lecture 1 in quantum mechanics

8/13/2019 Lecture 1 in Quantum Mechanics

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Module 1 : Quantum Mechanics

Chapter 1 : Prelude to Quantum theory

Prelude to Quantum theory

Nineteenth century saw the climax of the achievements of classical physics: laws of mechanics,electromagnetism, and statistical mechanics provide a very good description of many macroscopicproperties.

(a) Newtonian mechanics: The dynamics of a particle of mass is described by Newton's second law

(1.1)

where is the force acting on ith particle due to jth particle. Newton's second law that action and

reaction are equal and opposite leads to the conservation of momentum,

(1.2)

The isotropy of space, or being proportional to leads to the conservation of angular

momentum,

(1.3)

One also has the result that for conservative forces for which work done on a closed path vanishes, the totalenergy is constant,

(1.4)

For gravitational interaction, one has

(1.5)

The force proportional to is one of the most beautiful forms of interaction. Going over to the centre of

mass frame,

(1.6)

one gets


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(1.7)

This equation provides a very good description of the dynamics of planets and satellites. In particular we

note that Runge-Lenz vector is a constant,

(1.8)

Taking a scalar product with , it leads to

(1.9)

where is the total energy, which implies that we have closed orbits for bound planets with

negative . For the source of gravitational force being a mass distribution with mass density

, one gets for the gravitational potential

(1.10)

For we can expand in powers of to obtain

(1.11)

This leads to

(1.12)


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which may be regarded as a multi-pole expansion of the potential. For an ellipsoidal earth withrotational symmetry,

(1.13)

one obtains for the potential in Eq.(1.11),

(1.14)

correct to order , where we have included the term coming from the centrifugal force.Taking the potential to be constant on the surface, it leads to the results

(1.15)

for the bulge on the surface of the earth and the difference in the gravitational acceleration at and

. It may also be noted that the earth has an extra observed bulge of about 80 m in the southern

hemisphere.

(b) Electromagnetic fields and interaction: The Maxwell's equations govern the properties ofelectromagnetic fields. They are

(1.16)

The Lorentz force is (in MKS and CGS units),

(1.17)

The fields are expressed in terms of scalar and vector potentials, as


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(1.18)

or MKS and CGS units.

(c) Statistical mechanics:Supplement the basic laws of mechanics and electromagnetism, with statisticalmechanics to describe the properties of macroscopic bodies. Statistical mechanics is the underlyingmicroscopic theory of thermodynamics and kinetic theory.

The situation appeared to be very satisfying. However, there were some irritations and more irritationsstarted coming in very rapidly around the turn of the century. One was the non-invariance of ofelectromagnetic equations under Galilean transformations. Related problem was the invariance of the speedof light in different inertial frames. Another was the observation of discrete lines in atomic spectra. Thenthere was a deluge of new observations requiring fundamental changes in our ideas about the physicalworld. Though the classical laws continued to be useful, they are to be regarded as certain approximationsto the more general fundamental laws of nature.

lecture 1 in quantum mechanics

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