Quotes and Notes from Introduction to Quantum Mechanics: In Chemistry, Material Science, and Biology by S. M. Blinder
Einstein's
explanation of the photoelectric effect in 1905 appears trivially simple once
stated. (p. 8)
The structure of the level of kinetic energy of
electrons from photoelectric effects is the same as option payoff. For
electrons, the kinetic energy is zero when the energy of incoming lights is
below the threshold; after the threshold, the kinetic energy of the electron is
the difference between the energy of the light quanta and the threshold energy.
Similarly, the option payoff is zero when the stock price ends up below the
strike price; if stock price is above the strike price, the payoff is equal to
the difference between stock price and the strike price. No wonder Feynman's
mathematical work applies both to quantum field theory and the option theory.
I'll think more about it and try to bring a unified theory of physics, life and
social sciences.
For all its relevance, the quantum world differs quite dramatically from the world of everyday experience. To understand the modern theory of matter, challenging hurdles of both psychological and mathematical variety must be overcome. We are faced with the reality that the human brain, optimized for survival in subtropical forests and savannas, is simply not wired to deal with the conceptual environment of the subatomic world. (p. 17)
The best we can do is to describe the phenomena constituting the wave-particle duality. There is no widely accepted explanation in terms of everyday experience and common sense. Feynman referred to the “experiment with two holes” as the “central mystery of quantum mechanics.” It should be mentioned that a number of models have been proposed over the years to rationalize these quantum mysteries. Bohm proposed that there might exist hidden variables which would make the behavior of each photon deterministic, i.e., particle-like. Everett and Wheeler proposed the “many worlds interpretation of quantum mechanics” in which each random event causes the splitting of the entire universe into disconnected parallel universe in which each possibility becomes the reality. Needless to say, not many people are willing to accept such a metaphysically unwieldy view of reality. Most scientists are content to apply the highly successful computational mechanisms of quantum theory to their work, without worrying unduly about its philosophical underpinnings. (p. 23)
I have always been very curious about
the foundation problems in quantum mechanics. I have to think more about them.
Schrodinger in 1926 first proposed an equation for de Broglie’s matter waves. This equation cannot be derived from some othe principle since it constitutes a fundamental law of nature. Its correctness can be judged only by its subsequent agreement with observed phenomena (a posteriori proof). Nonetheless, we will attempt a heuristic argument to make the result at least plausible. (p. 24)
I read the Section 2.3. My main question
is that why the argument is so general as Schrodinger
equation is a very general physical law.
Think about the relation among
classic wave equations, which are second order in time direction, heat
equations, which are first order in time direction and Schrodinger equations,
which are first order in time direction but with an imaginary number attached.
Feynman-Kac formula takes the form of a heat
equation, not a wave equation. How come Feynman’s formulation of quantum
mechanics equivalent to Schrodinger equation, which is supposed to be a wave
equation? Read Feynman’s book on path integral and think about it.
Quantum entanglement --- a term introduced by Schrodinger
--- really happens! In drawing this conclusion we are actually glossing over a
number of still-unresolved hair-splitting metaphysical arguments. This
remarkable result is often summarized as
The local realistic model is violated by quantum mechanics.
Henry Stapp regards
Think more about it!
What are the main differences between quantum mechanics and classical mechanics? In quantum mechanics, it seems that additional restrictions are introduced by quantization of angular momentum. Can angular momentum be used in classical mechanics to determine, say, the orbits of planets?
In hydrogen, ground state is set from Bohr’s theory. How about other atoms? Can this quantum theory be applied to solar system as will to determine the orbits of planets? In solar systems, planets have different masses while in an atom, all electrons have identical mass and electric charge. But we may look for some more general patterns. We look for patterns in atoms because there are so many atoms but there is only one solar system. However, in the universe, there are billions of solar systems. But we cannot do much empirical test because we cannot observe planets of other stars.
How to determine E in Helium? In Hydrogen, it was determined by Bohr’s theory. Can the same be done in quantum mechanics? Work out all the calculations myself.
It seems obvious that density function has much more direct physical meaning than wave function. Why it takes so long to develop? What are the difficulties? Refer to Section 12.5.
Energy density in electromagnetic waves is represented by
rho(r,t) = fi(r,t)^2
This can be extended naturally to quantum mechanics wave function.
Even though the individual scintillations appear at random positions on the
screen, their statistical reproduces the original high-intensity diffraction
pattern. ... Evidently the statistical behavior of individual photons follows a
predicable pattern, even though the behavior of individual photons is
unpredictable. This implies that each individual photon, even though it behaves
mostly like a particle, somehow carries with it a "knowledge"
of the entire wave like diffraction pattern. In some sense, a single photon
must be able to go through both slit at the same time. This is what known as
the wave-particle duality for light: under appropriate circumstances light can
be either as a wave or as a particle. ... Feynman refers to the
"experiment with two holes" as "the central mystery of quantum
mechanics". (p. 22)
Each human particle seems free by itself. But together
they interact and influence each other. For example, there can be only one
president in one country, no matter how hard everyone else works. The ability
to develop a revolutionary theory is only up to the ability for the community
to accept such a theory. But in human cases, the interaction is more obvious.
The interaction of low density photon is less obvious. Think more about it.
Section
5.2 Quantum Harmonic Oscillator and 5.3. Harmonic-Oscillator Eigenfunctions
and Eigenvalues. These two sections
are amazing. By moving classical oscillator into Schrodinger equation, one
naturally get the quantized version. What is really
behind that? The ground state is the gaussian
distribution. So we could imagine a quantum system is a
ordinary system plus noise. Think more about it.
Section
5.5. Quantum Theory of radiation. In page 70, 71, The absorption probability is linear in nLamda,
which is proportional to the radiation density in the enclosure. By contrast,
the emission probability, varying as nLamda+1, is made up of two distinct
contributions. The part linear in nLamda is called
stimulated emission, while the part independent of nLamda
accounts for spontaneous emission. Remarkably, the probability for absorption
is exactly equal to that of stimulated emission. A detailed calculation gives
the following transition rates:
Wabs = Wstimem = ...
while
Wspont em = ...
... The
dependence on omiga^3 makes spontaneous emission significant only for
higher-energy radiation...
Comment: It
seems that this is a quantitative description of the second law. Definitely I
should read more about it. It seems that the diffusion rate increases as energy
level increases. The Spontaneous emission seems like the increase of entropy.
Think about it.
Maxwell believed that "... the effectual studies of the sciences must be
ones of simplification and reduction of the results of previous investigations
to a form in which the mind can grasp them." While "Clockwork
Universe" might be a metaphor which succinctly captures the essence of
classical physics, quantum mechanics lacks any such analog accessible to
everyday experience and common sense. (p. 265)
Comment:
Actually our economic theory provides such analog accessible to everyday
experience and common sense. Our theory shows that for a system to be viable,
fixed cost has to be larger than zero. For all organisms, the fixed costs are
significantly larger than zero for the sizes of even the simplest organisms are
of thousands of atoms. So all living organisms are truly
quantum. Our theory is not merely an analog. Living organisms live truly
from quantum phenomena. The Life cycle starts when plants receive a photon and
transform the quantum light energy into quantum chemical energy. The principle of quantum mechanics guide all living organisms.
It is not a coincidence that our economic theory is derived from a mathematical
method originally developed for quantum mechanics. I should go over the quantum
mechanics from the Feynman's method and see how we can integrate economics and
quantum mechanics into a single theory.
In
contemplating quantum mechanics, it is well to keep in mind the schema relating
appearance and reality central to the metaphysics of Immanuel Kant. Appearance
is determined by our observations and experiences, both external and internal.
Reality represents the ultimate causes of phenomena, which are forever hidden
from our perception. Theories are models we create in attempt to make
connections between appearance and reality. (p. 266)