Download Quantum Postulates

Overview of QM
Statics
 (r, t )   (r ) (t )
H n (r )  En n (r )
M.O. Calculations, Spectroscopy, and
Q. Stat. Mech.
2

ˆ 
ˆ  Vˆ (r)
Hˆ  K  Vˆ (r) 

2m
Translational Motion
2

ˆ 
ˆ
Hˆ 

2m
Rotational Motion
Dynamics
d
ˆ
H  (r, t )  i
 (r, t )
dt
(r, t )   (0)e  (r)
it
 (r, t   )  Uˆ ( ) (r, t )
Mol. dynamics, Q. Comp., Laser Pulse
Methods,2D NMR, and SS NMR, and
spectroscopy.
2

ˆ L
ˆ
Hˆ 
L
2I
Harmonic Motion
Cartesian
P. in Box
ex) STM, Devices
Spherical Polar
Rigid Rotor
Spin
ex) FTS, NMR
Centre of Mass
2

k
ˆ
ˆ
ˆ
H
   (r  req )2
2
2
Vibrations
ex) IR, Raman
Quantum Mechanics for Many Particles
(r1 , r2 , r3 ,..., rk , t )  (r1 , t )(r2 , t )(r3 , t )...(rk , t )
(ri , t )   (ri )i (t )
H n (r1 , r2 , r3 ,..., rk )  En n (r1 , r2 , r3 ,..., rk )
m1
z1
r1
m2
z2
r2
2

ˆ 
ˆ  Vˆ (r , r )
Hˆ  

i
i
ij i
j
i 2mi
i, j
2
zi z j e
ˆ
Vij (ri , r j ) 
r  rj 
2  i
4 o ri  r j
m3
z3
r3
r4
(0,0,0)
m4
z4
En – Energy Levels n – Wavefuntions
Electronic Structure of Mols.
The Wavefunction
 ( x, t )   ( x) (t )
Single Valued
 (t )   (0)eit
Finite and continuous
t
ro
o
 (0)
 (t ) R
e
Im

( x, t ) dx  
2
 ( x, t ) 
 ( x, t ) 
2
14_01fig_PChem.jpg
The Wavefunction
(r1 , r2 , r3 ,.., rk , t )  (r1 , t )(r2 , t )(r3 , t )...(rk , t )
(ri , t )   (r  xi , yi , zi ) )i (t )

(ri , t ) dvi  
2
3
ri  xi i  yi j  zi k 
3
dvi  dx i dyi dzi
 (ri , t ) 
Spherical Polar
Coordinates
3
 (ri , t ) 
2
3
ri  ri sin i cosi i  ri sin i sin i j  ri cosi k
dvi  ri 2 sin i dri di di
14_01fig_PChem.jpg
Probability Distribution
 (r, t )  * (r, t ) (r, t )
 (r, t ) 
Since
2
z  z* z  z 
P(r, t )  * (r, t )(r, t )
P ( R, t ) 
xj yj zj
rj
xi yi zi
ri
Probability of finding the particle at
exactly r, as a function of time.
   P(r, t ) dv   P(r, t ) dv
  P(r, t ) dv
R
   (r, t ) (r, t ) dv
*
R
Probability of finding the particle
between ri and rj, defining the region
R, as a function of time
Probability Distribution and Time
P( R, t )   * (r, t )(r, t ) dv
R
   (r ) (t )  (r) (t ) dv =   (t )* (r)* (r) (t ) dv
*
R
R


*
  (t )   (r)  (r) dv   (t )   (t )* P( R) (t )
R

*
  (t )* (t ) P( R)   (0)* eit (0)eit P( R)
  (0)* (0) P( R)  ro2 P( R)  P( R)
P( R)   (r)* (r) dv
R
Probability Distribution of Wavefunctions
Pn ( R, t )   Pn (r, t ) dv   *n (r, t ) n (r, t ) dv   n* (r ) n (r) dv

R
R
R
= Pn ( R)
Re( ( x, t ))
t
P( x)
Probability of finding a
particle in a given interval is
independent of time and is
determine only by the r.
Measurements are usually an
average over a long time on
the quantum mechanical time
scale and often reflect an
average over a large number
of particles.
In most experiments the
wavefunctions are incoherent.
Normalization of Wavefunctions
The probability of finding a particle in all space, S, must be 100 %.
Pn ( S , t )    *n (r, t ) n (r, t ) dv = 1
S
 P( S )   n* (r ) (r ) dv = 1
S
Therefore wavefunctions must be normalized.
If
 n ( x, t ) is a solution to the Schrödinger equation it must be normalized.
 n ( x, t ) 
 n ( x, t )

S
 n ( x, t )

N
 *n ( x, t ) n ( x, t ) dv
N is the normalization constant.
Probability Distributions and Averages
Observed Distribution of Measurements
Normal Distribution
P(x)
N measurements, xi, with ci repeats, of k possible outcomes.
1 k
x   ci xi
N i1
ci
P( xi )   ci
N
For continuous P( x)  c( x)
variables
k
c
i 1
 c( x)
R
i
k
k
x   P( xi ) xi
 P( x )  1
x   P( x) x dx
 P( x) dx  1
i 1
R
i 1
R
i
Expectation Values
*
*


(
x
,
t
)

x   Pn ( x) x dx  n
n ( x, t ) x dx
R
R
   n ( x)n (t )   n ( x)n (t ) x dx
*
Measurements are
averages in time and
large number of
particles of
observables.
R
 *
 *
    n ( x) n ( x) x dx  n (t )n (t )
R

n (t )*n (t )  ro2  1
*
   n* ( x) x n ( x) dx    n ( x) xˆ n ( x) dx  x
R
R
O    n* ( x)Oˆ  n ( x) dx
R
Expectation values of x.
Every observable has a
corresponding operator
14_01tbl_PChem.jpg
Operator Algebra
Linearity
Addition
Association
ˆ [af ( x)  bf ( x)]  aO
ˆ f ( x)  bO
ˆ f ( x)
O
ˆ  bPˆ ] f ( x)  aO
ˆ f ( x)  bPˆ f ( x)
[aO
ˆ ˆ ˆ  f ( x)  O
ˆ  Pˆ Q
ˆ f ( x)  
OPQ



 
Operator Algebra
Commutation
ˆ ˆ f ( x)  PO
ˆ ˆ f ( x)
OP
ˆ ˆ  PO
ˆ , Pˆ  f ( x) Commutator
ˆ ˆ  f ( x )  O
OP




ˆ , Pˆ ]  0  OP
ˆ ˆ  PO
ˆˆ
[O
Ex) Position and Momentum
ˆˆ x  pˆ x xˆ  ( x)  xp
ˆˆ x ( x)  pˆ x xˆ ( x)
 xˆ, pˆ x  ( x)   xp
 d
 d
 d ( x) d

 x i

(
x
)

i
x

(
x
)

i
x

x

(
x
)


 dx 


dx
dx
 dx 
d ( x) 
 d ( x) dx
 i x
  ( x) 
x   i  ( x)
dx
dx
 dx

  xˆ, pˆ x   i  0
Properties of Hermitian Operators
ˆAT*  A
ˆ
*
ˆ  ( x) dx   ( x) A
ˆ * * ( x) dx

(
x
)
A


S
For matrices
S
For functions
ˆ  ( x)  a  ( x)
A
n
n n
ˆ * * ( x)  a* * ( x)
A
n
n n
*
ˆ ( x) dx   * ( x)a  ( x) dx  a  * ( x) ( x) dx

(
x
)
A
n n
n
n
 n n n
 n
 an
Alternatively
*
ˆ * * ( x) dx   ( x)a* * ( x) dx
ˆ ( x) dx   ( x) A

(
x
)
A
n
n
 n
 n n n
 n
 an*  n ( x) n* ( x) dx
a
*
n
 an*  an  an 
Properties of Hermitian Operators
ˆ  ( x)  a  ( x)
A
n
n n
ˆ  ( x)  a  ( x)
A
m
m m
*
*
ˆ
ˆ

(
x
)
A

(
x
)
dx


(
x
)
A

n
m ( x) dx

 n
*
m
*
ˆ ( x) dx  a  * ( x) ( x) dx

(
x
)
A
n
n m
n
 m
*
*
* *
ˆ

(
x
)
A

(
x
)
dx


(
x
)
a
m
 n
 n m m ( x) dx
 am  n ( x) m* ( x) dx
Properties of Hermitian Operators
 an  m* ( x) n ( x) dx  am  n ( x) m* ( x) dx  0


(an  am )  m* ( x) n ( x) dx  0
  m* ( x) n ( x) dx   n ,m when an  am  n, m
Orthonormal set
  m* ( x) n ( x) dx  0
Not orthogonal
when an  am
Degenerate eigenvalues
Superposition Principle
H n (r )  En n (r )
En
Eigen Value
Eigen Relationship
 n (r)
Set of Eigenfunctions
Consider  m (r ) and n (r) share the same eigenvalue En= Em=E
Any linear combination of eigen functions of degenerate eigenvlaues is an
eigenfunction:
H  a n (r)  b m (r)   Ha n (r)  Hb m (r)
 aH n (r )  bH m (r )
 aEn n (r)  bEm m (r)
 aE n (r)  bE m (r)
 E  a n (r)  b m (r) 
The Momentum Operator is Hermitian
?
* *
ˆ
ˆ

(
x
)
p

(
x
)
dx


(
x
)
p
 ( x) dx


*
S
S

d

S  ( x) pˆ ( x) dx    ( x) i dx   ( x) dx
*
*

d
 i   ( x)  ( x) dx
dx

*
Integration by parts
b
 u dv  uv
a
b
b
a
  vdu
a
d
u   ( x) & dv   ( x)dx
dx
*
d *
 du   ( x)dx & v   ( x)
dx
The Momentum Operator is Hermitian


d
d *
*
 ( x) dx  ( x) dx   ( x) ( x)     ( x) dx ( x) dx
*

d *
 (0  0)    ( x)  ( x) dx
dx



d
i   ( x)  ( x) dx  i
dx

d *
  ( x) dx ( x) dx
*

 d *
   ( x ) i
 ( x) dx

 dx 



* *
ˆ

(
x
)
p
 ( x) dx


wavefunctions are
finite and therefore
converge to zero as
infinity
Operators with Simultaneous Eigenfunctions Commute.
ˆ  ( x)  a  ( x)
A
n
n n
ˆ  ( x)  b  ( x)
B
n
n n
ˆ ,B
ˆ ˆ  BA
ˆ  ( x)   AB
ˆ ˆ  ( x)  AB
ˆ ˆ  ( x)  BA
ˆ ˆ  ( x)
A
n
n
n
n




ˆ b  ( x)  B
ˆ a  ( x)
A
n n
n n
ˆ  ( x)  a B
ˆ  ( x)
 bn A
n
n
n
 bn an n ( x)  anbn n ( x)
  bn an  anbn  n ( x)
0
ˆ ,B
ˆ ˆ  BA
ˆ ˆ  ( x)  BA
ˆ   0  AB
ˆ ˆ  AB
ˆ ˆ  ( x)
A
n
n


Order of operations does not matter only if A and B commute.
Description of a Quantum Mechanical System
d
ˆ
H  n (r, t )  i
 n (r, t )
H n (r )  En n (r )
dt
 n (r, t )   n (r)n (t ) State
En Energy Level
n Quantum number
 3 (r )
E3
E2
E1
E0
0
 2 (r )
1st excited State
Ground State
 1 (r )
 0 (r )
Hˆ (t )
n
   * ( x) * (t ) Hˆ  ( x) (t ) dx
S
   n ( x)* Hˆ  n ( x) dx
S
   n ( x)* En n ( x) dx  En
S
Energy levels are independent of time.
Eigenfunctions are stationary states.
n (r, t   )  Uˆ ( )n (r, t )  ein n (t ) n (r)
The system stays in the
same state, even though
the phase of the function
is time dependent.
Expectation Values Revisited
O(t )   *n ( x, t )Oˆ  n ( x, t ) dx
R
*
  Uˆ (t ) n ( x,0)  Oˆ Uˆ (t ) n ( x,0)  dx
R

  e
R 

ˆ 
 iHt




*
   iHtˆ 

 n ( x,0)  Oˆ e    n ( x,0)  dx
 

 

ˆ 
ˆ 
 iHt
 iHt









   *n ( x,0)e    Oˆ e    n ( x,0)  dx
 

R 

 

  iHtˆ   iHtˆ  
ˆ     ( x,0)dx
  *n ( x,0)  e   Oe
n


R


Expectation Values Revisited
  iHtˆ   iHtˆ  
ˆ     ( x,0)dx
O(t )   *n ( x,0)  e   Oe
n


R


   n* ( x) * (0)Oˆ (t ) n ( x) (0)dx
R
   n* ( x)Oˆ (t ) n ( x)dx
R
Oˆ (t )  e
ˆ 
 iHt




ˆ
Oe
ˆ 
 iHt




ˆ (t )
 U (t )OU
1
Expectation Values Revisited
Consider
e
ˆ 
 iHt





U (t )Oˆ  e
1
ˆ 
 iHt




Oˆ
k
k
 1  iHt
ˆ  
1  it  ˆ k ˆ
ˆ
ˆ


O   
 O     H O
 k k ! 

k k !
 
ˆ ˆ  HO
ˆˆ
If [ Hˆ , Oˆ ]  0  OH
ˆ ˆ 2  OHH
ˆ ˆ ˆ  HOH
ˆ ˆ ˆ  HHO
ˆ ˆ ˆ  Hˆ 2Oˆ
 OH
ˆ ˆ k  Hˆ k Oˆ
 OH
k
Repeat k-1 times
k
1  it  ˆ k ˆ
1  it  ˆ ˆ k
     H O      OH


k k !
k k !
Expectation Values Revisited
e
ˆ 
 iHt





ˆ 
 iHt
k
k
 1  iHt

ˆ  


1
it


k
ˆ  
Oˆ     
  Oˆ  Oˆ     Hˆ  Oe
 k k ! 

k k !
 
1
1
ˆ
ˆ

U (t )O  OU (t )  U (t ), Oˆ   0
1
1
1
ˆ
ˆ
ˆ
 O(t )  U (t )OU (t )  OU U (t )  Oˆ
O(t )    n* ( x)Oˆ (t ) n ( x)dx =   n* ( x)Oˆ n ( x)dx= Oˆ
R
R
Only if [ Hˆ , Oˆ ]  0
Non Stationary States
If [ Hˆ , Oˆ ]  0
ˆ (t )
 Oˆ (t )  U 1 (t )OU
 Oˆ cos(t )  Pˆ cos(t )
Which means that the observable is time dependent.
Consider that an additional interaction is introduced modifying the
Hamiltonian:
Hˆ '  Hˆ  Oˆ
where
 Hˆ , Oˆ   0


Non Stationary States
The states under this new Hamiltonian are
En (t )
n (r, t )
The Energy Levels become time dependent
n (r, t )   an (t ) n (r, t )
n
The state can change quantum number with time under the influence of a
non-commuting operator.
Non-stationary states!!!
The act of measurement can cause the system to change state
Indeterminacy??
A non-commuting operators can therefore induce the state to change
over time. (i.e the state can be influenced externally!!!)