Here is a
summary revision of the development of Quantum
Theory from its inception so that the reader may
know the steps in the evolution of the theory.
I have avoided all rigor and derivations.
Interested readers can check any textbook for
the details. This is intended only as an
outline. It is not necessary that one
understand all the mathematical formalism to
take up the conclusions. Do not be disheartened
if you do not understand the equations or
derivations.
2.1 Discovery
of Electron
In April 30, 1897, Joseph John Thomson
announced that cathode rays were negatively
charged particles, which he called
'corpuscles’. Thomson proposed the existence of
elementary charged particles, which is now
called electrons, as a constituent of all atoms.
2.2 Discovery of Neutron
In 1932 Chadwick proposed the existence of
Neutron as a result of his studies in alpha
particle collisions.
2.3 Black Body
Radiation
1859 Gustav
Kirchhoff’s studies in blackbody radiation
showed that the energy radiated by a black body
depended on the temperature of the body.
Attempts to explain this shape of the energy and
the wavelength at which the maximum energy occur
continued for several decades
In 1884, Ludwig
Boltzmann derived Stefan’s Law theoretically
In
1896,
Wilhelm Carl Werner
Otto Fritz Franz Wien (1864-1928)
Prussia-Germany derived a distribution law of
radiation.
In 1900
Max Karl Ernst Ludwig
Planck (Germany 1858-1947), who
was a colleague
based his quantum hypothesis to explain the fact
that Wien's law, while valid at high
frequencies, broke down completely at low
frequencies.
“Moreover, it is necessary to
interpret UN [the total energy of a
blackbody radiator] not as a continuous,
infinitely divisible quantity, but as a discrete
quantity composed of an integral number of
finite equal parts”
Planck. On the Law
of Distribution of Energy in the Normal
Spectrum. Max Planck. Annalen der Physik 4
(1901): 553.
In 1900, Planck
devised a theory of blackbody radiation, which
gave good agreement for all wavelengths. In this
theory the molecules of a body cannot have
arbitrary energies but instead are quantized -
the energies can only have discrete values. The
magnitude of these energies is given by the
formula
E =
nhf
where n =
0,1,2,... is an integer, f is the
frequency of vibration of the molecule, and h
is a constant, now called Planck's constant:
h
= 6.63
x
10- 34 J s
Furthermore, he
postulated that when a molecule went from a
higher energy state to a lower one it emitted a
quanta (packet) of radiation, or photon, which
carried away the excess energy.
With this photon
picture, Planck was able to successfully explain
the blackbody radiation curves, both at long and
at short wavelengths. Using statistical
mechanics, Planck derived an equation similar to
the Rayleigh-Jeans equation, but with the
adjustable parameter h. Planck found that h =
6.63 x 10-34 J·s, fitted the data. As
we can see, h is a very very small number. Thus
the electromagnetic waves (light) consists not
of a continuous wave but discrete tine packets
of energy E = hf where f is the frequency of the
light.
2.4
Photoelectric Effect
1905 when Einstein
extended the photon picture to explain, another
phenomenon called photoelectric effect. In this
effect when light is allowed to fall on a metal
and electrons are released. However there is a
lower cut off frequency below which every
electron stopped. Einstein was able to explain
this assuming that photons are particles of
energy E=hf.
2.5 Hydrogen
Spectrum
1913 Niels Bohr
(1885-1962) was able to explain the discrete
spectrum of hydrogen atom with the assumption
that there are possible stable energy levels
where electrons can stay without emitting any
wave and the light is emitted when it falls from
a higher level to a lower level. The frequency
of the light so emitted was given by Energy of
the difference in levels = hf.
2.6 Compton
Effect
In 1923, Arthur
Compton showed that he could explain the
collision of a photon with electrons at rest
using the same idea. These phenomena came to be
known as Compton Effect
2.7
Wave-Particle Duality
Thus, it appears
that light could behave like a wave some time
(to explain reflection, refraction and
polarization, interference) while at other times
(Photoelectric effect, Compton effect) it
behaved like a particle. The wave-particle
duality of electromagnetic wave is a fact of
experience and seemed mutually exclusive without
compromise.
In 1924 in his
doctoral thesis, Prince Louis de Broglie argued
that if light waves exhibited the particle
properties, particles might exhibit wave
properties. The experiment to test was done on
a stream of electrons as particles at a double
slit and single slit and the pattern exhibited
fitted the interference pattern for a wave given
by
m= mass v =
speed of the electron thus: mv = momentum of
the electron
2.8
Schrodinger Equation
In 1926, Erwin
Schrödinger introduced operators associated with
each dynamical variable and the Schrodinger
equation, which formed the foundation of modern
Quantum Theory. A partial differential
equation describes how the wave function of a
physical system evolves over time. In the
Schrodinger picture differential calculus was
used.
The time-independent one-dimensional Schrödinger
equation is given by
The solution for
the value of E gives us a spectrum of values for
the Energy of the system.
Using the
spherical coordinates, this equation gives:
And using the
separable form of the wavefunction in terms of
the radial, angular parts in three dimensions
Using the
potential energy as:
it gave the
correct energy levels and correct spectral
frequencies because of transitions. This
indeed was the greatest success of Quantum
Theory and which gave it the impetus.
2.9 Operators and
Quantum Mechanics
In quantum
mechanics, physical observables (e.g., energy,
momentum, position, etc.) are represented
mathematically by operators. For instance, the
operator corresponding to energy is the
Hamiltonian operator
Where i is
an index over all the particles of the system.
Later Dirac
developed the Matrix method and is known as
Dirac Bracket formalism.
In this mechanism
the operators are replaced by matrices and the
wave equation then reduce to a matrix equation
2.10 Quantum Wave
Functions and State Vectors
While operators
represent the observables, the operand – the
function on which the operators act is known as
the wavefunction , which is a function of the
position for stationary solutions.
Postulates of
Quantum Mechanics were developed later as below:
Postulate 1.
The state of a quantum mechanical system is
completely specified by a function
that depends
on the coordinates of the particle(s) and on
time. This function, called the wave function or
state function, has the important property that
is the probability that the particle lies in
the volume element located at time t.
Postulate
2. All observables are
associated with a hermitian operator. In any
measurement of the observable associated with
operator
, the only values that will ever be
observed are the eigenvalues a, which
satisfy the eigenvalue equation
The solution to
the eigenvalue problem given above will give a
spectrum of possible values for a corresponding
to a spectrum of eigenfunctions
. These
eigenfunctions form a set of linearly
independent functions. At any point in time, we
could assume that the state of the system will
be a linear combination of these functions.
Some commonly used
operators are given below:
Postulate 3.
If a system is in a state described by a
normalized wave function
, then the average value of the observable corresponding
to is given by
Postulate 4.
The time evolution of system is
given by
The postulates of quantum mechanics, written in
the bra-ket notation, are as follows:
1. The
state of a quantum-mechanical system is
represented by a unit ket vector | ψ>, called a
state vector, in a complex separable
Hilbert space.
2. An
observable is represented by a Hermitian linear
operator in that space.
3. When
a system is in a state |ψ1>, a measurement of an
observable A produces an
eigenvalue a :
A| ψ1> = a | ψ1> so that < ψ|A| ψ1> = a
< ψ| ψ1> = a since the wavefunctions are
orthogonal
The probability of getting this value in any
measurement is
|<
ψ |ψ1>|2
where | ψ1 > is the eigenvector with
eigenvalue a. After the measurement is
conducted, the state is | ψ1 >.
4. There
is a distinguished observable H,
known as the Hamiltonian, corresponding to the
energyof the system. The time evolution of the
state vector |ψ(t)> is given by the Schrödinger
equation:
i
(h/2π) d/dt |ψ(t)> = H
|ψ(t)>
2.11
Heisenberg’s Uncertainty Principle
1927 Heisenberg discovered that there is an
inherent uncertainty if we try to measure two
conjugate observables. This is known as
Heisenberg’s Uncertainty Principle
The simultaneous
measurement of two conjugate variables (such as
the momentum and position or the energy and time
for a moving particle) entails a limitation on
the precision (standard deviation) of each
measurement. Namely: the more precise the
measurement of position, the more imprecise the
measurement of momentum, and vice versa. In the
extreme case, absolute precision of one variable
would entail absolute imprecision regarding the
other.
” The more precisely the position is determined,
the less precisely the momentum is known in this
instant, and vice versa.”
“I believe that the existence of the classical
"path" can be pregnantly formulated as follows:
The "path" comes into existence only when we
observe it.”
“In the sharp
formulation of the law of causality-- "if we
know the present exactly, we can calculate the
future"-it is not the conclusion that is wrong
but the premise. “
--Heisenberg, in
uncertainty principle paper, 1927
http://www.aip.org/history/heisenberg/
In 1929, Robertson
proved that for all observables (self-adjoint
operators) A and B
where [A,B] = AB - BA
In 1928, Dirac
introduced his Bracket notation and QT in terms
of matrix algebra
In 1932, von
Neumann put quantum theory on a firm theoretical
basis on operator algebra.
2.12 Quantum
Non-locality
In 1935 Einstein,
with his collaborators Boris Podolsky and Nathan
Rosen, published a list of objections to quantum
mechanics, which has come to be known as "the
EPR paper" One of this was the problem of
nonlocality. The EPR paper argued that "no real
change" could take place in one system because
of a measurement performed on a distant second
system, as quantum mechanics requires because it
will violate the relativity principles.
Einstein, B. Podolsky, N. Rosen: "Can
quantum-mechanical description of physical
reality be considered complete?"
[i]
For example, consider a neutral-pi meson
decaying into electron – positron pair. The
spin of Pi meson is zero. Therefore, the total
spin of electrons must be zero. Hence one of
the electron will have spin (1/2) and the other
spin (- ½). If the electron pair moves apart a
million light years and we measure the spin of
the electron on earth as ½, QM requires that the
other should have a spin (-1/2) if someone
measures it in his or her galaxy at the same
time. How would they know which spin should it
be since Relativistically it is impossible to
transfer any information with a speed greater
than that of light. This is the “spooky
action-at-a-distance” paradox of QM
There are two
choices.
You can accept the postulates of QM as is
without trying to explain it, or you can
postulate that QM is not complete, that there
was more information available for the
description of the two-particle system at the
time it was created, and that you just didn't
know it because QM does not properly account for
it.
So, EPR requires that there are hidden variables
in the system, which if known could have
accounted for the behavior. QM theory is
therefore incomplete, i.e. it does not
completely describe the physical reality. In
1952, David Bohm introduced the notion of a
"local hidden variable" theory, which tried to
explain the indeterminacy in terms of the
limitation of our knowledge of the complete
system.
[ii]
In 1964, John S. Bell, a theoretical physicist
working at the CERN laboratory in Geneva
proposed certain experimental tests that could
distinguish the predictions of quantum mechanics
from those of any local hidden-variable theory
These involved the use of entangled photons –
photons which interacted together at some point
before being separated. These photon pair can
be represented by one wave function. In 1982,
Aspect, Grangier and Roger at the University of
Paris experimentally confirmed that the
“preposterous” effect of the EPR Paradox, the
"spooky action-at-a-distance" is a physical
reality. All subsequent experiments established
the existence of non-locality as predicted by
Quantum Theory..
[iii]
In 1986, John G
Cramer of University of Washington presented his
Transactional Interpretation for Quantum
Mechanics.[iv]
In 1991,
Greenberger–Horne–Zeilinger (GHZ) sharpened
Bell's result by considering systems of three or
more particles and deriving an outright
contradiction among EPR's assumptions. They
showed a situation involving three particles
where after measuring two of the three, the
third becomes an actual test contrasting between
locality and the quantum picture: a local theory
predicts one value is inevitable for the third
particle, while quantum mechanics absolutely
predicts a different value. Bell-GHZ showed that
wave functions "collapse at a distance" as
surely as they do locally.[v]
