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De Broglie’s notion & Erwin Schrodinger

Late in November, 1925, Schrodinger gave a talk on de Broglie’s notion that a moving particle has a wave character.

A colleague remarked to him afterward that to deal properly with a wave, one needs a wave equation.

Schrodinger took this to heart, and a few weeks later he was struggling with a new atomic theory.

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Schrodinger & Paul Dirac, Nobel Prize in 1933

In January 1926 the paper on “Quantization as an Eigenvalue Problem” was published.

In this epochal paper Schrodinger introduced the equation that bears his name and solved it for the hydrogen atom.

By June Schrodinger had applied wave mechanics to the harmonic oscillator, the diatomic molecule, the hydrogen atom in an electric field, the absorption and emission of radiation, and the scattering of radiation by atoms and molecules.

The significance of Schrodinger’s work was at once realized. The Nobel Prize in Physics 1933 was awarded jointly to Erwin Schrodinger and Paul Adrien Maurice Dirac “for the discovery of new productive forms of atomic theory.”

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SCHRODINGER’S EQUATION:TIME-DEPENDENT FORM

We assume that for a particle moving freely in the +x direction is given by

(2)

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Differentiating Eq. (2) twice with respect to x,which gives

Differentiating Eq. (2) once with respect to t gives

(4)

(3)

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At speeds small compared with that of light, the total energy E of a particle is the sum of its kinetic energy P²/2m and its potential energy U(x,t):��E = P²/2m + U(x,t) (5)�Multiplying both sides of Eq. (5) by the wave function �

(6)

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Now Substitute for from Eq. (3) & (4) to obtain the time dependant form of Schrodinger equation

(7)

In three dimensions the time-dependent form of Schrodinger equation is

(8)

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SCHRODINGER’S EQUATION: STEADY-STATE FORM

One-dimensional wave function of an unrestricted particle may be written

(10)

(9)

Substituting the wave function of Eq. (9) into the time dependant form of Schrodinger equation, we find that

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�� (11)

In three dimensions it is

(12)

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Time-independent Schrodinger equation

Equation (12) can be written

(13)

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Standing waves in a stretched string fastened at both ends.

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REFERENCES:

Inroduction to Quantum Mechanics (2^nd Edition): David J. Griffiths.

(2) Concepts of MODERN PHYSICS (6^th Edition) :ARTHUR BEISER.

(3) THEORY AND PROBLEMS OF ADVANCED MATHEMATICS FOR

ENGIGINEERS & SCIENTISTS

[SI (METRIC) EDITION] : MURRAY R. SPIEGEL.

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