January 15, 2019 Tuesday
Bedtime Story
Eigenstate
Tonight we shall continue with some basic
tenets of the Copenhagen Interpretation of Quantum Mechanics from where we had
left it last night.
One of them was that any observation with
respect to the wave function is an act of its disturbance.
This means that any man-made device used
for observing a wave function necessarily interacts with the system.
When such a man-made device records any
measurement the wave function of the system is said to collapse.
The technical mathematical term that was
introduced here in quantum mechanics is the “eigenstate”.
The word eigenstate is derived from the
German/Dutch word “eigen” which means “inherent” or “characteristic”.
The idea of eigenstate had to be introduced
in the quantum mechanics precisely to differentiate between the two states we
have just described, the state and wave function of a system before its
measurement and after.
So when an electron around an atom is
described in the form of a probability cloud then it lacks any eigen value.
But once after observation it has been
“pinned down” in some respect then it is said to possess an eigen state.
If the eigen state can be prescribed
certain number to it then it will possess an eigenvalue.
Again it must be stressed that the term
eigenvalue arose in mathematics in the study of polynomials and differential
equations and was much latter borrowed by men such as Heisenberg and Bohr.
This is one thing that has very much
surprised the physicists about the uselessness of mathematical knowledge;
mathematics that was often done for no good reason finds itself useful later in
some theoretical physics.
This is the reason why one group of men considers
nature to be mathematical or perhaps computational.
This view is countered by the other group
who say that all we have is mathematics to apply to the understanding of the
universe and nothing better and that is why nature appears to be
mathematical.
So a collapse of a wave function during an
observation can be stated as an irreversible reduction of a wave function to an
eigenstate of an observable that is registered.
It was stressed by Bohr that the results
obtained by such laboratory measuring devices are essentially classical.
In contrast to this, the wave function is
probabilistic and this statement is called the Born Rule.
Born Rule is a law of quantum mechanics
that was stated in 1926 by Max Born as: the probability density of finding a
particle at a given point is proportional to the square of the magnitude of the
particle’s wave function at that point.
Mathematically it can be written as
p(x, y, z) = [
(x,y,z,t0)]2
Stay tuned to the voice of an
average story storytelling chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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