QUANTUM MECHANICS

WAVES: A wave is nothing but disturbance which is occurred in a medium and it is specified by its frequency, wavelength, phase, amplitude and intensity. PARTICLES: A particle or matter has mass and it is located at a some definite point and it is specified by its mass, velocity, momentum and energy.

Waves and Particles : What do we mean by them? Material Objects: Ball, Car, person, or point like objects called particles. They can be located at a space point at a given time. They can be at rest, moving or accelerating. Falling Ball Ground level

Waves and Particles: What do we mean by them ? Common types of waves: Ripples, surf, ocean waves, sound waves, radio waves. Need to see crests and troughs to define them. Waves are oscillations in space and time. Direction of travel, velocity Up-down oscillations Wavelength ,frequency, velocity and oscillation size defines waves

Particles and Waves: Basic difference in behaviour When particles collide they cannot pass through each other ! They can bounce or they can shatter Before collision After collision Another after collision state shatter

Collision of truck with ladder on top with a Car at rest ! Note the ladder continue its Motion forward ….. Also the small care front End gets smashed.

Head on collision of a car and truck Collision is inelastic – the small car is dragged along By the truck……

Waves can pass through each other ! Waves and Particles Basic difference: Waves can pass through each other ! As they pass through each other they can enhance or cancel each other Later they regain their original form !

Waves and Particles: Spread in space and time Waves Wavelength Frequency Spread in space and time Waves Can be superposed – show interference effects Pass through each other Localized in space and time Particles Cannot pass through each other - they bounce or shatter.

A SUMMARY OF DUAL ITY OF NATURE Wave particle duality of physical objects LIGHT Wave nature -EM wave Particle nature -photons Optical microscope Interference Convert light to electric current Photo-electric effect PARTICLES Wave nature Particle nature Matter waves -electron microscope Electric current photon-electron collisions Discrete (Quantum) states of confined systems, such as atoms.

The physical values or motion of a macroscopic particles can be observed directly. Classical mechanics can be applied to explain that motion. But when we consider the motion of Microscopic particles such as electrons, protons……etc., classical mechanics fails to explain that motion. Quantum mechanics deals with motion of microscopic particles or quantum particles.

Classical world is Deterministic: Knowing the position and velocity of all objects at a particular time Future can be predicted using known laws of force and Newton's laws of motion. Quantum World is Probabilistic: Impossible to know position and velocity with certainty at a given time. Only probability of future state can be predicted using known laws of force and equations of quantum mechanics. Tied together Observed Observer

Wave Nature of Matter Louis de Broglie in 1923 proposed that matter particles should exhibit wave properties just as light waves exhibited particle properties. These waves have very small wavelengths in most situations so that their presence was difficult to observe These waves were observed a few years later by Davisson and G.P. Thomson with high energy electrons. These electrons show the same pattern when scattered from crystals as X-rays of similar wave lengths. Electron microscope picture of a fly

de Broglie hypothesis: In 1924 the scientist named de Broglie introduced electromagnetic waves behaves like particles, and the particles like electrons behave like waves called matter waves. He derived an expression for the wavelength of matter waves on the analogy of radiation.

According to Planck’s radiation law Where ‘c’ is a velocity of light and ‘λ‘is a wave length. According to Einstein mass-energy relation From 1 & 2 Where p is momentum of a photon.

The above relation is called de Broglile’ s matter wave equation The above relation is called de Broglile’ s matter wave equation. This equation is applicable to all atomic particles. If E is kinetic energy of a particle Hence the de Broglie’s wave length

de Broglie wavelength associated with electrons: Let us consider the case of an electron of rest mass m0 and charge ‘ e ‘ being accelerated by a potential V volts. If ‘v ‘ is the velocity attained by the electron due to acceleration The de Broglie wavelength

Characteristics of Matter waves: Lighter the particle, greater is the wavelength associated with it. Lesser the velocity of the particle, longer the wavelength associated with it. For v = 0, λ=∞ . This means that only with moving particle matter wave is associated. Whether the particle is charged or not, matter wave is associated with it. This reveals that these waves are not electromagnetic but a new kind of waves .

It can be proved that the matter waves travel faster than light. We know that The wave velocity (ω) is given by Substituting for λ we get As the particle velocity v cannot exceed velocity of light c, ω is greater than velocity of light.

Experimental evidence for matter waves There was two experimental evidences Davisson and Germer ’s experiment. 2. G.P. Thomson Experiment.

DAVISON & GERMER’S EXPERMENT: Davison and Germer first detected electron waves in 1927. They have also measured de Broglie wave lengths of slow electrons by using diffraction methods. Principle: Based on the concept of wave nature of matter fast moving electrons behave like waves. Hence accelerated electron beam can be used for diffraction studies in crystals.

filament High voltage Target Anode cathode G cylinder S Circular scale Faraday cylinder Galvanometer Nickel Target Circular scale cathode Anode filament G S c

Experimental arrangement: The electron gun G produces a fine beam of electrons. It consists of a heated filament F, which emits electrons due to thermo ionic emission. The accelerated electron beam of electrons are incident on a nickel plate, called target T.

The target crystal can be rotated about an axis parallel to the direction of incident electron beam. The distribution of electrons is measured by using a detector called faraday cylinder c and which is moving along a graduated circular scale S. A sensitive galvanometer connected to the detector.

Results: When the electron beam accelerated by 54 volts was directed to strike the nickel crystal, a sharp maximum in the electron distribution occurred at an angle of 500 with the incident beam. For that incident beam the diffracted angle becomes 650 . For a nickel crystal the inter planer separation d = 0.091nm.

250 650 Incident electron beam Diffracted I θ V = 54v 500 C U R E N T

According to Bragg’s law For a 54 volts , the de Broglie wave length associated with the electron is given by This is in excellent agreement with the experimental value. The Davison - Germer experiment provides a direct verification of de Broglie hypothesis of the wave nature of moving particle.

G.P THOMSON’S EXPERIMENT: G.P Thomson's experiment proved that the diffraction pattern observed was due to electrons but not due to electromagnetic radiation produced by the fast moving charged particles. EXPERIMENTAL ARRANGEMENT: G.P Thomson experimental arrangement consists of (a) Filament or cathode C. (b) Gold foil or gold plate (c) Photographic plate (p) (d) Anode A. The whole apparatus is kept highly evacuated discharge tube.

G.P THOMSON EXPERIMENT: Discharge tube cathode Anode Gold foil Vacuum pump slit Photo graphic plate

Photographic film Diffraction pattern.

c Tan 2θ = R / L If θ is very small 2θ = R / L 2θ = R / L ………. (1) R Incident electron beam Gold foil radius θ B c θ A o θ L Brage plane Tan 2θ = R / L If θ is very small 2θ = R / L 2θ = R / L ………. (1)

When we apply potential to cathode, the electrons are emitted and those are further accelerated by anode. When these electrons incident on a gold foil, those are diffracted, and resulting diffraction pattern getting on photographic film. After developing the photographic plate a symmetrical pattern consisting of concentric rings about a central spot is obtained.

According to Braggs law

According to de Broglie’ s wave equation

The value of ‘d’ so obtained agreed well with the values using X-ray techniques. In the case of gold foil the values of “d” obtained by the x-ray diffraction method is 4.060A.

Heisenberg uncertainty principle: This principle states that the product of uncertainties in determining the both position and momentum of particle is approximately equal to h / 4Π. where Δx is the uncertainty in determine the position and Δp is the uncertainty in determining momentum. This relation shows that it is impossible to determine simultaneously both the position and momentum of the particle accurately.

This relation is universal and holds for all canonically conjugate physical quantities like 1. Angular momentum & angle 2. Time & energy

Consequences of uncertainty principle: Explanation for absence of electrons in the nucleus. Existence of protons and neutrons inside nucleus. Uncertainty in the frequency of light emitted by an atom. Energy of an electron in an atom.

Physical significance of the wave function: The wave function ‘Ψ’ has no direct physical meaning. It is a complex quantity representing the variation of a Matter wave. The wave function Ψ( r, t ) describes the position of a particle with respect to time. It can be considered as ‘probability amplitude’ since it is used to find the location of the particle.

ΨΨ* or ׀Ψ׀2 is the probability density function. ΨΨ* dx dy dz gives the probability of finding the electron in the region of space between x and x + dx, y and y + dy, z and z + dz. The above relation shows that’s a ‘normalization condition’ of particle.

Schrödinger time independent wave equation: Schrödinger wave equation is a basic principle of a fundamental Quantum mechanics. This equation arrives at the equation stating with de Broglie’s idea of Matter wave. According to de Broglie, a particle of mass ‘m’ and moving with velocity ‘v’ has a wavelength ‘λ’.

According to classical physics, the displacement for a moving wave along X-direction is given by Where ‘A’ is a amplitude ‘x’ is a position co-ordinate and ‘λ’ is a wave length. The displacement of de Broglie wave associated with a moving wave along X-direction is given by

If ‘E’ is total energy of the system E = K.E + P.E ---------(4) If ‘p’ is a momentum of a particle K.E = ½ mv2 = p2 / 2m According to de Broglie’ s principle λ = h / p p = h / λ K.E = p2 / 2m The total energy Where ‘V’ is potential energy.

Periodic changes in ‘Ψ’ are responsible for the wave nature of a moving particle

This is Schrödinger time independent wave equation in one dimension. In three dimensional way it becomes…..

Particle in a one dimensional potential box: Consider an electron of mass ‘m’ in an infinitely deep one-dimensional potential box with a width of a ‘ L’ units in which potential is constant and zero. X=0 X=L V=0

V + X Nuclei Periodic positive ion cores Inside metallic crystals. One dimensional periodic potential in crystal.

The motion of the electron in one dimensional box can be described by the Schrödinger's equation. Inside the box the potential V =0

To apply boundary conditions. The solution to equation (3) can be written as Where A,B and K are unknown constants and to calculate them, it is necessary To apply boundary conditions.

When x = 0 then Ψ = 0 i.e. |Ψ|2 = 0 ------1 x = L Ψ = 0 i.e. |Ψ|2 = 0 ------2 Applying boundary condition (1) to equation (4) A sin k(0) + B cos K(0) = 0 B = 0 Substitute B value equation (4) Ψ(x) = A sin kx

Applying second boundary condition for equation (4) Substitute k value in equation (5) To calculate unknown constant A, consider normalization condition.

Normalization condition

The normalized wave function is The wave functions Ψn and the corresponding energies En which are often called Eigen functions and Eigen values, describe the quantum state of the particle.

The electron wave functions ‘Ψn’ and the corresponding probability density functions ‘ |Ψn|2 ’ for the ground and first two excited states of an electron in a potential well are shown in figure.

X=0 X=L E2=4h2/8mL2 E1=h2 / 8mL2 E3=9h2 / 8mL2 n = 1 n = 2 n = 3 L / 2 L / 3 2L / 3 √ (2 / L)

Conclusions; 1.The three integers n1,n2 and n3 called quantum numbers are required to specify completely each energy state. since for a particle inside the box, ‘ Ψ ’ cannot be zero, no quantum number can be zero. 2.The energy ‘ E ’ depends on the sum of the squares of the quantum numbers n1,n2 and n3 and no on their individual values. 3.Several combinations of the three quantum numbers may give different wave functions, but of the same energy value. such states and energy levels are said to be degenerate.

A localized wave or wave packet: A moving particle in quantum theory Spread in position Spread in momentum Superposition of waves of different wavelengths to make a packet Narrower the packet , more the spread in momentum Basis of Uncertainty Principle

ILLUSTRATION OF MEASUREMENT OF ELECTRON POSITION Act of measurement influences the electron -gives it a kick and it is no longer where it was ! Essence of uncertainty principle.