*
 qi   e

iW
t
  qi   e


iW
iW
t
2
t

*
 qi    qi     qi 
2
Even if the wavefunction evolves in time, the probability density remains constant!
Expectation value
Consider a state which is a superposition of eigenstates of the t  indep. Hˆ
 Hˆ  k  qi   Wk k  qi  and
  qi , t    ck  e

iWk t
 k  qi 
k

  qi , t  Hˆ   qi , t 
H 

 cm  e

iWm t
 m  qi  Hˆ
m
 ck  e

iWk t
k
  cm*  e

iWm t
m
 ck  e

iWk t
 m  qi  Hˆ  k  qi 
k
  cm* ck e

i Wm Wk t
 m  qi  Hˆ  k  qi 
m,k
0 for k  m
1 for k  m
  cm* ck e

m,k
i Wm Wk t
  ck Wk
The expectation value of the
Hamiltonian does not depend
on time. A superposition of
eigenstsates has constant E (E≠t).
Wk  m  qi   k  qi 
0 for k  m
1 fo r k  m
2
k
 k  qi 
This is TRUE for any observable whose
[A,H]=0
Heisenberg Representation
In the Heisenberg (interaction) representation, the evolution of the system is given
by the evolution of an operator
d A
dt
A
  H , A 
t
i
But…
how do we get to this equation?
Consider the time derivative of an expectation value:
d A
dt

  A
t
We can evaluate the partial derivatives for each
state  and for the operator A
 

A
A  
   A
t
t
t
We recognize that 2 of these terms correspond to the evolution of states


i
H 
t
and
 
i
  H
t
cont
d A
dt
A
i
   A H 
t

i
 HA   

i
 HA    AH   

i
 HA  AH   

i
  H , A   

i
 H , A

A
t
A

t
A

t
A

t
Schrödinger equation of a free particle
Consider again a FREE PARTICLE. We know how to evaluate the

momentum, by solving the eigenvalue-eigenvector eq.
i
P p P
x
…what about the Energy of a free particle? We need to construct the Hamiltonian
operator to learn about the energy of the system
 Ĥ   E 
Hˆ  Tˆ  Vˆ
and for a free particle Vˆ  0
ˆ 2  2 2
P
 Hˆ  Tˆ 

2m 2m x 2
There are 2 ways to solve this problem
solve the 2nd order differential equation and find 
 2 2

 E
2
2m x
HARD WAY!!!!
How useful are []?
ˆ 2 Pˆ 2
ˆ 3 Pˆ 3
P
P
ˆ
ˆ
consider the  P, H   Pˆ 

 Pˆ 

0
2m 2m
2m 2m
 Pˆ , Hˆ   0,


 there is a complete set   which are eigenfunctions of both operators!
Since we already know the eigenfunction of P, we can try to see if it is also an
eigenfunction of H
1 ikx
for P 
e
 what about Hˆ P  ?
2π
2
2


Hˆ P 
2m x 2
1
2π
eikx

2 2
 2 eikx
 2
k
2 ikx


ik
e



2
2m
2m 2p x
2m 2p
2
1
2π
eikx
p2
EASY WAY!!

P
2m
The momentum eigenfunctions are also eigenfunctions of the Hamiltonian, with
eigenvalues = classical kinetic energy
Even though the quantum and classical energies coincide, the particle behavior is very
different, i.e. the Quantum particle can never be at rest, because DpDx≽h/4p
Good solutions of the Schrödinger eq.
Solving the Schrödinger eq  (most times) solve a differential eq., but not all
mathematical solutions will be good physical solutions for the probability density.
well - behaved functions :
 is finite (must not go to  at any point)
Single valued
continuous
1st derivative, also continuous