particle in a box- application of schrodinger wave equation

QUANTUM CHEMISTRY

Presented By:-Saurav K. Rawat

Department of Chemistry,St. John’s College, Agra

Introductory Quantum Mechanics

Bohr's Atom

Heisenberg

TranslationalVibrationalRotationalSpectroscopy (NM R)

Applications

Operator Algebra Postulates

Quantum T ools

W ave vs. Partic le De Brogile 's Hypothesis

Historical

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E = m c2

EH^

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Genealogy of Quantum Mechanics

Classical Mechanics(Newton)

Wave Theory of Light(Huygens)

Maxwell’sEM Theory

Electricity and Magnetism(Faraday, Ampere, et al.)

Relativity Quantum Theory

Quantum Electrodyamics

High

Velocity

Low

Mass

Energy and Matter

Size of Matter Particle Property Wave Property

Large – macroscopic Mainly Unobservable

Intermediate – electron Some Some

Small – photon Few Mainly

E = m c2

The Wave Nature of Light

c hE The speed of light is constant!

Classical Mechanics is based on the Newton’s Law of Motion – describes the dynamic proportion of the macroscopic world . It failed to describe the behavior of particles at atomic scale . The concept of quantum • The experiments of Young and Fresnal on light showed the latter behaved as waves.• But with Planck’s Quantum , Einstein's Photon and Bohr’s atom it confirmed by 1920 that despite of the wave like properties of light (interference and diffraction), when it came to transfer of energy and momentum light behaved like a particle . This led to the concept of Quantum which means a bundle or unit of any form of Physical Energy such as Photon which represents a discrete amount of electromagnetic radiant energy •In 1924 de Broglie made a formulation that particle behaves like waves

λ=h/p, where p is the momentum of the particle and Λ is the wave length.•All particles have a wave characteristics where they are moving with a moving momentum•The macroscopic objects which have a large mass have a wave with very small wave length •CONCLUSION:- I. The particle and wave aspects of electromagnetic

radiations .II. The wave aspect of the particle allows the calculation of

the probability of locating the particleIII. The prediction of the locations of Photons and sub-atomic

particle like electron , neutron , etc, probabilistic IV. The probability is given by |E(r,t)|2

THE NEED OF NEW MECHANICS FOR SUB-ATOMIC PARTICLES:-The concept of continuous energy absorption ( classical mechanics) and emission was in conflict with atomic and sub atomic phenomena ( black body radiation, photo electric effect, Compton effect ,diffraction of electron and atomic spectra of hydrogen)The explanation led to the new mechanics called quantum mechanics SCHRODINGER EQUATION (characteristics of Ψ )Ψ should be single valued Ψ should be continuousΨ should finish for a bound state

APPLICATIONS OF SCHRODINGER EQUATION

•PARTICLE IN A BOX

•Hydrogen atom

•Rigid rotator •Simple Harmonic Oscillator

Particle in a Box (1D) - Interpretations

● Plots of Wavefunctions

● Plots of Squares of Wavefunctions

● Check Normalizations

● How fast is the particle moving? Comparison of macroscopic versus microscopic particles.

Calculate v(min) of an electron in a 20-Angstrom box.

Calculate v(min) of a 1 g mass in a 1 cm-box

axn

ansin2

2

22

8 amhnEn

10

2 a

dx

V=0V=α V=α

Region -I Region-II Region -III

x = 0 x = a

Free particle – P.E. is same everywhere, i.e. V=0

Potential box – P.E. is 0 within the closed region and infinite (i.e. V=α) everywhere else

Particle in a Box

Region-II, V=0

For one dimensional box-

(1)

(2)

(3)

(4)

(5)

Schrodinger Equation-

Solution of Equation- Ψ= A cos kx + B sin kx

• Region I + II• Ψ=0, V=α• At, x=0 Ψ=0 from -• 0= A cos 0 + B sin 0 • A=0 • in

• Ψ= B sin kx (Ψ=0, x=0, x=a)• B sin kx=0, B sin ka=0

Sin ka=0, ka=nπ, k= nπ/a• n=0,1,1,3…….. allowed solution.• n=1,2,3……….. acceptable solution.

(3)

(6)

(7)

(8)

(9)

(8) (6)

(6)

• Ψ= Ψn= B sin nπx/a ; n=1,2,3,… • Wave Function for particle in a box-• From (5) and (9)• E= n2h2/ 8ma2

• E depends on quantum no. which can have integral value, the energy levels of the particle in a box are quantized.

(10)

Normalisation of ψ-

Normalisation Constant

• The solution of Schrödinger equation for a particle in a one dimensional box-

• Ψ= √2/a sin(nπx)/a• En= n2h2/8ma2 n=1,2,3• The particle will have certain discrete values of

energy, so discrete energy levels. Hence energy of the particle is quantized. These values, E depend upon n which are independent of x. These are called Eigen values. So a free particle can have all values of energy but when it is confined within a certain range of space, the energy values become quantized.

• n=1, E1=h2/8ma2

• n=2, E2=4h2/8ma2

Emin= h2/ma2

• Zero point energy (ZPE)- When the particle is present in the potential box, the energy of the lowest level n=1 is called zero potential energy.

• Eigen Function• n=1 Ψ1=√2/a sin[ πx/a]• n=2 Ψ2=√2/a sin[2 πx/a]• n=3 Ψ3 ==√2/a sin[3 πx/a]

Nodes- The points were the probability of finding the particle is zero in the particle wave.

(n -1) nodes • Greater the number of nodes,

more the curvature in the particle wave. For a potential box of fixed size, as the curvature in the wave function increases the number of nodes increases, the wavelength decreases and the total energy in the box, P.E.(V) has been assumed to be zero.

Ψ- Wave Function Ψ2 – Probability Function

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