Conserved quantities
Quantities that do not change in time are said to be
conserved
dp x
Examples
 0  p x is a conserved quantity
Physical Chemistry
dt
dE
dt
dp z
dt
Lecture 12
Mathematics of Quantum
Mechanics
 0  E is a conserved quantity
 0 
p z is not conserved
Also called constants of motion
Conserved quantities define a system’s state

They are unique characteristics of the system when in the
state
We seek conserved quantities as a way to describe a
state
Waves
Eigenvalue equations
Special relationship between a
function and an operation
Operation on the function yields
the same function multiplied by
a constant
Classical waves are
periodically varying
functions of time and
space



Described by an amplitude,
A, a wavelength, , and a
period, T.
Alternative to T is the
frequency, .
A wave’s speed depends on
the wavelength and
frequency.
 Speed of sound
 Speed of light


v  
Wave motion
The amplitude at a point varies as a
function of time
The amplitudes at a time vary in space
Convenient to define the wave vector,
the angular frequency, and the phase
k

2
  2

  x, t  
A sin kx  t   
Standing waves, which are the result of
pinning with boundary conditions, can be
written in a simple form
 ( x, t ) 
A ( x) f (t )
By taking the derivatives, one can show
that a wave obeys the wave equation:
1  
k 2 x 2
2

1  
 2 t 2
2
For any operator, only a certain
group of functions can satisfy
such an equation
Establishes the relationship
Example
d2
 A sin kx    k 2  A sin kx 
dx 2
ALL functions having this
relationship to a particular
operator form a complete set
 ( x, t ) 
x t 
A sin 2   
 T 


 K f
Oˆ f
Usually distinguished by a
number (often appended as a
subscript to the function
Each member of the set is a
distinct, special case
fn 
Eigenvalue equations give a
means to find states
corresponding to constants of
motion
Schroedinger’s equation
Means to solve for
wave functions of a
system corresponding
to constant energy
states
Obtain Hamiltonian
operator by
correspondence
Set up eigenvalue
problem for constantenergy states
Gives a differential
equation to solve for
the eigenfunctions of
the Hamiltonian
 T
H
 V

p2
2m
 V (r)
By correspondence
pˆ x
  i

x

pˆ x2
2
 
 

   i   i     2 2
x
x 
x 

Substitution gives, for a single particle in three dimensions
Hˆ
 
2
2 
2  2

  Vˆ ( x, y, z )


2m  x 2 y 2 z 2 
Schroedinger ' s equation
Hˆ   E

2
2  2
2 

   Vˆ ( x, y, z )  E


2m  x 2 y 2 z 2 
1
The wave function
Because of uncertainty, one
cannot use the position of a
particle as a unique
characteristic (constant of
motion) of the state of a
particle
The wave function depends
on position, so what is its
meaning?
Max Born’s interpretation


 * ( x) ( x)dx 
Expectation values
probability that
the particle is in
the region
x  x  dx
The wave function is related
to the probability that a
particle will be at x if the
system is in the state given
by the wave function
The square of the wave
function represents the
probability density
Only certain values of the
energy observed
When a system is in an
eigenstate of the Hamiltonian,
repeated measurements give
the same value
When the system is NOT in an
eigenstate of the Hamiltonian,
repeated measurements give
different values
The first derivative must be
continuous at all points in space.
Must be square-integrable over
spatial dimensions.


The square of the wave function
represents the probability per
unit length [in 1D], or area [in
2D] or volume [in 3D], of finding
the particle in the region around
the point.
Summing over all possible
positions must give a definite
finite value other than 0 that
represents the total probability


Normalization
The square of the wave
function is the probability
density
If one seeks the particle at
every possible location, the
sum of the probabilities must
total to 1. [i.e. the particle is
somewhere in the space]
This relation gives a way to
define one of the function’s
parameters – the amplitude
of the wave function –
through the process of
normalization.
  ( x) f ( x)  ( x)dx
*

The momentum

  ( x) pˆ ( x)  ( x)dx
 p( x)  
*

The energy (if in an eigenstate)

E 
  ( x) Hˆ  ( x)dx
*


  ( x) E

*

*


 ( x)dx

 E   * ( x) ( x)dx  E (1)

The set of values contain only
the eigenvalues of the
Hamiltonian
The fraction of measurements
giving a particular value is the
probability that the system’s
states “appears” like that
eigenstate
We call that fraction of times
the eigenvalue appears the
probability, cn*cn, of that state
being like the eigenstate
In an eigenstate
  n
En , En , En , En ,
In a state that is not an eigenstate
 
c 
n
n
n
E1 , E0 , E10 , E8 , E0 , E11 , E0 ,
Co-ordinate systems

  ( x) ( x)dx

 
  i   * ( x)  ( x)dx
 x 

Interpretation of quantum
calculations and measurement
Must be single-valued, so that
the square at any point in space
is a unique number
Eigenvalues must be real
Second derivative of the wave
function must be finite at all
points in space.


 f ( x)  
 E
Requirements on the wave
function

Functions of a particle' s position (in 1D)
Given a wave function,
how does one find the
value of a measurable
property?
Properties are defined
by integrals over the
probability distribution.
If the state is an
eigenstate, then the
expectation value IS the
eigenvalue.
 1
Most convenient co-ordinates
depends on the structure of the
Hamiltonian

Symmetry often determines the
appropriate co-ordinates
Common co-ordinate systems



Cartesian co-ordinates
 (x, y, z)
 Appropriate to rectilinear
motions
Spherical polar co-ordinates about a
unique origin
 (r, , )
 Appropriate to spherical motion
relative to an origin
Cylindrical co-ordinates relative
to an origin
 (, , z)
 Appropriate to motion relative
to a unique axis

Elliptical co-ordinates relative to
two foci
rA  rB
R
r r
 A B
R
 

 (, , )
 Appropriate to motion relative
to two fixed points
2
Volume elements for
integration
In the cartesian
representation
dV = dx dy dz
All space


 -  x  
 -  y  
 -  z  
In the spherical polar
representation
dV = r2dr sind d
All space


 0r
 0
 0    2
Complex numbers and
functions
z represents a quantity like a
two-dimensional vector
Use the imaginary
number
i 
1
Expressed as either
 The real (x) and imaginary
(y) parts
 The magnitude (r) and
phase ()
Can depend on other variables
 Complex functions
z(t ) 
x (t )  i y (t )  r (t ) exp(i (t ))
Summary
The quantum description relies on a wave equation
Construction of the Hamiltonian operator allows one to form equation for
constant-energy states: Schroedinger’s equation
Correspondence principle

Determination of form of operators
Wave functions have special properties


Related to the probability of finding the particle at a particular point when in a
particular state
Usually normalized to be congruent with the idea of the wave function as being
related to probability
Calculation of observables related to integrals over the wave function




Integration required to get expectation values of properties
Properties for which the wave function is an eigenfunction give a definite value
for the expectation value
Defining the operators in the appropriate system of co-ordinates simplifies the
mathematical problem
Integration requires volume elements for each type of co-ordinate system
In quantum systems, only certain values (eigenvalues) are seen


In an eigenstate, repeatedly measure the same number
In a state that is not an eigenstate, measure different numbers (only
eigenvalues), the fraction of times an eigenvalue is measured being determined
by the probability that the state is like the eigenstate
3