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THE
QUANTUM THEORY
The Quantum theory, originally formulated by Max Planck, was
used in 1913 by Neils Henrik David Bohr, one
of the greatest physicists of the 19th century
and a great profounder of quantum idea. He
devised the so-called Bohr’s model, where he assumed
the quantization of angular momentum. One of
the most important figures in the quantum world
was Werner Karl Heisenberg (Nobel Prize 1932).
Bohr and Heisenberg had a strong mentor/protégée
relationship. While analyzing the data of some
atomic spectra, he found a trouble with the
original theories. Then, by suggestion of Pauli
(often
a very critical one), he derived the so-called
matrix mechanics, which was further developed
with the assistance of Max Born, Pauli and
Pascal Jordan. |
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When Heisenberg presented his idea to Bohr, he rejected it
first. But, after considering its importance, he
joined the group of new thinkers. In 1925, he
completed his work. Then, in 1926, Erwin Schrodinger
formulated the very equivalent wave mechanics. It was
Dirac who, during the same time, presented a unified
and more general bra-ket formulation of quantum
mechanics. We start with Schrodinger approach.
For a wave
y
= A0sin(2px/l),
the wave equation is d2y/dx2
+ 4p2y/l2
= 0.
Now, from de-Broglie hypothesis,
l
= h/p and energy relation,
p2 = 2m (E-U),
the equation takes the form
:
In general, we replace the double derivative by Laplacian
operator. So, we have, 
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Now, for dependence on time, we have the equation:
(iћ)(dy/dt)
= Ey.
This directly yields plane wave solution
y(x,t)
=
y(x)
e-iwt, where
w
= 2pE/h.
This is the Hamiltonian form of wave equation.
Let U = 0. So, we have,
Ey
=
-(ћ2/2m)(d2y/dx2).
The quantity
-(h2/8p2m)(d2y/dx2)
is called the energy operator. It operates on
y
to give the energy of the system. You can get the momentum
operator from here, p2 = 2mE (bold
symbols represent operators). So, p2 =
-(h2/4p2)(d2/dx2).
So, we have,
p = (iћ)(d/dx).
The expression for p is called the momentum operator. p2
implies applying p twice.
For quantum mechanics, the position operator is simply x.
We define a term, mean value of observable (something that
is being observed),
áxñ
=
ò y*xyd3x (where
y* is a complex conjugate of the wave function).
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This definition comes from Statistics and the complex conjugate
is taken to remove ‘i’ from the equation. One
does not want a complex mean value. There is
another quantity, mean squared value áx2ñ = òy*x2ydx.
Now, uncertainty is defined as sx2 = áx2ñ - áxñ2.
There is a relation called Cauchy-Schwarz inequality,
which applies to various spaces. In quantum terms,
it says,
sx2sy2 ³ (1/4)½[òy*[xy - yx] ydx]2½
(the
terms inside the innermost square brackets do
not cancel each other as they are operators).
If x is the momentum operator, y is
the position operator.
[xy - yx] y =
(iћ)[(d/dx)xy - xdy/dx)]
=
(iћ)y |
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The operators discussed above are called Hermitian operators
and the result of Schwarz inequality follows for
these operators. From here, uncertainty relation sxsp > h/4p follows.
To prove it, you need the relation òy*y dx
= 1. This is called normalization relation and every
wave function requires this. All the above ideas
are statistical and have come from Max Born’s interpretation
of wave function as probability amplitude. The quantity y*ydx
is called the probability of observing wave in the
region dx as a function of x. Thus, òy*y dx
= 1. Total probability is 1.
The idea of statistical interpretation has a very direct link
to the very famous and the most fundamental experiment - Double
Slit Experiment. Read this account for information
in this site: http://en.wikipedia.org/wiki/Double-slit_experiment (click
here). ‘A particle has wave nature’ implies that
it can go through any of the two slits, with some
probability and the net probability is defined according
to Born’s interpretation. |
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