Download Physical Chemistry Postulates of quantum mechanics Origins of

Postulates of quantum
mechanics
Any state of a dynamical system of N particles is
described as fully as is possible by a function, ,
such that the quantity *d3r is proportional to the
probability of finding r between r and r + d3r.
For every observable property of a system, there
exists a corresponding linear hermitian operator
Physical Chemistry
Lecture 12
Wave Equations and States

The physical properties of the observable can be inferred
from the mathematical properties of its associated operator.
The measurement of a physical observable gives only
one of the eigenvalues corresponding to that
observable.
Expectation value of a
property
Origins of quantum mechanics
Hamilton’s equation for the energy of a
classical system
H
 E
Schroedinger’s equation for the wave function
of a system with a definite energy
Ĥ  E

Where  is the wave function corresponding to
the energy
x

x
px

 i
px2  p 2y  pz2

Requirement on the
wave function and
operator
The operator must be a
hermitian operator
  Oˆ dV
O 
*
all
space

 
 * Oˆ  dV

all
space
 Oˆ   dV
*
all
space
Orthogonality and
completeness
Choosing operators
Operators for various
observables are found by
correspondence to
classical equivalents
For position, the
operation is
multiplication by the
appropriate coordinate
For momentum, the
operation is related to
differentiation with
respect to the conjugate
coordinate
Other operators are
found by correspondence
through the momentum
and position
dependencies
If a system is in a state
with wave function, ,
the measured property is
given by an integral
Must know both the state
and the operator to find
the value
Expectation values of
measurables are real

x
p2

pˆ 2
 
  
 
  
 


   i   i     i   i     i   i 
x 
x  
y 
y  
z 
z 


 2
2
2 
   2  2  2  2     22
y
z 
 x
Eigenfunctions of a
hermitian operator
*
corresponding to
 a b dV  0 if a  b
different eigenvalues
all
space
are orthogonal.
The set of all
eigenfunctions of a
hermitian operator is
complete. Any function
f ( x, y, z )   ck k ( x, y , z )
of the coordinates on
k
which it depends can
be expressed as a
linear combination of
the members of the set.
1
Heisenberg’s uncertainty
principle
Finding basis sets
Simultaneous measurement of two quantities,
<> and <>
Define uncertainty in one as


  2     2
With some operator algebra, one can show
that


1
|  ,   |
2


V = 0 everywhere
The Hamiltonian
operator contains
only the kineticenergy operator
 

 d 
2m dx 2
2



May take on any
value
k (x)  Ae
ikx
k 
2mE
2
Find the Hamiltonian operator for a system
Set up Schroedinger’s equation
Solve for all solutions of Schroedinger’s equation consistent
with boundary conditions
Usually study model systems, the mathematics of
which are tractable

Use model solutions to specify more complex systems
approximately with various techniques.
The free particle in 1D
2 d 2
2m dx 2
2
s   ik
Can be used to express other arbitrary functions

E
Schroedinger’s
equation
The wave
function is found
by simple solution
of the differential
equation
d 2
dx 2
 
trial ( x ) 
s   
2mE

2
A e sx
2mE
2
Another look at the free
particle
The free particle in 1D
Wave function is
a sum of
particular
solutions
Wavevector, k,
defines the state
Expresses all possible conditions of the system


The free particle in 1D
Hˆ

Operational plan
Unless two operators commute, measurement
of each with absolute precision is not possible
when the system is in an arbitrary state.
A model for a
totally isolated
particle
The particle
senses no forces
The complete set of eigenfunctions of an operator is
special.
ikx
 Be
Determine the
linear momentum
of a particle in the
state with
wavevector k
This function is
an eigenfunction
of the linear
momentum
operator
pˆ k ( x ) 
p ( x )
 i
d
dx
 i
d
Aeikx  
dx

p
 i 2 k Aeikx   kAeikx 
2
Eigenvalue equation for the
particle in a 1D box
The particle in a 1D box
Schroedinger’s equation
A model for 1D
translation of a gas
molecule
Two regions



In the box, V(x) = 0
Outside the box,
V(x) = 
In the box, the
Hamiltonian is well
known

 
Hˆ
2
2
 d
2m dx 2
Inside the box
 2 d 2

 E
2m dx 2
Outside the box
   E
The only solution outside the box is
(x) = 0

Provides a boundary condition on (x) at
the box edges
Particle-in-a-1D-box
eigenvalue equation
Differential equation
 2 d 2
2m dx 2

Solution may be written in terms of sines and
cosines, which is more convenient
E  0
Use methods of differential calculus to solve
s ( x) 
A e sx
A exp(i
E
 2mE 
 2mE 
 C sin 
x   D cos
x 
2
2


 


Boundary conditions give a quantum number n
This gives a equation for s and :
2mE
sE   i
2
E ( x) 
Particle-in-a-1D-box
eigenvalue equation
2mE
2mE
x)  B exp(i
x)
2
2


(0) = 0 
(a) = 0 
D  0
2mE
a  n
2
 nx 
n ( x)  C sin 

 a 
Normalized wave functions
En

h2 2
n
8ma 2
Grotrian diagram
The coefficient, C, is so far undetermined

Use the requirement that the wave function be
square integrable with an integral of 1
  ( x)( x)dx
*
 1

all
space
This gives the final specification of the wave
function

The particle in a 1D box
Quantized energy levels
 n 
  ( x) ( x)dx  C 0 sin  a x dx  1

Zero point energy

a
*
2
2


C

2
a
Quantum numbers are
the positive integers
Lowest-energy state is
not at E = 0
Particle is moving, even in
the lowest-energy state
Consistent with
Heisenberg’s principle
3
Summary
Operator eigenvalue equations give the
wave functions from which one obtains all
system information
Schroedinger’s equation results in
differential equations because of the
presence of the momentum operators
Solution requires finding Hamiltonian
operator
Boundary conditions determine quantum
conditions
Example: particle in a one-dimensional
box
4