Download Lecture Notes, Feb 29

Lecture IX ( Feb 29, 2016)
Review from Last Class
Wave Particle Duality
(1) Radiations have duality personality– can exhibit both wave and particle ( called photons)
characteristics...
This is based on Planck’s Theory of Black Body Radiation, Einstein’s theory of Photoelectric
effect and the Compton’s Effect
(2) Atomic Spectrum and Stability of Atoms led to Bohr model...
(3) This wave-particle duality is summarized by de Broglie theory:
h
h
, p=
p
λ
E
f = , E = hf
h
λ =
(1)
(2)
SOME QUESTIONS ....
(a) What is the quantum equivalent of Newton’s law, F = ma
(b) Bohr model explained the colors or the frequencies of the atomic spectrum, but how to
explain the intensities of the discrete lines – some colors are brighter than others and what
determines it ???
(c) Understand atoms with more than one electron.
Note H-atom has only one electrons,
He-atom has two electrons, Sodium has 13. How are they arranged in an atom?? Do they line up
in the same orbit ???
Quantum Mechanics
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Heisenberg and Schödinger independently developed two distinct but equivalent frameworks
to describe microscopic world that replaced Newton’s classical equations with new equations of
quantum theory. The new theory is called Quantum Mechanics. As we will see, the equation of
quantum mechanics is completely different from classical mechanics.
Schrödinger Equation — Equivalent of Newton’s Equation F = ma
In December 1925 while on vacation, Schro”odinger ( Univ of Zurich physics professor ),
looked at de Broglie’s thesis. He worked out a single equation, explaining the behavior of particles
in terms of de Broglie waves. The lead player in the equation is a quantity called Ψ ( pronounced
”sigh” ) which is called the wave function.
• Instead of describing particle by its position and velocity, in Schr”odinger’s equation, the
particle is described by wave function Ψ.
• Even in classical physics, waves are described by wave functions, which gives the amplitude,
wavelength and shape of the wave. However, Schr”odinger wave function is not a “real”
quantity. In other words, unlike water or sound or even electro-magnetic waves, matter
waves are not described by ordinary real numbers. Since it is not real , called complex
number, we cannot determine the shape of the wave from its wave function.
• Known as the Schrödinger equation (SE), this partial differential equation for a
non-relativistic particle of mass m in a potential V is given by
2
−~ 2
∂
∇ + V (r, t) Ψ(r, t)
i~ Ψ(r, t) =
∂t
2m
(3)
Here r in general represents all the spatial variables (x, y, z). i is a complex number equal to
√
−1. The function Ψ(r, t), known as the wave function of the particle, encodes complete
information about the state of the particle at a location r at time t.
• It was Max Born, who successfully interpreted the wave function Ψ as the probability
amplitude of the wave associated with the particle and was awarded the Nobel prize in 1954.
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Unlike classical wave functions, Ψ is a complex number and is not altogether a measurable
quantity. Therefore, unlike water waves, or waves in a string, or electromagnetic waves,
where the wave function is an observable entity describing oscillations of the medium or
electromagnetic fields, the wave function for a matter wave is an abstract quantity .
It is the absolute square , namely |Ψ(r, t)|2 , that is a physical entity describing the probability
of finding the particle at location r at time t. While the probability amplitude encodes all the
information about the state of the particle, taking the absolute value (the modulus) destroys
some information (called the phase). This subtle distinction is the ultimate source of all
quantum mechanical “weirdness”.
• Schrödinger equation determines Ψ and the energy and predicts the atomic orbits identical
to that of Bohr model...if we interpret the |Ψ|2 as the probability. It shows only certain
values of energy are permissible and have the form En = 13.6/n2 .
Heisenberg Formulation of Quantum Mechanics
Unlike SE, Heisenberg formulation was more complex and abstract and he himself did not
know how to use it. In trying to refine Bohr theory, he came across mathematical quantities called
matrices.. In this formulation, for example... there were weird things like: xp 6= px
In trying to understand atomic transitions ( quantum jumps), Heisenberg was interested in
a sort of bookkeeping method for all atomic transitions, and matrices are a natural way to do so.
Matrices are square array of numbers and in quantum mechanics, each entry represents a possible
atomic transition. Wolfgang Pauli took forty pages to calculate energy levels of H-atom using
Heisenberg theory.
SE and Heisenberg matrix theory were shown to be equivalent.
Heisenberg’s Uncertainty Principle:
Unlike classical physics, in quantum world uncertainties in the measurements of position and
momentum cannot be reduced to zero. Heisenberg was able to give his ideas precise mathematical
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4
form:
~
2
~
∆E∆t ≥
2
∆x∆p ≥
(4)
(5)
The idea of the position of an object seems so obvious that the concept of position is
generally taken for granted in classical physics. Knowing the position of a particle means
knowing the values of its coordinates in some coordinate system. The precision of those values, in
classical physics, is limited only by our skill in measuring. In quantum mechanics, the concept of
position differs fundamentally from this classical meaning. A particle’s position is summarized by
its wavefunction. To describe a particle at a given position in the language of quantum mechanics,
we would need to find a wavefunction that is extremely high near that position and zero elsewhere.
The wavefunction would resemble a very tall and very thin tower. None of the wavefunctions
we have seen so far look remotely like that. Nevertheless, we can construct a wavefunction that
approximates the classical description as precisely as we please.
Myths about the uncertainty principle
Heisenberg’s uncertainty principle is among the most widely misunderstood principles of
quantum physics. Non-physicists sometimes argue that it reveals a fundamental shortcoming in
science and poses a limitation to scientific knowledge. On the contrary, the uncertainty principle
is seminal to quantum measurement theory, and quantum measurements have achieved the highest
accuracy in all of science. It is important to appreciate that the uncertainty principle does not limit
the precision with which a physical property, for instance a transition frequency, can be measured.
What it does is to predict the scatter of results of a single measurement. By repeating the
measurements, the ultimate precision is limited only by the skill and patience of the experimenter.
Should there be any doubt about whether the uncertainty principle limits the power of
precision in physics, measurements made with the apparatus shown in Figure 24 (in web
lectures[1]) should put them to rest. The experiment confirmed the accuracy of a basic quantum
mechanical prediction to an accuracy of one part in 1012 , one of the most accurate tests of theory
in all of science.
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——————————————————————————–
Characters Needed for the Play... Choose your part before spring break
(1) Planck...
(2) Einstein...
(3) Compton
(4) Bohr .....
(5) de-Broglie...
(6) Schrödinger
(7) Heisenberg
(8) Bohm
(9) Pauli...
(10) Narrator...
(11) Chairman
[1] http://www.learner.org/courses/physics/
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