Quantum Mechanics: Important part-A 2 marks Short Questions and Answers

ANNA UNIVERSITY PART A QUESTIONS AND ANSWERS

1. State Kirchoff's law of radiation

Kirchoff's law: Ratio of emissive power to the coefficient of absorption, of any given wavelength is the same for all bodies at a given temperature and is equal to the emissive power of the black body at that temperature.

i.e., e_λ / a_λ = E

2. Explain Planck's hypothesis (or) What are the postulates of Plancks quantum theory? (or) What are the assumptions of quantum theory of black body radiation?

(i) The electrons in the black body are assumed as simple harmonic oscillators.

(ii) The oscillators will not emit energy continuously.

(iii) They emit radiation in terms of quantas of magnitude 'hv', discretly.

i.e. E=nhv

where n = 1, 2, 3, 4...

3. State de‒Broglie's hypo thesis (or) Explain the concept of wave nature. (or) What is meant by matter waves? Give the origin of this concept.

We know that, nature loves symmetry, since the light exhibits the dual nature (ie) it can behave both as a particle and a wave, debroglic suggested that an electron, which is particle can also behave as a wave and exhibits the dual nature.

Thus, the wave associated with a material particle (electron) are called matter waves.

If v is the velocity and m is the mass of the particle then

de Broglie wavelength λ = h / mv

4. What is the physical significance of a wave function?

1. The probability of finding a particle in space, at any given instant of time is characterised by a function ᴪ(x, y, z), called wave function.

2. It relates the particle and the wave statistically.

3. It gives the information about the particle behaviour.

4. It is a complex quantity.

5. |ᴪ|² represents the probability density of the particle, which is real and positive.

5. What is a black body and What are its characteristics?

A perfect black body is the one which absorbs and also emits the radiations Completely.

In practice, no body is perfectly black. We have to coat the black colour over the surface to make a black body.

Black body is said to be a perfect absorber, since it absorbs all the wavelengths of the incident radiation. The black body is a perfect radiator, because it radiates all the wavelength absorbed by it. This phenomenon is also called black body radiation.

6. Define Rayleigh‒Jeans law. Give its limitation.

It is defined as, "The energy is directly proportional to the absolute temperature and is inversely proportional to the fourth power of the wavelength"

(ie) E_λ ∝ T/λ₄

(or)

E_λ = 8πK_BT / λ₄

where K_B → Boltzmann constant,

Limitation: It holds good only for longer wavelengths.

(ie) E=nhv

where n=0,1,2, 3,4,….

7. Define Wien's displacement law. Give its limitation.

It is defined as, “The product of the wavelength (λ_m) of maximum energy emitted and the absolute temperature (T) is a constant".

(ie) λ_mT = constant

Also the E_maximum ∝ T⁵.

(or) E_m = constant . T⁵

Limitation: It holds good only for shorter wavelength.

8. For a free particle moving within a one dimensional potential box, the ground state energy cannot be zero, Why?

For a free particle moving within a one dimensional potential box, when n=0 the wave function is zero for all values of x i.e., it is zero even within the potential box. This would mean that the particle is not present within the box. Therefore the state with n=0 is not allowed. As energy is proportional to n² the ground state energy cannot be zero since n=0 is not allowed.

9. What is quantum tunneling? Give its significance.

Quantum Tunneling

Quantum tunneling is a phenomenon in which the particles penetrate through a potential energy barrier with a height greater than the total energy of the particles. In quantum tunneling, the particles will have a finite probability to cross an energy barrier, so that as the energy needed to break a bond with another particle is attained, even though the particle's energy is less than the energy barrier.

Example: The emission of alpha rays in radioactive decay is an example for quantum tunneling.

10. Define normalisation process and write down the normalised wave function for an electron in a one dimensional potential well of length 'a' metre.

Normalisation is the process by which the probability of finding a particle inside any potential well can be done.

For a one dimensional potential well of length 'a' metre, the normalised wave function is given by

ᴪ_n = √(2/a) sin(nπx / a)

11. Define Eigen value and Eigen function.

Eigen value is defined as energy of the particle and is denoted by the letter (E_n)

Eigen function is defined as the wave function of the particle and is denoted by the letter (ᴪ_n).

ADDITIONAL PART 'A' QUESTIONS AND ANSWERS

1. What is meant by energy spectrum of a black body? What do you infer from it?

The distribution of energy for various wavelengths at various temperatures is known as energy spectrum of a black body.

Inference:

(i) The energy distribution will not be uniform for any particular temperature.

(ii) When temperature increases, the wavelength decreases.

(iii) The total energy emitted at any particular temperature can be found with the help of the area traced by the curve.

2. Define Stefen‒Boltzmann's law.

It is defined as "The radiant energy (E) of the body is directly proportional to the fourth power of the temperature (T) of the body"

(ie) E ∝ T⁴

(or) E = σT⁴.

where σ → Stefan constant.

3. What is meant by photon? Give any two properties.

Definition: Photons are discrete energy values in the form of small quantas of definite frequency (or) wavelength.

Properties:

1. They do not have any charge and they will not ionise.

2. The energy and momentum of the photon is given by

E = hv and p = mc

where

v→ frequency

m→ mass of photon

c→ velocity of photon

h→ Planck's Constant

4. What are the properties of matter waves?

(i) Matter waves are not electromagnetic waves.

(ii) Matter waves are new kind of waves in which due to the motion of the charged particles, electromagnetic waves are produced.

(iii) Lighter particles will have high wavelength.

(iv) Particles moving with less velocity will have high wavelength.

(v) The velocity of matter wave is greater than the velocity of light.

5. What do you understand by the term 'wave function'.

Wave function (ᴪ) is a variable quantity that is associated with a moving particle at any position (x, y, z) and at any time 't'. It relates the probability of finding the particle at that point and at that time.

Since ᴪ is a complex quantity, it has no meaning and hence the probability function |ᴪ|² = ᴪ*ᴪ is found, which is real and positive and has physical meaning, which is a measurable quantity too.

6. Write down the schroedinger wave equation and give any two applications of it.

There are two types of schroedinger wave equations, viz.

(i) Schroedinger time dependent wave equation, given by

Eᴪ = Hᴪ

where

E→ Total energy of the particle

H→ Hamiltonian operator

ᴪ→wave function

(ii) Schroedinger time independent wave equation, given by

 ∇²ᴪ + 2m/ḧ² = [E‒v]ᴪ = 0         [3‒dimensional]

Where

E → Total energy of the particle

V→ Potential energy of the particle

m→ mass of the particle

 ḧ^-→ h/2π (h→ Planck's constant)

Applications

1. It is used to find the electrons in metals.

2. It is used to find the energy levels of an electron in an infinite deep potential well.

7. Write down the one dimensional schroedinger time independent equation and write the same for a free particle.

The one dimensional (along x axis) schroedinger time independent equation is given by

 d²ᴪ/dx² + 2m/ḧ².[E‒V]ᴪ = 0

For a free particle, the potential energy is zero i.e., V=0, Therefore the schroedinger equation becomes

d²ᴪ/dx² + 2m/ḧ².Eᴪ = 0

8. Define Work Function.

In metals, there are large number of free electrons. These free electrons can more freely to a highest energy level, so called Fermi energy level E_F. No electron at E_F can escape from the metal because of the presence of an energy barrier at the surface of the metal. If an electron has to escape beyond the Fermi energy level, we have to supply some additional energy in order to overcome the energy barrier (E_B) of the metal. This additional energy required to make an electron to escape from the metal surface is called work function, denoted by ϕ.

 ϕ = E_B‒E_F

9. Obtain Einstein's Photo‒electric equation.

Einstein proposed a quantum theory on the basis that, "when a photon collides with an electron in a metal, it transfers the energy to the electron in an "all (or) none" process", i.e., either the photon gives out its total energy (hv) to the electron (or) no energy is transferred to the electron.

According to Einstein, the total energy of the photon, which is completely given to the electron is used in two ways viz,

1. A part of energy is used to eject the electron from the surface of the metal. This energy is known as Photo‒electric work function (ϕ).

2. The other part of energy is supplied to the electron as the kinetic energy ( ½ mv²) for it to move with the velocity 'v'.

∴ Total energy of the photon (E) = ϕ + ½mv²

Since E=hv, we can write, hv = ϕ +  ½mv²       ……....(1)

Equation (1) is called Einstein's photo‒electric equation.

10. What do you understand by the term "Probability of finding the particle”? Give examples.

• |ᴪ|² represents the probability density (or) probability of finding the particle per unit volume (or) Normalization of the wave function.

• For a given volume dτ, the probability of finding the particle is given by

Probability (P) = ∫∫∫ |ᴪ|² dτ

where dτ = dx . dy . dz

• The probability will have any value between zero to one. (i.e.,)

(i) If P=0 then there is no chance for finding the particle (i.e.,) there is no particle, within the given limits.

(ii) If P=1 then there is 100% chance for finding the particle (i.e.,) the particle is definitely present, within the given limits.

(iii) If P=0.7, then there is 70% chance for finding the particle and 30% there is no chance for finding the particle, within the given limits.

Example: If a particle is definitely present within a one dimensional box (x‒direction) of length 'l', then the probability of finding the particle can be written as

 P = ₀∫¹ |ᴪ|² dx = 1

11. What is meant by degenerate and non‒degenerate states.

(i) Degeneracy: It is seen from equation (28) and equation (29), for several combination of quantum numbers we have same energy eigen value but different eigen functions. Such states and energy levels are called Degenerate State.

The three combination of quantum numbers (112), (121) and (211), which gives same eigen value but different eigen functions are called 3 fold degenerate state.

(ii) Non‒Degeneracy: For various combinations of quantum number if we have same energy eigen value and same (one) eigen function then such states and energy levels are called Non‒Degenerate State.

Example

For n_x = 2n_y = 2n_z = 2 we have E₂₂₂ = 12h² / 8ma² and

ᴪ₂₂₂= √(8/a³) . sin(2πx/a).sin(2πy/a).sin(2πz/a)

12. What is meant by quantum tunnelling (or) tunnelling effect?

In quantum mechanics a particle having lesser energy (E) than the barrier potential (V) can easily cross over the potential barrier having a finite width 'l' even without climbing over the barrier by tunnelling through the barrier. This process is called Tunnelling.

13. Mention any four occurrences of tunnelling effect.

1. The tunnelling effect is observed in Josephson junction, in which electron pairs in the super conductors tunnel through the barrier layer, giving rise to Josephson current.

2. This effect is also observed in the case of emission of alpha particles by radioactive nuclei.

Here, though the 'α' particle has very less kinetic energy they are able to escape from the nucleus whose potential wall is around 25 MeV high.

3. Tunnelling also occurs in certain semiconductor diodes called resonant tunnelling diodes.

4. Electron tunnels through insulating layer and act as a switch by tunnelling effect.

14. What do you understand by the term Transmission and reflection co‒efficient?

Transmission Co‒efficient

We know that the probability density is the square of the amplitude of that function. Therefore the barrier transmission co‒efficient (T) is the ratio between the square of the amplitudes of the transmitted wave |C|² and the incident wave |A|².

The transmission co‒efficient T = |C|² / |A|²             …………....(1)

The above equation is also called as the "Penetrability" of the barrier.

Reflection Coefficient

The reflection co‒efficient (R) for the barrier surface at x=0 is the ratio between the square of the amplitude of the reflected wave |B|² and the square of the amplitude of the incident wave |A|².

The reflection co‒efficient R = |B|² / |A|²             …………....(2)