Post on 20-Dec-2015
Overview of QM
( ) ( )n n nH E r r
ˆ ( , ) ( , )d
H t i tdt
r r
( , ) (0) ( )i tt e r r
( , ) ( ) ( )t t r r
ˆ( , ) ( ) ( , )t U t r r
2
ˆ ˆ ˆ ˆ ˆ( ) ( )2
H K V Vm
r r
Translational Motion2
ˆ ˆ ˆ2
Hm
Rotational Motion2
ˆ ˆ ˆ2
HI
L L
Vibrations
22ˆ ˆ ˆ ( )
2 2 eq
kH
r r
Cartesian
Spherical Polar
Centre of Mass
Statics
Dynamics
P. in Box
Rigid Rotor
Angular Mom.&Spin
Harmonic Motion
ex) STM, Devices
ex) FTS, NMR
ex) IR, Raman
Mol. dynamics, Q. Comp., Laser Pulse Methods,2D NMR, and SS NMR, andspectroscopy.
M.O. Calculations, Spectroscopy, andQ. Stat. Mech.
Quantum Mechanics for Many Particles
1 2 3 1 2 3( , , ,..., ) ( , , ,..., )n k n n kH E r r r r r r r r
2
,
ˆ ˆ ˆ ˆ ( , )2 i i ij i j
i i ji
H Vm
r r
1 2 3 1 2 3( , , ,..., , ) ( , ) ( , ) ( , )... ( , )k kt t t t t r r r r r r r r
( , ) ( ) ( )i i it t r r
1r
(0,0,0)
3r
2r4r
m1m3
m2m4
z1z3
z4z2
2
2ˆ ( , )
4
i jij i j i j
o i j
z z eV
r r r r
r r
En – Energy Levels n – Wavefuntions
Electronic Structure of Mols.
14_01fig_PChem.jpg
Properties of the Wavefunction
( , ) ( ) ( )x t x t Single Valued Finite and continuous
( ) (0) i tt e
2( , ) xx t d
2( , )x t
Im
Re
t
( )t
( , )x t
(0)o
ro
Complex Valued
14_01fig_PChem.jpg
The Wavefunction in 3D
3i i i ix y z r i j k
3( , )i t r
sin cos
sin sin
cos
i i i i
i i i
i i
r
r
r
r i
j
k2v sin i i i i i id r drd d
( , ) ( , , ) ) ( )i i i i it x y z t r r 3
2( , ) vi it d r
v x y z i i i id d d d
Spherical Polar Coordinates
Cartesian Coordinates
2 3( , )i t r
Probability Distribution
2 *( , ) ( , ) ( , )t t t r r r
( , )t r *z z z z Recall
*( , ) ( , ) ( , )P t t t r r r Probability of finding the particle at exactly r, as a function of time.
*
( , ) ( , ) v ( , ) v
( , ) v
( , ) ( , ) v
j j j j
i i i i
x y z
x y z
R
R
P R t P t d P t d
P t d
t t d
r
r
r r
r
r r
Probability of finding the particle between ri and rj, defining the region R, as a function of time
Probability Distribution and Time
*( , ) ( , ) ( , ) vR
P R t t t d r r
*( ) ( ) ( ) ( ) v
R
t t d r r
* *
*
( ) ( ) ( ) v ( )
( ) ( ) ( )
R
t d t
t P R t
r r
*
*
( ) ( ) ( )
(0) (0) ( ) ( )i t i t
t t P R
e e P R P R
*( ) ( ) ( ) vR
P R d r r
Probability is independentof time!
Probability Distribution of Wavefunctions*( , ) ( , ) v ( , ) ( , ) vn n n n
R R
P R t P t d t t d r r r *( ) ( ) v
= ( )
n n
R
n
d
P R
r r
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.
t
Re( ( , ))x t ( )P x
In most experiments the wavefunctions are incoherent.
Normalization of Wavefunctions
*( ) ( ) ( ) v = 1n
S
P S d r r
The probability of finding a particle in all space, S, must be 100 %.
*( , ) ( , ) ( , ) v = 1n n n
S
P S t t t d r r
Therefore wavefunctions must be normalized.
( , )n x tIf is a solution to the Schrödinger equation it must be normalized.
*
( , ) ( , )( , )
( , ) ( , ) v
n nn
n nS
x t x tx t
Nx t x t d
N is the normalization constant.
Probability Distributions and AveragesObserved Distribution of Measurements Normal Distribution
N measurements, xi, with ci repeats, of k possible outcomes.
1
1 k
i ii
x c xN
1
( )k
ii i i
i
cP x c c
N
1
( )k
i ii
x P x x
1
( ) 1k
ii
P x
( ) ( ) ( )R
P x c x c x ( )R
x P x xdx ( ) 1R
P x dx
P(x)
For continuous variables
Expectation Values
( )n
R
x P x xdx * *( , ) ( , )n n
R
x t x t xdx
*( ) ( )n n
R
x x x dx * ˆ( ) ( )n n
R
x x x dx x * ˆ( ) ( )n n
R
O x O x dx
Measurements are averages in time and large number of particles of observables.
Every observable has a corresponding operator
Expectation value of x.
*( ) ( ) ( ) ( )n n n n
R
x t x t xdx * *( ) ( ) ( ) ( )n n n n
R
x x x dx t t
*( ) ( ) 1n nt t
Operator Algebra
ˆ ˆ ˆ[ ( ) ( )] ( ) ( )af x bf x a f x b f x O O O
ˆ ˆˆ ˆ[ ] ( ) ( ) ( )a b f x a f x b f x O P O P
Linearity
Addition
Association ˆ ˆ ˆ ˆˆ ˆ( ) ( )f x f x OPQ O P Q
( ) ( )f ax bx af x bf x Analogy
( ) ( ) ( )af bg x af x bg x Analogy
( )fgh x f g h xAnalogy
Commutation ˆ ˆˆ ˆ( ) ( )f x f xOP PO
ˆ ˆ ˆˆ ˆ ˆ( ) , ( )f x f x OP PO O PCommutator
ˆ ˆ ˆˆ ˆ ˆ[ , ] 0if O P OP PO
Order matters !!!
Operators Commute
ˆ ˆ ˆˆ ˆ ˆ[ , ] 0if O P OP PO Operators Do Not Commute
Translation then rotation or rotation then translation
Commutation
CommutationEx) Position and Momentum
ˆ ˆ ˆˆ ˆ ˆ, ( ) ( )
ˆ ˆˆ ˆ( ) ( )x x x
x x
x x
x x
x p xp p x
xp p x
( ) ( )d d
x i x i x xdx dx
( )( )
d x di x x x
dx dx
( ) ( )( ) ( )
d x dx d xi x x x i x
dx dx dx
ˆˆ , 0x i x p
Properties of Hermitian Operators
*ˆ ˆA AT
ˆ ( ) ( )n n nx a x A
* *( ) ( ) ( ) ( )n n n n n n
n
x a x dx a x x dx
a
* * * * * *
*
ˆ( ) ( ) ( ) ( ) ( ) ( )n n n n n n n n
n
x x dx x a x dx a x x dx
a
A
*n n na a a
* ˆ( ) ( )n nx x dx A
RHS
For matrices
* * *ˆ ˆ( ) ( ) ( ) ( )S S
x x dx x x dx A A
For functions
* * * *ˆ ( ) ( )n n nx a x A
LHS
ˆ ( ) ( )n n nx a x A
Properties of Hermitian Operators
ˆ ( ) ( )m m mx a x A
** *ˆ ˆ( ) ( ) ( ) ( )m n n mx x dx x x dx A A
* * * *ˆ( ) ( ) ( ) ( )n m n m mx x dx x a x dx A
*( ) ( )m n ma x x dx
* *ˆ( ) ( ) ( ) ( )m n m n nx x dx x a x dx A
Consider two eigenfunctions of A with different eigenvalues:
If A is Hermetian then:
* ( ) ( )n m na x x dx
RHS
LHS
Properties of Hermitian Operators
* *( ) ( ) ( ) ( ) 0n m n m n ma x x dx a x x dx
*( ) ( ) ( ) 0n m m na a x x dx
*,( ) ( ) when ,m n n m n mx x dx a a n m
* ( ) ( ) 0 whenm n n mx x dx a a
Orthonormal set
Degenerate eigenvaluesNot orthogonal
Superposition Principle
( ) ( )n n n H r r
( )n r
Eigen Relationship
n Eigen Value Set of Eigenfunctions
Any linear combination of eigenfunctions of degenerate eigenvalues is an eigenfunction:
( ) ( ) ( ) ( )n m n ma b a b H r r H r H r
( ) ( )n ma b H r H r
Consider ( ) and ( )m n r r share the same eigenvalue En= Em= E
( ) ( )n n m ma b r r
( ) ( )n ma b r r
( ) ( )n ma b r r
The Momentum Operator is Hermitian
* * *ˆ ˆ( ) ( ) ( ) ( )S S
x x dx x x dx p p
* *ˆ( ) ( ) ( ) ( )S
dx x dx x i x dx
dx
p
Integration by parts
?
b bb
aa a
udv uv vdu
*( ) ( )d
i x x dxdx
*( ) & ( )d
du x dx v xdx
*( ) & ( )d
where u x dv x dxdx
The Momentum Operator is Hermitian
* * *( ) ( ) ( ) ( ) ( ) ( )d d
i x x dx i x x i x x dxdx dx
wavefunctions are finite and therefore converge to zero as x goes to infinity
*(0 0) ( ) ( )d
i i x x dxdx
*( ) ( )d
x i x dxdx
* *ˆ( ) ( )x x dx
p
ˆ ( )x is Hermetian p
Operators with Simultaneous Eigenfunctions Commute.
ˆ ( ) ( )n n nx a x A ˆ ( ) ( )n n nx b x B
ˆ ˆ ˆˆ ˆ ˆ, ( ) ( )n nx x A B AB BA
ˆ ˆˆ ˆ( ) ( )n nx x AB BA ˆ ˆ( ) ( )n n n nb x a x A B
ˆ ˆ( ) ( )n n n nb x a x A B
( ) ( )n n n n n nb a x a b x
( )
0n n n n nb a a b x
ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ, 0 ( ) ( )n nx x A B AB BA AB BA
Order of operations does not matter only if A and B commute.
* *ˆ ˆ( ) ( ) ( ) ( ) ( )n
S
t x t x t dx H H
Description of a Quantum Mechanical System
( ) ( )n n n H r r
n
( , ) ( ) ( )n n nt t r r
Energy Level
State
* ˆ( ) ( )n n
S
x x dx H
Energy levels are independent of time.Eigenfunctions are stationary states.
*( ) ( )n n n n
S
x x dx 01
3
2
0 ( ) r
1( ) r
2 ( ) r
3( ) r
The system stays in the same state, even though the phase of the function is time dependent.
Ground State
1st excited State
0
n Quantum number
ˆ( , ) ( ) ( , ) ( ) ( )nin n n nt U t e t r r r
Expectation Values Revisited
* ˆˆ ˆ( ) ( ,0) ( ) ( ,0)n n
R
t x t x dx U O U
* ˆ( ) ( , ) ( , )n n
R
t x t x t dx O O
*ˆ ˆ
ˆ( ,0) ( ,0) xi t i t
n n
R
e x e x d
H H
O
ˆ ˆ
* ˆ( ,0) ( ,0)i t i t
n n
R
x e e x dx
H H
O
ˆ ˆ
* ˆ( ,0) ( ,0) xi t i t
n n
R
x e e x d
H H
O
Expectation Values Revisitedˆ ˆ
* ˆ( ) ( ,0) ( ,0) xi t i t
n n
R
t x e e x d
H H
O O
* * ˆ( ) (0) ( ) ( ) (0) xn n
R
x t x d O
* ˆ( ) ( ) ( ) xn n
R
x t x d O
ˆ ˆ
1ˆ ˆ ˆˆ ˆ( ) ( ) ( )i t i t
where t e e t t
H H
O O U OU ??
Expectation Values Revisited
1 ˆˆ!
kk
k
it
k
H O
ˆ ˆ ˆˆ ˆ ˆIf [ , ] 0 H O OH HO
ˆ1 ˆ!
k
k
i t
k
H
O
2ˆ ˆˆ ˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ
2
OH OHH
HOH
HHO H O
ˆ ˆˆ ˆk k OH H O
Repeat k-1 times
1 ˆ ˆ!
kk
k
it
k
OH
ˆ
1 ˆ ˆ( )i t
t e
H
U O O
ˆ
11ˆ ˆ ˆˆ ˆ ( )!
i tkk
k
ite t
k
H
O H O OU
Expectation Values Revisited
1
1
ˆ ˆˆ ˆ( ) ( ) ( )
ˆ ˆˆ ˆ( ) ( )
t t t
t t
O U OU
OU U O
*
*
ˆ( ) ( ) ( ) ( )
ˆ ˆ( ) ( ) x=
n n
R
n n
R
t x t x dx
x x d
O O
O O
1 1
1
ˆ ˆˆ ˆ( ) ( )
ˆˆ ( ), 0
t t
t
U O OU
U O
ˆˆTherefore if [ , ] 0H O
Non Stationary States
1ˆ ˆˆ ˆ( ) ( ) ( )t t t O U OUˆˆIf [ , ] 0H O
Which means that the observable is time dependent.
Consider that an additional interaction is introduced modifying the Hamiltonian:
' ˆˆ ˆ H H O ˆˆ , 0 H Owhere
ˆ ˆcos( ) cos( )t t O P
Non Stationary States
( , )n t r
( )n t The Energy Levels become time dependent
( , ) ( ) ( , )n n nn
t a t t r r
The state can change quantum number with time under the influence of a non-commuting operator. Non-stationary states!!!
A non-commuting operator can therefore induce the state to change over time. (i.e the state can be influenced externally!!!)
Indeterminacy??
The states under this new Hamiltonian are
The act of measurement can cause the system to change state