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- Wave-Matter Duality of Radiation
- De-Broglie Wavelength of Relativistic Particle
- Calculation of De-Broglie Wavelength of Neutrino, Diatomic Gas and Poly-Atomic Gas
- Superposition of Plane Waves
- Particles Wave Packet Spreads in Time
- Heisenberg's Uncertainty Principle and its Application
- Application of Heisenberg's Uncertainty Principle
- Need of Quantum Theory (Black Body Radiation)
- Inadequacies of Classical Mechanics
- Derivation for Group Velocity
- Relation between Group Velocity and Particle Velocity

- Step of Normalization
- Equation of Continuity in Quantum Mechanics
- Expectation Value of Dynamical Quantity
- Development of Time Dependent and Independent Schrodinger Equation
- Eigen Value Equation
- Importance of Normalization
- Solution of Schrodinger Equation
- Orthogonality of Wavefunction and Eigen Value
- Example of Normalization

- Commutation Relation between Different Components
- Introduction of Operator and Commutation Relation
- Commutation Relation between Position and Linear Momentum
- Square of Angular Momentum Operator, Ladder Operator
- Square of Angular Momentum Operator
- Hermitian Operators
- Equation of Motion in Quantum Mechanics
- Physical Significance of Ehrenfest Theorem and Properties of Parity Operator
- Properties of Hermitian Operator
- Properties of Parity Operator and Hermitian Operator
- Proof Associated with Hermitian Operator

- Heisenberg's Uncertainty Principle
- Probability of Finding One Dimensional Simple Harmonic Oscillator
- Energy Eigen Value of One-Dimensional Simple Harmonic Oscillator
- Characteristics of Energy Eigen Value
- Normalized First Four Wave Function for Simple Harmonic Oscillator
- Differential Equation
- Central Potential Problem
- Energy Eigen Value of Rigid Rotator
- Schrodinger Time-Independent Equation in Spherical Polar Co-ordinate System

In certain experimental observation radiation behaves as wave, i.e it has certain wavelength, frequency and energy.

For example: Experiments associated with

1. Interference

2. Diffraction

3. Polarization etc

In some experimental observation radiation behaves as a stream of particle ( i.e photon ) having dynamic mass, linear momentum and energy.

For example:

1 Compton effect

2. Photoelectric effect

3. Discrete emission and absorption of radiation by black body etc.

The some entity (radiation ) behaves as wave as well as particle. This nature of radiation is known as wave-matter duality of radiation.

According to Plank's theory of radiation energy of photon of frequency \(\nu\) is given by:

$$E= h\nu\dotsm(1)$$

Where, h= Planck's Constant (\( 6.626\times 10^{-34} JS\))

According to Einstein mass energy equivalent relation:

$$E= mc^2\dotsm(2)$$

Where,

m= dynamic mass of photon

c= Speed of photon in free space

From equation (1) and (2)

$$h\nu=mc^2$$

$$or\;\; h\frac{c}{\lambda}= mc^2$$

$$or,\;\; \frac{h}{\lambda}= mc$$

$$\lambda= \frac{h}{mc}= \frac{h}{p}\dotsm(3)$$

Equation (3) connects the wave and particle nature of radiation.

de-Broglie proposed that a moving material particle also has dual nature. The wave associated with moving particle is known as matter- wave or de-Broglie wave.

For a particle of mass m moving with velocity ( Speed ) v is also associated with wave of wavelength \(\lambda\).

$$\lambda=\frac{h}{mv}=\frac{h}{p}$$

Where, p is a linear momentum of moving particle.

A particle which has kinetic energy only is known as free particle. It's potential energy is zero.

\(\therefore\) Total energy of free particle is given by

$$E= K.E$$

$$=\frac12 mv^2$$

Where,

m= mass of particle

v= speed of particle

$$\frac{ m^2 v^2}{2m}$$

$$= \frac{p^2}{2m}$$

$$\therefore \;\; p \sqrt{2mE}\dotsm(1)$$

The de-Broglie wavelength of free particle is

$$\lambda = \frac{h}{p}$$

$$=\frac{h}{\sqrt{2mE}}\dotsm(2)$$

$$\Rightarrow \lambda\propto\frac{1}{\sqrt{m}}$$

$$\lambda\propto \frac{1}{\sqrt{E}}$$

the measurement of dde-Broglie wavelength of matter wave is significant only if value of \(\lambda\) is comparable to linear dimension of system (particle itself or region where particle is located ).

For example:

Particle of mass (ice) = 1kg

Speed of ice= 10m/s

$$\therefore \lambda= \frac{h}{mv}= \frac{6.62\times 10^{-34}}{1\times 100}$$

$$= 6.62\times 10^{-36} m$$

\(v= d^3\;, \frac{m}{v} =\rho\)

\(m=\rho v\)

\(1=1000\times d^3\)

\(d= \biggl(\frac{1}{1000}\biggr)^{\frac13}\)

\(d= 0.1 m = 10\; cm= 1\times 10^{-1} m\)

\(\lambda <<< d \) So calculation of \(\lambda\) is not important.

Example 2:

de-Broglie wavelenth of electron in 1st orbit of H-atom

Speed of electron \((V_1)=\frac{c}{137}=\frac{3\times 10^8 m/s}{137}\)

$$=2.18\times 10^6m/s<<c$$

$$\lambda=\frac{h}{mv}$$

$$=\frac{6.625\times 10^{-34}}{9.1\times 10^{-31}\times 2.18\times 10^6}= 3.15\times 10^{-10}m= 3.15 A^0$$

Radius of first orbit= \(0.529\times 10^{-10}m= r\)

Circumference =\( 2\pi\times r= 3.32A^0\)

In this case the de-Broglie wave length is comparable to circumference of electric orbit. Hence the measurement of de-Broglie wave length is significant.

de-Broglie wavelength of bounded (non-relativistic particle)

If a particle is moving inside force field it is known as bounded particle. It has both K.E and potential energy.

The total energy of particle is given by

$$E= K.E+ P.E$$

$$or,\;\; E= \frac12 mv^2+ V$$

$$or,\;\; E-V= \frac{p^2}{2m}$$

$$or,\;\; p= \sqrt{2m (E-V)}$$

\(\therefore\) de-Broglie wave length (\(\lambda)= \frac{h}{p}\)

$$\lambda=\frac{h}{\sqrt{2m(E-v)}}$$

**Reference**

*Mathews, P.M and K Venkatesan. A Text Book of Quantum Mechanics. New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.**Merzbacher, E. Quantum Mechanics . New York: John Wiley, 1969.**Prakash, S and S Salauja. Quantum Mechanics. Kedar Nath Ram Nath Publishing Co, 2002.**Singh, S.P, M.K and K Singh. Quantum Mechanics. Chand & Company Ltd., 2002.*

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