Quantum Mechanics

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BSc Physics notes

Text of Quantum Mechanics

QUANTUM MECHANICSFailure of classical mechanics:

Classical mechanics (Newtons laws and electromagnetic theory) fail to explain the phenomenon like radioactivity, photoemission, x-rays etc. New ideas were introduced to explain the parade axis. The new formalism, used to explain the microscopic phenomenon is known as quantum mechanics.

1. Black body radiation:A perfectly black body is one which absorbs all the radiations incident on it and emits the same when it is heated to various temperatures. The distribution of energy in the spectrum of a black body is shown in the figure below.The following conclusions are drawn from figure.

(i) At a given temperature the energy is uniformly distributed.(ii) For a given temperature the intensity of radiation increases with wavelength reaches a maximum, for a particular wavelength beyond which intensity decreases.(iii) As the temperature increases maximum shifts towards shorter wavelength region.(iv) The area under each curve represents the total energy emitted.The classical law proposed by Wien is max = constant or Emax = constant where max is the wavelength corresponding to Emax or Imax. T is absolute temperature.

This law succeeded in explaining the shorter wavelength region of the black body spectrum. The law proposed by Rayleigh and Jeans is where K is Boltzmanns constant, T is absolute temperature, is the wavelength of the emitted radiation. These laws succeeded in explaining the longer wavelength region of the black body spectrum. None of the classical laws succeeded in explaining the u-v region of the black body spectrum, which is popularly known as ultra violet catastrophe.

Max plank introduced a new hypothesis to explain the entire region of the black body spectrum. He assumed the particles emitted are considered as (simple) harmonic oscillators and the energy associated with each photon (quanta) is E = h, where h is planks constant, is the frequency of emitted radiation.

2. Stability of the atom and atomic spectra:According to Rutherford the entire mass and positive charge of the atom is concentrated in a small region called the nucleus. The electrons revolve round the nucleus in orbits. The electrostatic force of attraction between electron and the nucleus is compensated by centrifugal force.

where Ze = charge on the nucleus, r = radius of orbit, e = electric charge

v = velocity of electron, permitivity of free space.

The orbital motion of the electron cause continuous emission of radiation. The above facts support the stability of the atom. The emission of energy is possible only if an electron is an accelerated charge, according to classical physics. Then the electron cannot stay in a circular orbit but it follows a spiral orbit of decreasing radius and finally falling to the nucleus. It is contradiction to the observed stability of the atom. According to classical mechanic an exited atom continuously emit radiations of all wavelengths. This is contradictory to the observed discrete spectra of Hydrogen atom. Bohr introduced Quantum ideas to explain the stability of the atom and the discrete spectral lines. According to him stability can be achieved by (angular momentum) where mass of electron, velocity of electron, radius of the atom, integer (number of energy levels), planks constant.

The spectral lines of hydrogen atom can be explained by using the relation where Rydbergs constant, and various energy state.

3. Photoelectric effect:According to classical ideas, radiation consists of oscillating electric and magnetic fields. The intensity of radiation is being proportional to . The force exerted on the electron is i.e., where is the electronic charge. Therefore the kinetic energy of the emitted electron should depend on the intensity of radiation. This is contradiction to the experimental facts. Also classical physics fails to explain the existence of threshold frequency and to explain the instantaneous nature of photo emission. All the above facts were successfully explained by Einstein using the relation

max = hh0 where max = Kinetic energy of an emitted electron, planks constant, = frequency of incident radiation, 0 = threshold frequency.

4. Compton scattering:According to classical theory when a beam of incident on matter. The scattered should possess the same wavelength as the incident one. The scattering constant is found to be independent of the incident radiation. The distribution of scattered intensity should be symmetrical. The experimental observations revels the following:

(i) The scattering constant was found to depend on the wavelength of the incident .(ii) The scattered radiation was found to consists of two wavelength. One same as that of incident wavelength and the other a longer wavelength.(iii) The distribution of scattered intensity is not symmetrical.5. Specific heat of Solids:Specific heat of a solid is defined as

It is the amount of heat required to raise the temperature of unit mass of a substance through one degree centigrade.

The classical physics predicts that all solids have the same and temperature independent molar specific heats (= 3R).

Dulong and Pettits law showed that at room temperature, Atomic weight Specific heat = 6.4 (Atomic specific heat). But in the case of diamond and silicon it is not true (For these two specific heat increases with temperature). The experiment showed specific heat decreases slowly with fall of temperature and finally tends to zero at absolute zero at 0 K. This is contradictory to Dulong and Pettits law.

The experimental observations were successfully explained quantum mechanically by Einstein.

deBroglie waves or matter waves:deBroglie suggested that particles like electrons, protons, neutrons etc exhibit dual nature i.e., material particle can behave both as wave as well as particle. He proposed the following assumptions.

(i) The Universe is made up of particles and radiations. These entities must be symmetrical.

(ii) Nature loves symmetry.

According to deBroglie, moving particle can be associated with a wave. The waves associated with (material) moving particle are known as deBroglie waves or matter waves.

Expression for deBroglie wave:

According to Quantum theory of radiation, the energy associated with a photon is given by

E = h where planks constant, = frequency of incident radiation

We know that =

wavelength, velocity of light.

The equation becomes,

.....................(1)

According to Einstein, mass and energy are inter-convertibles.

.....................(2)

where mass of the particle.

From equations (1) and (2)

where the momentum of photon.

If the material particle of mass moving with a velocity then the wavelength associated with the particle is given by

...................(3)

Different forms of deBroglie equations for wavelength:Consider a beam of electrons travelling through a potential difference , then the electrons acquire kinetic energy

where is the electronic charge.

Equation (3) becomes

According to kinetic theory of gases,

where Boltzmanns constant, absolute temperature

Equation (3) becomes

deBroglie wavelength for a particle moving a wave with very high speed or velocity. The mass of moving particle is given by

where mass of moving particle, rest mass of particle, velocity with which particle moving.

Equation (3) becomes,

Note: Wavelength associated with an electron of mass Kg

GP Thomsons experiment:

Construction and working:

High energy electrons produced by a cathode are passed through two perforated discs (D1, D2) to get a narrow beam and then passed through an aluminium cylinder which serves as an anode. While passing through the aluminium cylinder electrons get accelerated to very high velocities. This high energy electron beam is made incident on a gold foil, which act as a diffraction grating. The diffraction pattern is obtained on the screen.

Theory:

High energy electrons on passing through a thin poly crystalline gold foil undergo Braggs reflection and form a series of rings on the photographic plate (screen).

According to Braggs law

where order of the diffraction pattern, spacing between the layers where the atoms are accommodated. The incident beam AB passes through the film at B. BP is the reflected beam. Let OP= R & BO = .From figure , PBO= 2, where = glancing angle

P

R

A B O

For small angle,

..............(1)

From le BOP

Equation (1) becomes,

For the order

The wavelength calculated from this relation is in good agreement with wavelength calculated using the relation

DavissonGermer experiment:

Construction:

DavissonGermer experiment consists of an electron gun in which thermal electrons are produced by heating the filament using low tension battery. Two aluminium diaphragms D, D are used to collimate the electron beam, which is then accelerated by an anode (A). Electrons ejected from the electron gun are made to be incident on a single crystal Nickel (Ni) which can be rotated horizontally by means of a handle H. The scattered electrons are received by the ionisation chamber, which is connected to a sensitive galvanometer. T