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Mechanics

It is a branch of science dealing with the behaviour of physical bodies when subjected to forces

and displacements, and the effect of the bodies on their environment.

It has several 2 branches - 1.Classical Mechanics2. Quantum Mechanics

I. Classical Mechanics

It deals with(kinematics) and the mechanical behaviour of macroscopic bodies

To study classical mechanics we use 2 techniques - a. Continuum Mechanics

b. Statistical Mechanics

The mathematical formulations involve(positions, momentum evolve in space and time

by) -

Newtons mechanics

Lagrangian mechanics Hamiltonian Mechanics

The Physics remains the same in all three. It is just different ways of looking at the same

problem suited to what system you are looking. The last two are grouped into Analytical

Mechanics.

- For CM, Newtons mechanics is extensively used.

- For SM, Lagrangian and hamiltonian Mechanics is extensively used.

a. Continuum Mechanics(CM)- In here materials are modelled as a continuous mass rather

than as discrete particles. In here, an object is modelled in such a way, that the substance that

constitutes the object fills the space occupied by it. These techniques are used in fluid and solid

mechanics.1. Therefore it ignores the fact that matter is actually made up of atoms, which are discrete,

but at the length scales much larger than the interatomic distances( ~ 10 nm), this model

gives highly accurate results.

2. To use these models(i.e. to quantify it), we use Tensors. Tensors are mathematical

objects which are independent of the coordinate system in which the object is

represented. This is essential because CM deals with the physical properties of the

object which are independent of the coordinate system.

3. Scales of operation -

Microscale - QM

Mesoscale - Statistical Volume Element.Macroscale - Representative Volume Element

b. Statistical Mechanics(SM)- In here materials are treated as having discrete particles, due

to the length scales it operates in(~ 10nm). It is used to study the thermodynamic behaviour of

systems, by providing a framework for relating microscopic properties of individual atoms and

molecules with the bulk properties that can be observed in everyday life. And how does it do

that? - by utilizing the principles of probability theory.

http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensor

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1. It basically provides a molecular level interpretation of thermodynamical properties such

as - pressure, heat, work, entropy. The laws of Thermodynamics can be derived from

SM.

2. Objective - To calculate the distribution of a given amount of energy(E) over N particles

in a system.

3. Goal - To understand and interpret macroscopic properties of materials in terms of theproperties of their constituent particles and the interactions between them

4. How? - Connect Thermodynamic functions with quantum-mechanical

equations(this is essentially done via the 2nd law, where entropy connects the two

realms)

II. Quantum Mechanics

It deals with physical phenomena at the microscopic scale. It has action at the order of

Plancks constant(h).

It departs from classical mechanics in the realm of atomic and subatomic distances.

The mathematical formulation of Quantum Mechanics are abstract. It defines something

called a wave functionwhich describes the properties of matter, probabilistically. It

treats the object as a quantum harmonic oscillator, due to which it starkly departs from

classical mechanics. The main thing being -

At the ground state - CM - Object has zero energy(which make sense)

QM - Object has non-zero energy

The wavefunction is evolves in space and time by the Schrodinger equation.