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 THE QUANTUM THEORY

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.

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,

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).

 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

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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