Transcript Presentation #2

MODULE 1
FROM PLANCK TO EINSTEIN TO DE BROGLIE TO SCHRÖDINGER…
You should be familiar with the major rethinking about the physics of
electromagnetic radiation and matter that occurred at the opening of the 20th
century.
If not, I suggest that you read and digest an account that you will find in any
advanced text on modern physics, physical chemistry, and the like.
In Module 1 you will find a very brief and hardly sufficient overview of the
historical background to the birth and early growth of quantum mechanics, the
concept of wave-particle duality, and the contributions of the “greats”.
I urge you to read this historical material and to supplement it from other
sources. It will help you ease into the course material.
12
8
7
6
1
2
13
3
4
14
9
10
5
1: Langmuir
7: Bragg
13: Pauli
2: Planck
8: Dirac
14: Heisenberg
3: Mme Curie
9: de Broglie
15: Brillouin
4: Lorentz
10: Born
5: Einstein
11: Bohr
6: Debye
12: Schrödinger
15
11
Solvay Conference
Brussels, Belgium, 1927
As a result of his contemplation of the PHOTOELECTRIC EFFECT, Einstein
proposed that when light was emitted from a source, it was not a continuous
wave, but as a beam of minute energy packets (quanta) and this was the way
that it generated electrons on impinging on the sensitive metal surface.
Eventually these light quanta became known as photons.
In 1924 Louis de Broglie proposed that if light (a classical wave) can display
particle-like behavior, then matter (classical particles) should likewise display
wave-like behavior under particular experimental conditions.
This proposal has brought an intriguing symmetry to natural phenomena.
de Broglie argued that the equation that Einstein had arrived at for photons, viz.
l  h/ p
where l is the wavelength, p is the momentum and h is Planck's constant,
should be applicable also to matter.
On this concept, l would be the wavelength of matter waves that are
associated with the particle.
Consider a particle of mass m moving with velocity v.
The momentum is given by p = mv
And its de Broglie wavelength is given by l = h/mv.
For masses in the kg range, moving with everyday velocities (e.g.,
a fast moving baseball) l is of the order of 10-34 m, a value that is
not observable and of no consequence in everyday life.
However, for an electron traveling at 1% of the speed of light
l~ 10-10 m, i.e. 100 pm
This is in the x-ray spectrum, and clearly measurable.
In fact, it is now commonplace that electron beams of such
character can be employed for diffraction studies (a specific
property of waves) to obtain information on the structure of
crystalline materials.
Thus it seems we have a paradoxical situation:
an electron shows properties of both a particle (which we think of
as having a particular set of spatial coordinates, i.e. localized)
and it shows wave-like properties (in which it cannot be pinned to
a particular set of coordinates).
The paradox is resolved by deciding that an electron (and
electromagnetic radiation) is neither a particle nor a wave but can
show properties of either (but not simultaneously both), depending
on how we carry out the observations.
So electrons and photons resemble one another but they are not
twins.
electrons have rest mass and travel at any velocity less than
3x108 m s-1
photons have no rest mass and always travel at 3x108 m s-1 in
vacuo.
MOTION, MOTION, MOTION...
The motions of macroscopic bodies (bullets, baseballs, and
planets) are perfectly described by the laws of Newton.
In classical mechanics we deal with what are called dynamical
variables. These are position, momentum (linear and angular),
and energy.
A dynamical variable that can be measured is called an
observable.
The state of a macroscopic body at any time is specified
completely by its position coordinates (x, y, and z) and the
coordinates of its momenta (px, py, and pz) at the time in question.
Since momentum is the product of mass with velocity we could
use the velocity coordinates (vx, vy, and vz) instead of the
momenta.
The three-dimensional path that is taken by a body in motion as a
function of time is called its trajectory.
This trajectory is the full description of the motion of the particle
Newton's Second Law enables us to calculate the trajectory of a
particle in terms of the forces acting on it.
Thus the entire history and the entire future of the body's motion,
point by point, can be computed from the knowledge of the current
position of the particle and the magnitude and direction of the
forces acting upon it.
Newton's laws are in this sense deterministic.
This is the property that enables our Astronomy colleagues to
compute the trajectory of the planets and to accurately predict the
occurrence of future phenomena such as solstices and eclipses.
Newton’s Second Law is a Law of Motion and is usually
formulated as
F  ma  mx
where a (or x ) is the acceleration imparted on the mass m under
the application of the force F.
The Uncertainty Principle (to be discussed later) is of no practical
importance for the motions of macroscopic bodies. Because we
can determine both their position and momentum simultaneously
with arbitrary precision.
However, as we shall see, this is not the case when we wish to
compute the motions in the sub-microscopic world of atoms,
molecules, nuclei, and electrons.
See Module 1 for some discussion on the Bohr model of the H
atom.
The laws that govern our everyday human world such as
Newton's laws of motion and the laws of Thermodynamics are
postulates.
This means that they cannot be derived from other, more
fundamental laws.
Postulates cannot be proven, but if other rules or properties that
are derived from postulates are shown by experiment to be true,
then we can say that the postulates are consistent with our
experience (e.g., the laws of Thermodynamics)
The laws of quantum mechanics can also be posed in the form of
postulates.
The difference being that the postulates that govern our everyday
world are apparent and sensible to us; those that govern the
micro-world are much more abstract.
Some postulates of quantum mechanics are now presented.
POSTULATE 1
"The state of a system of particles is completely specified by a function (Y)
)that depends on the coordinates of the various particles, and on time.
All possible information about the system can be derived from this function."
POSTULATE 1A
“The probability that a particle will be found at a specific time t, in a volume
element dt at the point r is proportional to Y (r,t)*Y (r,t)dt
POSTULATE 2
"For every observable (dynamical variable that is experimentally measurable)
there exists a linear, Hermitian operator."
POSTULATE 3
“The wavefunction Y(r1, r2, …, t) changes in time according to
Y
Hˆ Y  i .
t
Where Ĥ is the hamiltonian operator
POSTULATE 1
"The state of a system of particles is completely specified by a function that
depends on the coordinates of the various particles, and on time. All possible
information about the system can be derived from this function."
this function is commonly termed the wavefunction, or state function.
the phrase "all possible information" refers to the information about the system
that is open to experimental determination (the observables).
the wave function is usually represented by Y(r1, r2,…, t), where r1, r2,… are the
spatial coordinates (with respect to some origin) of particles 1, 2, …, that make
up the system, and t is the time.
the spatial and temporal components of Y can be separated. By convention
we denote the time-independent wavefunction by y (lower case, italic).
Often chemists are concerned only with the spatial characteristics of a system
(the shape of a compound) and are unconcerned with how it changes with time.
Not so for Photoscientists!!!
Y is a mathematical function that carries all the information.
What we need to do is to find some means to unlock the
information from Y.
Much in the same way that we can unlock information about real
compounds, such as benzene, by carrying out some appropriate
experiments.
We must find some way to work on Y to cause it to give us the
same information that we could obtain through experiments.
Later we shall see how to do this.
A chemical system can exist in many stationary states.
The symbol y (time-independent) denotes a math function that
carries information on all stationary states of the system.
But we may want more specific information, e.g. the energy of a
particular stationary state (position variables only) of a molecule or
atom.
In such cases wavefunctions are qualified by quantum numbers
that can specify the appropriate wave function for particular
variables.
Such wavefunctions may appear as ym,n,… where m, n, etc are
quantum numbers that represent the particular subsets of y that
interest us.
We can also refer to the particular state of a system using its
quantum numbers only, with y implied.
E.g. the wavefunction ya, b, c can be simply denoted by [a,b,c].
In the Dirac bra-ket system ym and ym* (the complex conjugate of
ym) are
represented by m and m , respectively.
Acceptable Wavefunctions
Wavefunctions are mathematical entities that denote real systems
and therefore they must make some kind of physical sense.
Because of this there are certain limitations that need to be placed
on the wavefunctions.
Such wavefunctions are said to be well behaved.
The criteria for acceptability (using only the x dimension) are:
At any x, y(x) must have only one value (single-valued)
y(x) and dy(x)/dx must be continuous
y(x) must not be infinite over a finite range (but it can be
infinite over an infinitesimally small range)
y(x) must be square integrable, i.e.
0
y *( x)y ( x)dx  
allspace
must be a finite quantity.
The square integrable condition requires that the integral
above must not be equal to zero or to infinity, it must be finite
(more later).
About complex conjugates:
Note that if y = a + ib, then y* = a – ib
and yy* = (a + ib)(a - ib) = a2 + b2
and yy* is thus real and non-negative
POSTULATE 1A
This concerns the interpretation of the wavefunction itself.
The probability that a particle will be found at a specific time t, in a
volume element dt at the point r is proportional to
Y (r,t)*Y (r,t)dt
In one-dimensional motion this product represents the probability
of finding the particle between x and x + dx, at time t.
In a 3-d situation, the postulate tells us that the probability of
finding the particle in some small volume element, dt, is given by
Pr = Y2 dt
(or Pr = Y*Y dt if Y is complex)
the quantity Y2 can be thought of as a probability density in the
sense that it generates a probability when it is multiplied by the
volume element, dt.
The wavefunction itself is a probability amplitude and has no
physical meaning, per se.
This interpretation is due to Max Born (1926)
He argued that probability is measurable (at least in principle) and
as such it must be a real, non-negative quantity, and it therefore
must relate to a real, non-negative quantity arising out of the
wavefunction.
Wavefunctions must be well behaved but they can be real or
complex, positive or negative.
Born postulated that the correct identity was given by the product
of the wavefunction with itself, or with its complex conjugate when
the function contains imaginary quantities.
The Born postulate highlights the difference between classical and
quantum mechanics.
In classical terms we can precisely locate a particle as having unit
probability of being at a particular narrow range of x-values
(between x and dx). Everywhere else on the x-dimension the
probability is zero.
In quantum terms we equate the measured probability of finding
the particle at any value of x to Y2 or the product of Y*Y.
Since the wavefunction can have non-zero values for finite values
of x, we find ourselves unable to locate the particle at a definite
point on the axis.
All we can do is to predict the probability of finding the particle at a
given value of x, if we know Y(x,t) .
The nature of quantum science is that it is statistical, not
deterministic.
Note, however, that particles are not behaving like waves, but it is
their probability patterns that satisfy a wavefunction
One important implication of the Born postulate is that

y * ( x)y ( x)dx  
allspace
(the square integrable condition)
there must be a finite probability of finding the particle somewhere
within the region defined as "all space".

y * ( x)y ( x)dx  
allspace
If, in a given case, the integral above equals a real constant N,
then dividing y(x) by N1/2 results in the integral having the value
1.0, i.e.

y *( x)y ( x)dx  1
allspace
Now the wave function is said to be normalized and the probability
is unity for finding the particle in "all space“, as it must be.
POSTULATE 2
"For every observable (dynamical variable that is experimentally
measurable) there exists a linear, Hermitian operator."
An operator is a symbolic form that tells us to perform a
mathematical transposition (operation) on the function that follows
the symbol.
Thus
,
d
, x, sin
dx
are all operators that tell us to take the square root, take the
derivative with respect to x, multiply by x, and take the sine of,
respectively, whatever follows the operator symbol.
Barrante, chapter 10 and Math Module B on the our web site
develop operator algebra and define linear and Hermitian
operators.
operators are usually written with a caret “^” over the symbol.
Thus it is the application of the appropriate operator that unlocks
the related observable from the wavefunction.
We can bring postulates 1 and 2 together as a diagram
EXPERIMENT
SYSTEM
OBSERVABLE
WAVEFUNCTION
OPERATOR
The postulates examined to date enable us to calculate
observables for a few simple systems. But first we need to know
how to construct operators for the observables
Constructing Quantum Mechanical Operators for Observables
write down the classical expression for the observable we are
concerned with in terms of its classical position coordinates and
momentum.
Replace the position-dependent quantities by the appropriate form
of the multiplication operator.
Replace the momentum quantities by

i q
where q is the position variable.
(Do not try to understand this. It can be demonstrated that it is
legitimate by employing a mathematical device called a Poisson
bracket. We simply accept this transform).
Other variables (e.g. time, dielectric constant) retain their classical
form
For example let us find the operator for energy
Newton’s equations of motion are convenient to use if the problem
to be solved is cast in the form of Cartesian coordinates, but in
other coordinate systems (e.g. spherical) Newton’s equations
become cumbersome.
In the 19th century Joseph Lagrange and William Rowan Hamilton
independently developed equations of motion that were
independent of the coordinate system.
The approach of Hamilton is the one that best allows the
transformation from classical to quantum mechanics.
In conservative systems (in which the total energy remains
constant with time) the Hamilton function is equal to the total
energy of the system expressed in terms of the coordinates of the
particles and their conjugate momenta.
In a Cartesian system these are x,y, and z, and px, py, and pz.
Assume that we are interested in a single particle only that is
constrained to move on the x-axis, then the Hamilton function (H)
is the sum of the kinetic (T) and potential (V) energies of the
particle and
T V  H
if m = mass and v = velocity, then T ( x)  mvx 2 / 2
And in terms of momentum
px  mvx
T ( x)  px / 2m
2
The potential energy (V) depends only on the position on x and
thus the Hamilton function is given by
2
px
H 
 V ( x)
2m
Now we have an expression for total energy (kinetic and potential)
in terms of position and momentum
using our rules for the construction of operators we form a
Hamiltonian operator.
In the x dimension, the operator for kinetic energy is thus
2
2
1




Tˆ ( x) 

 
2m  i x 
2m x 2
2
In 3 dimensions this becomes
2
2
2
2





Tˆ  
. 2  2  2   
2
2m  x y z 
2m
2
Where  (del) is the Laplacian operator
The potential energy of a particle in one dimension is a function of
its position with respect to some origin/baseline.
Thus V(x), becomes multiplication by the operator Vˆ ( x )
The same is true for 2 or 3 dimensions.
Thus the Hamiltonian operator for a particle of mass m, with
motion restricted to the x dimension can be written as
Hˆ  Tˆ  Vˆ
dˆ 2 ˆ

 V ( x)
2
2m dx
2