## CHAPTER 4

## Elements of Statistical Mechanics and Principles of Quantum Mechanics

#### 4.1 Introduction

Statistical mechanics mainly deals with the distribution of identical, distinguishable and indistinguishable particles of a system in different states of the system. The number of ways, in which the particles can be arranged in different energy states depends on the distinguishability of the particles. Here we are going to see three different types of distributions: (i) Maxwell–Boltzmann distribution, (ii) Fermi–Dirac distribution and (iii) Bose–Einstein distribution.

#### 4.2 Phase space

Let us consider a system consisting of a large number of particles. The state of a particle at an instant can be represented with three position and three momentum coordinates. To specify the position and momentum of a particle in a system, a six-dimentional space called phase space or *µ*-space is used. Any point in this space can be represented with three position coordinates *x*, *y*, *z* and with three momentum coordinates *P _{x}*,

*P*,

_{y}*P*. The points in this space are called phase points or representative points. The phase space is considered to be divided into large number of small elements called cells or groups. The volume of each element is d

_{z}*x*, d

*y*, d

*z*, d

*P*, d

_{x}*P*, d

_{y}*P*and this is equal to

_{z}*h*

^{3}. Each group possesses as a large number of phase points.

#### 4.3 Maxwell–Boltzmann distribution

This distribution is applied to a macroscopic system consisting of a large number *n* of identical but distinguishable particles, such as gas molecules in a container. This distribution tells us the way of distribution of total energy *E* of the system among the various identical particles. Let us consider that the entire system is divided into groups of particles, such that in every group the particles have nearly the same energy.

Let the number of particles in the 1^{st}, 2^{nd}, 3^{rd},…i^{th},…groups be *n*_{1}, *n*_{2}, *n*_{3},…*n*_{i},…respectively. Also assume that the energies of each particle in the 1^{st} group is *E*_{1}, in the second group is *E _{2}* and so on. Let the degeneracy parameter be denoted by

*‘g’*[or the number of electron states] in the 1

^{st}, 2

^{nd}, 3

^{rd},…i

^{th},…groups be

*g*

_{1},

*g*

_{2},

*g*

_{3},

*…g*

_{i}

*…*and so on respectively. In a given system the total number of particles is constant.

The total energy of all particles present in different groups is equal to the energy of the system (E).

The probability of given distribution *W is* given by the product of two factors. The first factor is, the number of ways in which the groups of *n*_{1}, *n*_{2}, *n*_{3},…*n _{i}…* particles can be choosen. To obtain this, first we choose

*n*

_{1}particles which are to be placed in the first group. This is done in

*nC*

_{n}

_{1}ways.

The remaining total number of particles is *(n − n*_{1}). Now, we rearrange *n*_{2} particles in the second group. This is done in *(n — n*

_{1})

*C*

_{n2}ways

The number of ways in which the particles in all groups are choosen is

where **II*** _{i}* is the multiplication parameter

The second factor is the distribution of particles over the different states and is independent of each other.

Of the *n _{i}* particles in the

*i*

^{th}group the first particle can occupy any one of the

*g*states. So there are

_{i}*g*ways, and each of the subsequent particles can also occupy the remaining states in

_{i}*g*ways. So, the total number of ways the

_{i}*n*particles are distributed among the

_{i}*g*states is

_{i}*g*ways.

_{i}^{ni} The probability distribution or the total number of ways in which *n* particles can be distributed among the various energy states is *W*_{2}

The number of different ways by which *n* particles of the system are to be distributed among the available electron states is

where **II**_{i} represents the multiplication parameter

Taking natural logarithms on both sides of equation (4.5) we have,

Applying Stirling's theorem, ln *x*! = *x* ln *x – x*, on equation (4.7)

For the most probable distribution, *W is* maximum provided *n* and *E* are constants. Differentiate equation (4.8) and equate to zero for maximum value of *W.*

Multiplying equation (4.1) by –*α* and equation (4.2) by –*β* and adding to equation (4.9), we get

Taking exponential on both sides,

Equation (4.11) is called Maxwell-Boltzmann (M-B) law. The value of *β* has been extracted separately and is equal to , where *k*_{B} = Boltzmann constant and

*T* = absolute temperature

Equation (4.11) becomes

#### 4.4 Fermi–Dirac distribution

The Fermi–Dirac distribution is applicable to indistinguishable particles like electrons, They have a spin in the order of half-integral multiples of *n*. They obey Pauli exclusion principle (no two electrons in an atom have the same quantum state).

Hence occupation number is 0 or 1. Let the system contains ‘*n*’ number of indistinguishable particles possessing different energies *E*_{1}, *E*_{2},…*E _{i}*…Let the system be divided into groups. The

*i*

^{th}group contains

*n*number of particles, distributed in

_{i}*g*quantum states, all these particles have nearly the same energy

_{i}*E*

_{i}

*.*First we find out the number of ways in which these

*n*

_{i}particles can be distributed in

*g*

_{i}states with not more than one particle in a state, as follows.

First, the particles can be arranged in *g*_{i} ways. Secondly, the remaining particles are arranged in the ( *g _{i} –* 1) states in (

*g*1) ways. Thirdly, in the (

_{i}–*g*2) states in (

_{i}–*g*2) ways and so on. Therefore the total number of ways of arranging

_{i}–*n*particles is

_{i}

Of these arrangements, the permutation *n _{i}*! of the

*n*particles is not relevant, because of exclusion principle. Hence we have

_{i}

Here, the number of possible ways in which *n*_{1} particles may have energy *E*_{1} and *n*_{2} particles may have the energy *E*_{2} and so on for the other groups of the system. Therefore the total distribution for the complete system is given as

Taking logarithms on both sides of equation (4.14) we have

Applying Stirling's formula, ln *x*! = *x* ln *x – x*, on equation (4.15) we have

For most probable distribution, the derivative of equation (4.16) is zero. We have

At equilibrium, the total number of particles in the system n, and the total energy of the system, *E* are constant, hence we have

and

Applying the Lagrange method of undetermined multipliers, multiply equation (4.18) by − *α* and equation (4.19) by −*β* and add to equation (4.17), we obtain

Since *δ* _{ni} is arbitrarily choosen, the term in the bracket must be zero for each value of *i*.

Taking exponential on both sides,

The Fermi–Dirac distribution function is

Here represents the average number of particles in each of the quantum states of that energy.

Dropping the subscript ‘*i* ’ and substituting the values of equation (4.21) becomes

#### 4.5 Bose–Einstein distribution

The Bose–Einstein distribution deals with the distribution of identical indistinguishable particles like photons or phonons called bosons. They possess spin of integral multiple of *n*. Let the system contains *n* number of particles. The system is divided into groups. Let *n*_{1} number of particles have each of energy *E*_{1} are present in 1^{st} group has *g*_{1} states and *n*_{2} number of particles have each of energy *E*_{2} present in the 2^{nd} group which has *g*_{2} states and so on.

The number of ways of distributing *n _{i}* particles among the g

_{i}states is as follows. The

*g*states will have (

_{i}*g*– 1) partitions. The (

_{i}*g*– 1) partitions and

_{i}*n*particles constitute (

_{i}*g*+

_{i}*n*– 1) objects. They can be arranged in (

_{i}*g*+

_{i}*n*– 1)! ways among themselves. Whereas

_{i}*n*particles can be arranged in

_{i}*ni*! ways among themselves and (

*g*– 1) partitions in (

_{i}*g*– 1) ways among themselves. The effective number of ways of arranging them is

_{i}

Similar expressions can be written for other quantum states. Therefore, the total number of ways of distinct arrangement of all the ‘*n*’ particles of the system in various available states is *W*

Taking logarithms of equation (4.24), we have

Using Stirling's approximation, ln *x*! = *x* ln *x − x*, equation (4.25) becomes

Here, we neglected 1 in comparison to *n _{i}* and

*g*as they are very large numbers.

_{i}Differentiating equation (4.26)

For maximum probability the condition is *δ* ln *W = 0*

As the total number of particles in the system *n* and the total energy of the system *E* are constant, hence their derivatives are equal to zero.

and

Applying the Lagrange method of undetermined multipliers i.e., multiplying equation (4.29) by –*α* and equation (4.30) by –*β* and adding to equation (4.28), we get

Taking exponential on both sides of equation (4.31), we have

Hence, the Bose–Einstein's distribution function is

When *k _{B}* = Boltzmann constant

#### 4.6 Comparison of Maxwell–Boltzmann, Fermi–Dirac and Bose–Einstein distributions

#### 4.7 Photon gas

Einstein proposed the concept of localized small packets of light energy. This is similar to Planck's idea of quanta and named such packets as photons. The energy of a photon is given by *E = hγ*, where *h* = Planck's constant.

According to Einstein, light is transmitted in terms of particles like photons. As the intensity of light beam increases, the photon density increases. Photons have particle character as well as wave character. Light contains a very large number of photons and when the particle character of light is considered, photons may be visualized as moving similar to gas molecules in a container, or free electrons in a metal (i.e., electron gas). Hence light photons is considered as photon gas.

#### 4.8 Concept of electron gas and Fermi energy

#### (a) Electron gas

A metal consists of immobile positive ions and free electrons. These free electrons are very large in metals. They move in random directions inside a metal as we see the gas molecules in a container. Hence they are referred to as the free electron gas or electron gas in short.

When an electric field is applied on metals, the free electrons drift in a direction opposite to the applied field. Free electrons participate in thermal and electrical conductivity. They obey gas laws.

#### (b) Fermi energy

The electron gas obey Fermi–Dirac distribution. Let *g* (*E*) be the density of electron states. i.e., the number of available electron states per unit volume of metal in unit range of energies *E*. Let the number of electron-filled states be *N* (*E*) in *g* (*E*). Then the Fermi–Dirac distribution function *f* (*E*) is

The distribution function *f* (*E*) is defined as the probability that an energy level *E* is occupied by an electron. Suppose, if the level is empty then *f (E)* = 0 or if the level is filled, then *f* (*E*) = 1. In general the value of *f(E)* lies in between 0 and 1.

This shows that all states below *E _{F}* are completely filled and all states above

*E*are completely empty. This function is plotted in Fig. 4.1 for T = 0K and for higher temperatures. As temperature increases, the

_{F}*f (E)*decreases below

*E*At higher temperatures the curves pass through a point at which the probability of the electron being in the conduction or valence band is 0.5. The energy at which the probability of occupation is 0.5 at all temperatures is called Fermi energy. Alternatively, the highest energy possessed by an electron at absolute zero of temperature (0K) in a metal is called Fermi energy.

_{F}.

**Figure 4.1** Fermi–Dirac distribution function for electrons

#### 4.9 Density of electron states

The number of available electron states present per unit volume of a material in unit energy range at energy *E* is the density of electron states. To obtain an expression for the density of electron states, let us consider all possible energies of electrons in a material. The electrons are distributed in various electron states in three dimensional space. The electron states are considered in a quantum space. Let the coordinate axes of this space be represented by *n _{x}*,

*n*with origin at ‘

_{y}, n_{z}*O*’ as shown in Fig. 4.2. In this space, every point with integral values of coordinates represent an energy state or the unit volume of the space contains one electron state.

To find the density of states at energy, *E*, let us consider a sphere of radius *n* such that the origin of the sphere coincides with the origin of the coordinate system. The energy of a state on the surface of the sphere is *E.* A point on the surface of the sphere can be represented with *n _{x}, n_{y}, n_{z}* such that

The numbers of electron states inside a sphere is equal to the volume of the sphere in the positive quadrant.

The number of available electron states within a sphere of radius *n* is

**Figure 4.2** Density of electron states sphere

Similarly, the number of available electron states in a sphere of radius *(n +* d*n)* is

The number of electron states whose energies lie between *E* and *E* + d*E* is obtained by subtracting equation (4.36) from equation (4.37).

(neglecting higher power terms, because they are very small)

The expression for energies of electrons in a cubical box of side *a* is given by

Differentiate equation (4.39) with respect to *n.* We get,

Substituting equation (4.40) and equation (4.41) in equation (4.38) gives

According to Pauli's exclusion principle, each state can accommodate two electrons of opposite spin, hence the number of electron energy states available for electron occupancy is

Density of states is given as the number of energy states per unit volume.

Density of states between *E* and *E + dE* is

This is a parabolic function.

#### 4.10 Black body radiation

A body that completely absorbs all wave lengths of radiation incident on it at low temperatures or emits different wave lengths of radiation at higher temperatures is known as a black body. A black body may be idealized by a small hole drilled in a cavity. A graph has been plotted between intensity (or energy density spectral) versus wave length of radiation from a black body. The temperature of the body is raised to different values and distribution curves are plotted for different temperatures as shown in Fig. 4.3. From the graph it has been observed that:

- the intensity of radiation increases for each wave length as the temperature of the body increases.
- At any given temperature, the intensity of radiation from the body is maximum for a particular wave length represented as
*λ*_{1m},*λ*_{2m},*λ*_{3m},…This wave length shifts towards shorter wave length region with increase of temperature. - The area under the curve is proportional to the total radiation energy emitted by the body in unit time.

The spectral energy distribution of black body has been explained by many scientists as given below.

**(a) Wien's law:** Wien showed that the maximum energy, *E*_{m} of the emitted radiation from black body is proportional to fifth power of absolute temperature (*T* ^{5}).

**Figure 4.3** Graphs drawn between intensity versus wavelength of radiation from black body at different temperatures

Wien deduced the relation between the wave length of emission and the temperature of the body as

where *U _{λ}* dλ is the energy per unit volume in the wave length range λ and λ + dλ. Here

*C*

_{1}and

*C*

_{2}are constants.

Wien's law is valid at lower wave length region, where as it deviates from experimental values at higher wave length regions. This is shown in Fig. 4.4.

**(b) Rayleigh–Jeans law:** Rayleigh deduced an equation for the black body radiation based on the principle of equipartition of energy. According to equipartition of energy, each mode of vibration has assigned an average energy of *k*_{B}*T.* The number of vibrations per unit volume in the wave length range λ and λ + dλ is given by 8πλ^{–4}dλ. The vibration energy per unit volume in the range of λ and λ + dλ is

This is the Rayleigh−Jeans equation. This law correctly predicts the fall of intensity in the longer wave length side. However, it fails to explain the lower wave length side of the spectrum.

**Figure 4.4** The three laws of black body radiation

**(c) Planck's law:** Planck assumed that the walls of the black body consists of a large number of electrical oscillators, vibrating with their own natural frequencies. An oscillator possesses an energy equal to *h*γ. Where *h* is Planck's constant and γ is the frequency of the oscillator.

An oscillator may lose or gain energy by emitting or by absorbing photons respectively. Planck derived an equation for the energy per unit volume of black body in the entire spectrum of black body radiation. It is given by

This is Planck's law.

#### 4.11 Waves and particles—de Broglie hypothesis—Matter waves

Classical theory of Newtonian mechanics successfully explains the motion of macroscopic particles, but fails to explain the motion of microscopic particles such as electrons. Whereas, the quantum theory successfully explains the motion of microscopic particles, interference, diffraction and polarization of electromagnetic waves, black body radiation (1901), photoelectric effect (1905), line spectra (1913) and Compton effect (1924), etc. Explanation of the above effects by quantum theory shows the dual nature of waves [wave nature and particle nature]. To explain some of the above facts, we consider packets of energy [photons] and waves. For example, in case of photoelectric effect when photons of sufficient energy or radiation of frequency above a certain value incident on an alkali metal, then electrons are emitted. In this case, absorption of energy does not takes place continuously but in the form of packets of energy called quanta (photons). These photons have particle nature. In case of Compton effect, a photon of certain energy makes collision with a stationary electron, after collision the electron and photon get scattered with lesser energy (or longer wavelength). To explain the collision of photon and electron, we consider the particle nature of light wave. With this back ground, a French scientist de Broglie in the year 1924, proposed the dual nature of matter.

According to him, moving objects and particles possess wave nature. The dual nature of matter was explained by combining Planck's equation for energy of a photon, *E* = *hν* and Einstein's mass and energy relation *E* = *mc*^{2}

where *h* = Planck's constant, *ν* = frequency of radiation and *c* = velocity of light.

Substituting equation (4.46) in (4.45) gives:

where *p* = momentum and λ is the wavelength of photon.

The above equation indicates that a photon is associated with a momentum *p*. From this, de Broglie proposed the concept of matter waves. According to de Broglie, a particle of mass *m*, moving with velocity ‘*v*’ is associated with a wave called matter wave or de Broglie wave of wavelength λ, given by:

This is known as de Broglie equation.

According to the theory of relativity, the mass *m* used in the above equation is not constant but varies with its velocity, given by:

where *m*_{0} is the rest mass of the particle.

Suppose an electron is accelerated to a velocity ‘*v*’ by passing through a potential difference *V*, then work done on the electron, *eV is* equal to increase in its K.E.

and *mv* = (2*meV* )^{1/2} = momentum of an electron.

Substituting this momentum in de Broglie equation,

taking m ≈ m_{0}, rest mass of an electron, the above equation becomes:

Suppose an electron is accelerated through a potential difference of 100 V, it is associated with a wave of wavelength equal to 0.1227 nm.

#### Matter waves

The de Broglie concept that a moving particle is associated with a wave can be explained by using one of the postulates of Bohr's atomic model.

The angular momentum (*L*) of a moving electron in an atomic orbit of radius ‘*r*’ is quantized interms of *ħ.*

So, we have:

where *v =* Linear velocity of an electron

*n =* an integer

Equation (4.50) can be written as:

In the above equation, 2*πr* is the circumference length of the orbit and it is equal to *n* times the wavelength of the associated wave of a moving electron in the orbit. This can be diagramatically represented for *n =* 10 in Fig. 4.5.

**Figure 4.5** Bohr's orbit and de Broglie waves of an electron in the orbit

According to de Broglie, a moving particle behaves as a wave and as a particle. The waves associated with a moving material particles are called matter waves or de Broglie waves. They are seen with particles like electrons, protons, neutrons, etc.

#### Properties of matter waves

- de Broglie waves are not electromagnetic waves; they are called pilot waves, which means the waves that guide the particle. Matter waves consist of a group of waves or a wave packet associated with a particle. The group has the velocity of particle.
- Each wave of the group travel with a velocity known as phase velocity given as
- These waves cannot be observed.
- The wavelength of these waves,

#### 4.12 Relativistic correction

When an electron is accelerated through a high potential difference (*V* ), then the mass of electron varies with its velocity. Hence, we have to consider its relativistic mass. Hence, we calculate its relativistic wavelength and total energy in the following way.

**(a) Relativistic wavelength is calculated as follows:**

The momentum of an electron is:

Divide Equation (4.52) by *m*_{0}*c*, then:

Add and subtract 1 to the numerator,

Squaring and rearranging Equation (4.53),

The kinetic energy (*E*) of an electron is given by:

Squaring Equation (4.55) gives:

Equating Equations (4.54) and (4.56), we get:

Substituting Equation (4.57) in de Broglie equation,

As the electron is accelerated through a potential *V*, then its kinetic energy *(E) = eV.* So, Equation (4.58) becomes:

Equation (4.15) represents, the relativistically corrected wavelength.

**(b) Relativistic formula for total energy is calculated as follows:**

The rest mass (*m*_{0}) equivalent energy of a particle is *m*_{0}*c*^{2} i.e., *E*_{0} = *m*_{0}*c*^{2}

The mass equivalent energy of a particle when it is in motion is *mc*^{2} and this is equal to its total energy.

The total energy (*E*) when it is in motion is:

Squaring Equation (4.60),

Cross-multiplying Equation (4.61),

Equation (4.62) represents the relativistic total energy of the particle. Hence, kinetic energy of the electron = total energy – rest mass equivalent energy

#### 4.13 Planck's quantum theory of black body radiation

A body that absorbs all wavelengths of radiation at low temperatures and emits all wavelengths of radiation at high temperatures is known as black body. Figure 4.6 shows the graphs plotted between the intensities of emitted light and wavelengths at different temperatures. The area under the plot indicates total radiation (*R*), the power emitted per unit area. According to Stefan–Boltzmann law, the radiation is proportional to T^{4}.

where *σ* is the Stefan–Boltzmann constant.

**Figure 4.6** Plots of black body radiation

Expression for the radiated energy density per unit wavelength range (*R*_{λ}) was derived by Wien based on thermodynamics is:

where *C _{1}* and

*C*are constants. This formula explains the black body radiation in short wavelengths as shown in Fig. 4.3.

_{2}Rayleigh–Jean's derived another formula for *R*_{λ} based on statistical mechanics as:

where *K*_{B} = Boltzmann constant

The above formula could partly explain in the longer wavelength region as shown in Fig. 4.3.

In 1901, Max Planck proposed the particle character of radiation similar to Newton's corpuscular theory known as Planck's quantum theory. According to this theory,

- The black body walls contain large number of oscillators having different frequencies.
- The energy radiated by an oscillator during transition from one quantum state to another is:

where *n* is an integer and *h* = Planck's constant

*hν* is the quantum of energy. This shows that energy is radiated in the form of wave-packets. This energy packet has both wave and particle character. Based on this concept, Planck derived an expression for *R*_{λ} known as Planck's radiation law.

The above equation exactly fits the experimental graph shown in Fig. 4.7. The Planck's law, reduces to Wien's law when *hν>> K*_{B}*T* and to Rayleigh–Jeans law when *hν<< K*_{B}*T.*

Based on this Planck's quantum theory, Eienstein developed the theory of relativity and successfully explained photoelectric effect. Raman effect of light, Compton effect of X-rays, etc., support Planck's quantum theory.

**Figure 4.7** Comparison of the three radiation laws with the experimental curve [shown with dots]

#### 4.14 Experimental study of matter waves

de Broglie proposed matter waves but he did not prove it experimentally. Many scientists proved the existence of matter waves individually. In 1927, Davisson and Germer in the United States and in 1928 Thomson proved experimentally the existence of matter waves. Also, Stern and others showed the existence of matter waves in connection with molecular and atomic beams.

**(a) G.P. Thomson Experiment:** The diffraction of electrons by metal foil in G.P. Thomson experiment showed the wave nature of electrons and hence supports the de Broglie hypothesis. Now, we will study in detail the experimental set-up and theory of G.P. Thomson experiment. From the theory, we can estimate the wavelength of the waves associated with the moving electrons.

**Experimental set-up:** As shown in Fig. 4.8, the apparatus consists of a highly evacuated cylindrical tube ‘C’. Inside the tube, electrons are produced by heating the filament ‘F’ with low-voltage source. The emitted electrons are attracted by the anode ‘A’ to which high positive voltage has been applied and the beam is allowed to pass through a fine hole in a metallic block ‘B’. A fine narrow electron beam, which comes out from ‘B’ is allowed to fall on a polycrystalline thin gold foil ‘G’ of thickness 10^{−8} m. The gold foil consists of a large number of micro-sized crystallites, which are oriented in random directions. Hence, the crystal planes of these crystallites are oriented in all possible directions in the gold foil. Some of the electrons incident on the crystal planes, which satisfy Bragg's law (2*d* sin *θ* = *nλ*) gets reflected by the planes (or diffracted). In the Bragg's equation, *d* = interplanar spacings of crystal planes, *θ* = diffraction angle, *λ* = wavelength of the waves associated with electrons and *n* = order of diffraction. The diffracted electrons will go in the form of concentric cones and fall on fluorescent screen (S) present at the end surface of the evacuated tube. So, we can see concentric circles of diffraction pattern on the fluorescent screen. To record the diffraction pattern, a photographic plate (P) can be inserted in front of fluorescent screen in the tube as shown in Fig. 4.8. We can see the diffraction pattern on the photographic plate after processing it. The diffraction pattern consists of a series of concentric diffracted rings corresponding to different diffraction orders. The diameter of these rings are measured.

**Theory:** Figure 4.9 shows the diffraction of an electron beam by a crystal plane and the diffracted rings on photographic plate. In the theory of this experiment, we derive expressions for interplanar spacing and the de Broglie wavelength of waves associated with electrons.

**Expression for interplanar spacing:** As shown in Fig. 4.9, Let QR be an electron beam, which undergoes diffraction in the gold foil ‘G’ and falls on a photographic plate at a point E, at a distance *r* from the central point ‘O’ of the concentric circles. Let the incident and the first-order diffracted electrons make an equal angle ‘*θ*’ with the crystal plane YZ. Let RO = *L*, be the distance between gold foil and photographic plate.

**Figure 4.8** G.P. Thomson experimental set-up

**Figure 4.9** Schematic representation of electron diffraction in gold foil

Bragg's law is:

2*d* sin *θ* = *nλ* where *n* = 1, 2, 3,…

For first-order diffraction (*n* = 1)

Substituting Equation (4.69) in (4.68) gives:

**To find the de Broglie wavelength of an electron:** In G.P. Thomson's experiment, the particles [electrons] are accelerated by a potential difference of about 25 to 60 KV. Let an electron be accelerated to a velocity ‘*v*’. Then the moving electron is associated with a wave. The de Broglie wavelength of this electron is given by:

The quantity is obtained by equating the increase in kinetic energy of the electron to workdone on it by the accelerating potential [i.e., *eV* ]

According to the theory of relativity, the increase in kinetic energy of an electron (*E*) is:

*E* = *mc*^{2} – *m*_{0}*c*^{2}

where *m*_{0} = rest mass of an electron,

*m =* relativistic mass when it is moving with velocity ‘*v*’ and

*c =* velocity of light

Since this gain in kinetic energy is equal to *eV.*

Equation (4.73) is a part of Equation (4.71)

Again ‘*v*’ can be evaluated from Equation (4.73) in the following way. In Equation (4.73), put

Squaring and inverting Equation (4.74),

Multiplying Equations (4.74) and (4.75),

Substituting Equation (4.76) in (4.71), we have:

Substituting the value of ‘*x*’, we have:

Equation (4.77) represents the relativistic expression for de Broglie wavelength of an electron accelerated through a high potential difference of ‘*V*’ volts. If the relativistic effect is ignored, then Equation (4.77) reduces to:

Substituting Equation (4.77) in (4.70), we get:

The value of ‘*d*’ calculated using the above equation agree very well with that of the value obtained using X-ray method. This suggests the validity of this experiment. For example, the values of ‘*d*’ obtained by G.P. Thomson and X-ray method are 4.08 Å and 4.06 Å, respectively for gold foil.

**(b) The Davisson and Germer experiment:** This experiment proved the de Broglie hypothesis of matter waves of electrons in 1927. The original aim of this experiment was to find the intensity of scattered electrons by a metal target in different directions. The experimental arrangement of this experiment is shown in Fig. 4.10. The apparatus consists of an evacuated chamber ‘C’; inside this chamber, electrons are produced by heating the filament ‘F’ with a low-voltage battery B_{1}. The emitted electrons are attracted by a cylindrical anode ‘A’ to which high variable positive voltage is applied with battery B_{2} and a potential divider. A narrow fine beam of electrons is obtained by passing the electrons through a series of pin hole arrangement present inside the cylindrical anode. This beam of electrons is allowed to incident on a single nickle crystal. This nickel crystal acts as a target material. The target is rotated about an axis perpendicular to the plane of the paper to bring various crystal planes for electron scattering. Nickel crystallizes in cubic system so that the crystal possess three-fold symmetry. The intensities of the scattered electrons are measured with the help of electron collector by moving it along a circular scale. The counter can be rotated about the same axis as the target. The counter receives the scattered electrons ranging from 20° to 90° with respect to the incident beam. The accelerating potential to anode is varied in the range of 30 to 600 V. The electrons received by the counter are allowed to pass through a galvanometer and earthed. The deflection in the galvanometer is proportional to the number of scattered electrons received by the counter in unit time. The galvanometer readings are noted when the counter is at different angles with respect to the incident beam, as the crystal is rotated through 360° in its own plane for different accelerating voltages. There are three variables in the experiment, these being the potential applied to the anode, the position of the collector and number of electrons collected by the counter in unit time or galvanometer reading.

**Figure 4.10** Davisson and Germer experimental arrangement

Graphs are plotted between galvanometer readings [or the number of electrons collected per unit time] against the angles of scattered electrons with incident beam [i.e., angle of galvanometer with incident beam] for different accelerating voltages as shown in Fig. 4.11.

The graph remains fairly smooth till the accelerating voltage is less than 44 V. When the accelerating voltage is 44 V, then a spur is observed on the curve. The spur becomes more clear as the voltage reaches 54 V. The spur diminishes afterwards, above 68 V the spur disappears as shown in Fig. 4.11. The voltage and position of the collector are kept fixed at values corresponding to the largest spur and the crystal is rotated. The spur appears thrice in a complete rotation of the crystal corresponding to the three-fold symmetry of the crystal. Subordinate maxima occurs at the intermediate positions. From this, we know that the intensity of scattered [or diffracted] electrons is maximum at an angle of 50° with incident beam, when the accelerating voltage is 54 V in case of nickel crystal. The accelerating voltage sets up the correct wavelength of waves associated with the electrons incident on the nickel crystal for diffraction to take place with 50° scattering angle in this experiment. We know that incident and diffracted electron beam makes an angle [*θ*] of 65° with the family of Bragg's planes [a set of parallel crystal planes] as shown in Fig. 4.12.

**Figure 4.11** Curve showing the development of diffracted beam in the setting of crystal face

The above diffracted angle (*θ*) is substituted in Bragg's diffraction formula 2*d* sin *θ = nλ*, where *d =* interplanar spacing, *n* = order of diffraction and *λ* is wavelength. The interplanar spacing in nickel crystal can be determined using X-rays, it comes to 0.091 nm. Substituting the experimental values for first-order diffraction [*n* = 1] in Bragg's law,

we have 2*d* sin *θ* = 1*λ*

2 × 0.091 × 10^{-9} × sin 65° = *λ*

*λ* = 0.165nm.

**Figure 4.12** Electron diffraction in nickel crystal

Wavelength of the waves associated with the incident beam of electrons in this experiment can also be obtained by applying de Broglie equation:

In the above experiment, the electron diffraction is maximum for an accelerating voltage of 54 V, so the wavelength associated with these electrons is:

This value is in very good agreement with the experimental value [0.165 nm]. Thus, this experiment proves the de Broglie hypothesis of the wave nature of moving particles.

#### 4.14 Schrödinger's time-independent wave equation

Based on de Broglie's idea of matter waves, Schrödinger derived a mathematical equation known as Schrödinger's wave equation. To derive Schrödinger's wave equation, consider a particle of mass ‘*m*’ moving freely along X-direction [one dimensional] with velocity *v*. This moving particle is associated with a de Broglie wave of wavelength ‘*λ*’ and has frequency ‘*v*’. The expression for the displacement of a de Broglie wave associated with a moving particle is similar to an expression for undamped harmonic waves:

where *ω* = 2*πV =* angular frequency and *v = vλ=* velocity of the wave. ψ is called wave function, it is function of *x* and *t*. Substituting the values of *ω* and *v* in Equation (4.80), we have:

The energy of the wave can be represented by Planck's equation *E = hv* (or) *v = E/h* and the de Broglie wavelength, *λ* = *h/p*. The values of *v* and *λ* are substituted in Equation (4.81).

where in quantum mechanics. The above equation represents wave function for a freely moving particle along X-direction. If the particle is subjected to external fields or forces, then Equation (4.82) is not valid. In such cases, we have to obtain a differential equation and solving that differential equation in specific situations give ψ. To obtain the differential equation, differentiate Equation (4.82) twice with respect to ‘*x*’ and once with respect to ‘*t*’ and substitute in the energy equation for the particle.

The total energy, *E* of the particle is the sum of kinetic energy and potential energy, *V (x).*

Multiplying both sides of the above equation with ψ, we get:

Substituting Equations (4.83) and (4.84) in Equation (4.85), we get:

The above equation is known as time-dependent, one-dimensional Schrödinger's wave equation. In three dimensions, it can be represented as:

Here, ψ is a function of *x, y, z* and *t*.

[The differential operator where are unit vectors along X, Y and Z directions]

In many cases, the potential energy depends on the position only and independent of time. To obtain time-independent wave equation, Equation (4.82) can be represented as:

where

Here, ψ is a function of *x* and *t* whereas *ψ* is a function of *x* alone and *φ* is a function of ‘*t*’ alone. Equation (4.90) can be represented as:

Differentiating Equation (4.91) twice w.r.t. ‘*x*’ and once w.r.t. ‘*t*’ and substituting in Equation (4.86), we have:

Substituting Equations (4.92) and (4.93) in Equation (4.87), we get:

Equation (4.94) is the time-independent one-dimensional Schrödinger's wave equation. In three dimensions, it is represented as:

Here, *ψ* is a function *of x, y* and *z* only and independent of time.

#### 4.15 Heisenberg uncertainty principle

Heisenberg proposed the uncertainty principle in connection with the dual nature of waves and particles. The uncertainty principle has been explained in the following way: suppose if a particle is moving along X–direction, then according to uncertainty principle, it is impossible to measure accurately simultaneously its position (*x*) and also its momentum ( *p _{x}* ). If Δ

*x*is the uncertainty in measuring its position then Δ

*p*is the uncertainty in measuring its momentum.

_{x}Then,

where *h* is Planck's constant

The above equation is applicable in all directions. Along Y– and Z– directions, it is:

The above uncertainty has been already proved using diffraction of electrons by a long narrow slit.

The uncertainties Δ*x* and Δ*p*_{x} associated with the simultaneous measurement of *x* and *p*_{x} can be explained by considering the diffraction of electrons by a narrow slit as follows:

As shown in Fig. 4.13, let us consider a beam of electrons pass through a long narrow slit of width *d*, let the momentum of electrons along X-direction is negligible.

As the electrons enter the slit, there will be spreading due to diffraction. Let the diffraction angle is Δ*θ*, such that:

So, that an electron acquires momentum along X–direction given by:

As the electron pass through the slit, the uncertainly in simultaneous measurement of position along X-direction is Δ*x* ≈ d, then the above equation becomes:

Thus, the uncertainty principle is explained.

Similar to uncertainty in position and momentum, we have uncertainty in measuring time and energy of a wave packet. The uncertainty relation is:

Suppose the uncertainty in the energy determination of a wave packet is then the maximum time available for energy determination is Δ*t.*

The time–energy uncertainty can be explained by considering the wave packet moving with velocity *v* along X–direction and let it occupies a region Δ*x*. The uncertainty in passing this particle at a given point is:

**Figure 4.13** Diffraction of electron beam by a narrow slit

As the packet is localized to a region Δ*x*, then the spread in the momentum is Δ*p*, so,

Substitute Equation (4.101) in (4.99). Then,

Substitute Equation (4.103) in (4.102)

The above equation shows that the spread in the energy of a particle is ΔE, then the uncertainty in passing that particle through a point is Δ*t*

#### 4.16 Physical significance of the wave function

The wave function ψ associated with a moving particle is not an observable quantity and does not have any direct physical meaning. It is a complex quantity. The complex wave function can be represented as ψ(*x, y, z, t*) = *a +* *i*b and its complex conjugate as ψ^{*}(*x, y, z, t*) = *a —* *i*b. The product of wave function and its complex conjugate is ψ(*x, y, z, t*)ψ^{*}(*x, y, z, t*) = (*a + i*b) (*a — i*b) = *a* ^{2} + *b*^{2} is a real quantity. However, this can represent the probability density of locating the particle at a place in a given instant of time. The positive square root of ψ(*x, y, z, t*) ψ^{*} (*x, y, z, t*) is represented as |ψ(*x, y, z, t*)|, called the modulus of ψ. The quantity |ψ(*x, y, z, t*)|^{2} is called the probability density, denoted as *P.*

If a particle is moving in a volume *V*, then the probability of the particle in a volume element d*V*, surrounding the point *x, y, z* at an instant ‘*t*’ is *P*d*V.*

Integrating this probability throughout the volume *V*, is equal to 1

If the particle is not present in that volume, then

For a particle moving along X-direction [one dimensional] the quantity, *P*d*x* = ψ(*x*, *t*) ψ^{*}(*x*, *t*) d*x* = |ψ (*x*, t)|^{2} d*x*, represent the probability of the particle over a small distance ‘d*x*’, centred at *x*, at time ‘*t*’. The probability per unit distance [i.e., d*x* = 1] is called the probability density represented as |ψ(*x*, *t*)|^{2}.

The wave function that satisfies time-independent wave equation has probability independent of time.

#### 4.17 Particle in a potential box

A free electron (particle) in a metallic crystal may move freely inside the crystal from one place to another place but will not come out of the crystal because at the surface of the crystal, the electron experiences very large (infinite) potential [called potential barrier]. The potential barrier present at the surface [covering the metal surface] will act as a three-dimensional potential box for the free particle [electron]. This potential box can also be called as potential well because the electron will remain in that region only. For simplicity, first we see one-dimensional potential box [or potential well] and extend it to three-dimensional box.

#### (a) Particle in a one-dimensional box [or one dimensional potential well]

Suppose an electron (particle) of mass ‘*m*’ moves back and forth in a one-dimensional crystal of length ‘*L*’ parallel to X-direction. At the ends of the crystal, i.e., at *x* = 0 and at *x* = *L*, two potential walls of infinite height exist, so that the particle may not penetrate the walls. Due to collisions, the energy of the particle does not change. Throughout the length ‘*L*’ of the box, the potential energy *V* of the particle is constant and this constant potential energy of the particle inside the box is considered to be equal to zero for all practical purposes. A plot of potential energy of an electron versus distance is shown in Fig. 4.14. As the particle is inside the box, then the probability of the particle inside the crystal, *P* = *ψψ*^{*} is equal to 1 and outside the well probability is equal to zero, hence *ψ* must be zero when 0 ≥ *x* ≥ *L*.

Inside the box, *V* = 0, by solving one-dimensional Schrödinger's time-independent wave equation gives the motion of the particle inside the box. The study will show quantum numbers, discrete values of energy, zero-point energy and the wave function associated with the particle.

**Figure 4.14** One dimensional potential box with potential walls of infinite height at *x* = 0 and at *x* = L

One-dimensional Schrödinger's time-independent equation is:

The *K* in Equation (4.107) is the wave vector, this can be shown easily using de Broglie hypothesis in the total energy of the particles. The total energy *E* is equated to K.E because P.E of an electron is considered as zero in this case.

From de Broglie hypothesis,

From this,

Differentiating Equation (4.109) twice with respect to *x* and substituting in Equation (4.60) gives:

Substituting in Equation (4.106), we have:

The general solution will be of the form:

On expansion, we get:

where A = (*a + b*) and B = *i* (*a – b*) are again constants.

Equation (4.111) represents a general solution for Equation (4.106). The values of constants in Equation (4.111) can be obtained by applying boundary conditions at the ends of the crystal.

- At
*x*= 0, ψ(*x*) = 0, applying this on Equation (4.111), we get:

0 = A × 1 + B × 0 => A = 0

Substituting A = 0 in Equation (4.111) gives: - At
*x = L*ψ(*x*) = 0

Substituting this in Equation (4.112), we have:

0 = B sin KL, we cannot take B = 0, because for B = 0,*ψ(x)*= 0 (from Equation 4.112)

So, sin K*L*= 0 (or)*KL = nπ*

Substituting Equation (4.113) in (4.112) gives:

Here, *ψ*(*x*) is changed to *ψ _{n}*(

*x*) because wave function takes different values as ‘

*n*’ changes.

If *n =* 0, then *K* = 0, *E* = 0 and *ψ*(*x*) = 0 for all values of *x* in the well, so *n ≠* 0. This means that a particle with zero energy cannot be present in the box.

Substituting Equation (4.113) in (4.107) gives:

For different values of *n, E* also takes different values and hence *E* can be written as *E _{n}*

The lowest energy of the particle is obtained by putting *n =* 1 in Equation (4.115) and it is:

Equations (4.115) and (4.116) indicates that a particle in the box can take discrete values of energy, for *n =* 1, 2, 3,…i.e., the energy is quantized. These discrete energy values are called eigen values of energy. The number *n* is called the quantum number. Figure 4.11. shows the energy level diagram for a particle in a box. For the same value of quantum number *n*, the energy is inversely proportional to the mass of the particle and square of the length of the box. The energy is quantized and so it cannot vary continuously. But according to classical mechanics, there is a continuous range of possible energies. The increase in spacing between *n*th energy level and the next higher level is:

**Figure 4.15** Eigen values of energy

The wave function *ψ _{n}* corresponding to

*E*is called eigen function of the particle.

_{n}*Determination of B by normalization*

The value of *B* in Equation (4.114) can be obtained by equating the total probability of finding the particle inside the potential well is equal to unity, and this process is called normalization. Let *P _{n}*(

*x*) is the probability density of the particle at

*x*along X-direction:

Then,

Using Equation (4.114)

The second term of the above equation becomes zero at both the limits.

Substituting Equation (4.118) in (4.114) gives the normalized wave function:

The first three wave functions for *n =* 1, 2 and 3 are shown in Fig. 4.16. The shapes of wave functions shown in Fig. 4.16 have been obtained by substituting different values for *x* for each *n* value.

From Fig. 4.16, it is seen that the wave function *ψ*_{1} has two nodes at *x =* 0 and at *x = L*, the wave function *ψ*_{2} has three nodes at *x =* 0, *L*/2 and *L*. The wave function *ψ*_{3} has four nodes at *x =* 0, *L*/3, 2*L*/3 and *L*. Thus, the wave function *ψ*_{n} will have (*n +* 1) nodes.

**Figure 4.16** Eigen functions

*Probability of location of the particle*

The probability of finding a particle in a small distance d*x* centred at *x* is given by:

Probability density is:

This is maximum when,

For *n* = 1, the most probable positions of the particle is at *x* = *L*/2.

For *n* = 2, the most probable positions are at *x = L*/4 and 3*L*/4.

For *n* = 3, the most probable positions are at *x = L*/6, 3*L*/6 and 5*L*/6.

These positions are shown in Fig. 4.17.

The wave mechanical result is quite contradictory to the classical concept. According to classical mechanics, a particle in a potential box would travel with a uniform velocity from one wall to the other and at the walls it would be perfectly reflected. Therefore, the probability of finding the particle within a small distance d*x*, any where in the box is same and is equal to d*x/L.*

#### (b) Particle in a rectangular three-dimensional box

Consider a particle [electron] of mass *‘m’* constrained to move freely in the space of the rectangular metallic crystal with edges of length *a, b* and *c* along X, Y and Z-axes as shown in Fig. 4.18. Potential barrier which exists at the surface of the crystal will be in the form of rectangular box for the free electron inside this crystal. We take the origin of coordinate system at one corner of the box. We will solve three-dimensional time-independent Schrödinger's wave equation in the box.

**Figure 4.17** Probability density of particle in well

**Figure 4.18** Three-dimensional potential box

The potential energy of the particle is considered to be equal to zero inside the box and it is infinity (*V = α*) at the boundaries (surface) and in the remaining space.

*ψ* is function of the three variables *x, y* and z. One way of solving Equation (4.120) is to write *ψ* as the product of three functions as:

Here, the wave function is equal to the product of three wave functions X, Y and Z. Again, X is function of *x* only, Y is function of *y* only and Z is a function of *z* only. In simple, Equation (4.121) is represented as:

If *Ψ* is a solution of Equation (4.121), then differentiate Equation (4.122) with respect to *x, y* and *z* twice and substitute in Equation (4.120), we get:

Similarly,

Substituting these in Equation (4.120), we get:

dividing throughout by XYZ, we have:

Where

We, therefore, write Equation (4.123) as:

Similar to the solution for one-dimensional Schrödinger's wave equation, the general solution of Equation (4.126) will be of the form:

Applying boundary conditions, we have:

- at
*x*= 0, the wave function along X-direction is zero i.e., X = 0 applying this on Equation (4.128) gives A = 0.So, Equation (4.128) becomes:

- at
*x = a*, X = 0 = B_{x}sin*K*_{x}a

B* _{X}* cannot be zero, since B

*= 0 gives the wave function along X-direction is zero [X = 0]. i.e., the wave function does not exist.*

_{X}So sin *K _{x} a* = 0, therefore

*K*

_{x}a = n_{x}π

where *n _{x}* = 1, 2, 3,…….,

*n*≠ 0, because if

_{x}*n*= 0 gives, X = 0 for all values of

_{x}*x*in the box. Substituting Equation (4.130) in (4.129) gives:

Applying the normalization condition on Equation (4.131) between the limits *x* = 0 and *x = a*, we have:

Solving this, we get:

Substituting Equation (4.132) in (4.131) gives:

With similar treatment on Equation (4.126) and (4.127), we obtain:

Substituting Equations (4.133), (4.134) and (4.135) in Equation (4.122) gives:

Equation (4.124) is:

As *n*_{x}, *n _{y}* and

*n*takes different values, so

_{z}*E*takes the form:

where *n*_{x} = *n*_{y} = *n*_{z} = 1, *2*, 3,…

Equation (4.136) gives the total normalized wave functions inside the rectangular box for the stationary states. Equation (4.137) gives the eigen values of energy of the particle. These values are called the energy levels of the particle.

If the particle is confined in a cubical box i.e., *a = b = c*, the eigen values of energy are given by:

where *n*^{2} =*n*_{x}^{2} + *n*_{y}^{2} + *n*_{z}^{2} and the normalized wave functions are:

From equation (4.138), we know that several combinations of the three integers give different stationary states or different wave functions, in which some energies remain same, then they are said to be degenerate states and energy levels.

**Formulae**

**Solved Problems**

*1. Calculate the average energy of Planck's oscillator of frequency 5.6 × 10 ^{12} Hz at 330K.*

**Sol:** The average energy of planck's oscillator

*k _{B}* Boltzmann constant = 1.38 × 10

^{-23}J/K

*h* = Planck's constant = 6.625 × 10^{-34} JS

*γ* = frequency of oscillator = 5.6 × 10^{12} Hz

*T* = Temperature of the oscillator = 330 K

Substituting the above values in the expression

Average energy of Planck's oscillator

*2. A black body emits radiation at a temperature of 1500K. calculate the energy density per unit wave length at 6000Å of black body radiation.*

**Sol:** Temperature of the black body = 1500K

The wave length at which energy density is to be determined, λ = 6000Å

Planck's equation for energy density in the wave length range between *λ* and *λ+dλ*, *Uλ dλ* =

For one unit range of wave lengths, it is

*3. Calculate the wavelength associated with an electron with energy 2000* eV*.*

(Set-1–Sept. 2006), (Set-1–May 2004), (Set-1–May 2003), (Set-4–Nov. 2003)

**Sol:** *E =* 2000 eV = 2000 × 1.6 × 10^{-19} J

Kinetic energy

*4. Calculate the velocity and kinetic energy of an electron of wavelength 1.66* × *10 ^{−10}* m

*.*

(Set-1, Set-3–May 2006), (Set-2, Set-3–June 2005), (Set-2-May 2004), (Set-4-May 2003)

**Sol:** Wavelength of an electron (λ) = 1.66 × 10 ^{−10}m

*5. An electron is bound in one-dimensional infinite well of width 1* × *10 ^{-10} m. Find the energy values in the ground state and first two excited states.*

(Set-4–June 2005), (Set-2–Nov. 2004), (Set-3–May 2003)

**Sol:** Potential well of width (*L*) = 1 × 10^{−10} m

For ground state *n* = 1,

*6. An electron is bound in one-dimensional box of size 4 × 10*

^{−10}m

*. What will be its minimum energy?*

(Set-2–Sept. 2006), (Set-1–Nov. 2004), (Set-2–May 2003)

**Sol:** Potential box of size (*L*) = 4 × 10^{−10}m

*7. An electron is moving under a potential field of 15* kV. *Calculate the wavelength of the electron waves.*

**Sol:** *V =* 15 × 10^{3}V λ = ?

*6. Find the least energy of an electron moving in one-dimensional potential box (infinite height) of width 0.05nm*

**Sol:**

*8. A quantum particle confined to one-dimensional box of width ‘a’ is known to be in its first excited state. Determine the probability of the particle in the central half.*

(Set-1–Nov. 2003)

**Sol:** Width of the box, *L = a*

First excited state means, *n* = 2

Probability at the centre of the well, *P*_{2} (*L*/2) = ?

The probability of the particle at the centre of the box is zero.

*9. An electron is confined in one-dimensional potential well of width 3 × 10 ^{−10}*m.

*Find the kinetic energy of electron when it is in the ground state.*

(Set-2–Nov. 2004), (Set-1–May 2004), (Set-1–Nov. 2003)

**Sol:** One-dimensional potential well of width, *L* = 3 × 10 ^{−10}m

Electron is present in ground state, so *n =* 1

*10. Calculate the de Brogile wavelength of neutron whose kinetic energy is two times the rest mass of electron (given m _{n} = 1.676* ×

*10*kg

^{−27}*, m*×

_{e}= 9.1*10*kg

^{−31}*, C = 3*×

*10*×

^{8}m/s and h = 6.63*10*J.S

^{−34}*).*

**Sol:** Kinetic energy of neutron,

where *m _{n} =* mass of neutron

*m _{e}* = mass of an electron

de Brogile wavelength of neutron, λ_{n} = ?

*11. An electron is confined to a one-dimensional potential box of length 2* Å*. Calculate the energies corresponding to the second and fourth quantum states (in* eV*).*

(Set-2–Nov. 2003)

**Sol:** Length of the one-dimensional potential box, *L* = 2Å = 2 × 10^{−10}m

Energy of electron in *n*th level,

Energy corresponding to second and fourth quantum states is:

*E*_{2} = 2^{2}*E*_{1} = 4 × 9.43 eV = 37.72 eV

and

*E*_{4} = 4^{2}*E*_{1} = 16 × 9.43 eV = 150.88 eV

*12. Electrons are accelerated by 344* V *and are reflected from a crystal. The first reflection maximum occurs when the glancing angle is 60 °. Determine the spacing of the crystal.*

(Set-4–Nov. 2004), (Set-3–Nov. 2003)

**Sol:** Accelerating potential, *V =* 344 V

Order of diffraction, *n* = 1

Glancing angle, *θ =* 60°

Interplanar spacing, *d* = ?

We know that

So,

The interplanar spacing is 0.382 Å.

*13. Calculate the energy required to pump an electron from ground state to the 2nd excited state in a metal of length 10 ^{−10}*m

*.*

**Sol:** The energy of an electron of mass ‘*m*’ in *n*th quantum state in a metal of side *‘L’* is:

*n =* 1, corresponds to ground state

*n* = 2, corresponds to first excited state and

*n* = 3, corresponds to second excited state

Energy required to pump an electron from ground state to 2nd excited state

*14. Calculate the minimum energy of free electron trapped in a one-dimensional box of width 0.2* nm*. (Given, h = 6.63 -10 ^{-34}* J-S

*and electron mass*×

*9.1*×

*10*

^{−31}kg )**Sol:** One-dimensional box of width, *L =* 0.2 nm = 2 × 10^{-10} m

Minimum energy of the electron, *E1 =* ?

*15. Calculate the wavelength associated with an electron raised to a potential 1600* V*.*

(Set-1, Set-4–May 2008), (Set-3–May 2004)

**Sol:** Potential (*V* ) = 1600 V

#### Multiple Choice Questions

- Phase space is a __ dimensional space. ( )
- 3
- 4
- 5
- 6

- Maxwell–Boltzmann statistics is applicable to ( )
- identical distinguishable particles
- identical indistinguishable particles
- fermions
- bosons

- A point in phase space or
*μ*-space can be represented with ( )- Six position coordinates
- Six momentum coordinates
- Three position and three momentum coordinates
- Six position and six momentum coordinates

- At temperature
*T*K, if a particle possess energy*E*then the Maxwell–Boltzmann distribution for it can be represented as [a = constant, KB = Boltzmann constant] ( )_{i} - Fermi-Dinac distribution is applicable to ( )
- The Fermions possess a spin of ( )
- integral multiples of
*ђ* - half-integral multiples of
*ђ* - They possess any spin
- None of the above

- integral multiples of
- The Fermi-Dirac distribution function of a particle possessing energy
*E*at temperature*Tk*is ( ) - Bose-Einstein statistics is applicable to ( )
- identical distinguishable particles
- identical indistinguishable particles
- Both a and b
- None of the above

- Bosons possess a spin of ( )
- integral multiples of
*ђ*. - half-integral multiples of
*ђ*. - They can have any spin.
- They would not possess spin.

- integral multiples of
- For a particle of energy
*E*, the Bose-Einstein distribution can be represented as [_{i}*a*,*β*= const.,*K*= Boltzmann constant] ( )_{B} - The free electrons in a metal follow ________ distribution ( )
- Maxwell-Boltzmann
- Fermi-Dirac
- Bose-Einstein
- None of the above

- At 0K, the probability of an electron in Fermi energy level is ( )
- 0
- 0.5
- 1.0
- less then 1

- The probability of an electron in Fermi energy level at non-0K temperature is ( )
- 0
- 0.5
- between 0.5 and 1
- 1.0

- The expression for the density of electron states at energy
*E*can be represented as*[h =*Planck's constant] ( ) - Black body is ( )
- a body that absorbs all wave lengths of radiation incident on it at low temperatures
- a body that emits different wave lengths of radiation at high temperatures
- Both a and b
- None of the above

- If the temperature of a black body is increased then the intensity of radiation for each wave length of radiation ( )
- remains constant
- decreases
- increases
- None of the above

- In black body radiation, the wave length corresponds to maximum intensity moves towards ________ wave length
region with increase of temperature. ( )
- shorter
- longer
- both a and b
- none of the above

- Wien's law explains the black body radiation in ________ region of the spectrum. ( )
- the short wave length region
- the medium wave length region
- the longer wave length region
- None of the above

- The planck's black body radiation distribution can be represented as ( )
- Rayleigh–Jeans explained black body radiation based on ________ of energy ( )
- equipartition
- kinetic theory
- fifth power of wave length
- All the above

- Rayleigh–Jeans law is applicable to ________ wave length region of the spectrum ( )
- Planck's low is applicable to ________ spectrum of black body radiation ( )
- entire
- only a part of the
- shorter wave length
- longer wave length

- Quantum theory successfully explains: ( )
- interference and diffraction
- polarization and black body radiation
- photoelectric effect and Compton effect
- All the above

- Dual nature [particle and wave] of matter was proposed by: ( )
- de Broglie
- Planck
- Einstein
- Newton

- The wavelength associate with a particle of mass
*m*moving with a velocity*v*is [*h =*Planck's constant] ( ) - The wavelength of de Broglie wave associated with an electron when accelerated in a potential difference
*V*is [*h*= Planck's constant,*e*= charge on an electron] ( ) - When an electron is accelerated in a potential difference
*V*, then the de Broglie wave associated with it in nm is: ( ) - If
*m*_{0}is the rest mass of an electron, accelerated through a potential difference*V*, then its relativistically corrected wavelength is [c = velocity of light] ( ) - The existence of matter waves is proved by: ( )
- Davisson and Germer
- G.P. Thomson
- O. Stern
- all

- The gold foil used in G.P. Thomson experiment is: ( )
- single crystal
- polycrystalline
- amorphous
- none

- The interplanar spacing in gold foil obtained by G.P. Thomson and by X-ray method is: ( )
- 4.08 Å and 4.06 Å
- 4.06 Å and 4.08 Å
- 4.8 Å and 4.6 Å
- 4.6 Å and 4.8 Å

- The target material in Davisson and Germer experiment is: ( )
- gold
- nickel
- tungsten
- copper

- The spur in the curve drawn between the number of electrons collected against the angles of galvanometer with an
incident beam in Davisson and Germer experiment is more clear for a anode voltage of: ( )
- 40 V
- 44 V
- 54 V
- 68 V

- The diffraction angle for nickel crystal in Davisson and Germer experiment is: ( )
- 50°
- 65°
- 25°
- 130°

- Schrödinger's wave equation for a particle of mass
*m*have energy*E*, moving along X-axis is: ( ) - The wave function ‘
*ψ*’ associated with a moving particle: ( )- is not an observable quantity
- does not have direct physical meaning
- is a complex qu antity
- all

- By solving one-dimensional Schrödinger's time-independent wave equation for a particle in the well gives: ( )
- quantum numbers
- discrete values of energy and zero point energy
- wave function associated with the particle
- all

- The energy possessed by a particle of mass ‘
*m*’ in*n*th quantum state in a one-dimensional potential well of width ‘*L*’ is: ( ) - In G.P. Thomson's experiment ___________ particles are used for diffraction. ( )
- slow neutrons
- fast neutrons
- slow electrons
- fast electrons

- In photoelectric effect, absorption or emission of energy takes place: ( )
- in the form of packets of energy called quanta
- continuously
- both a and b
- none

- When an electron is accelerated through a potential difference of 100 V, then it is associated with a wave of wavelength equal to: ( )
- 0.112 nm
- 0.1227 nm
- 1.227 nm
- 12.27 nm

- ___________ proposed matter waves but he did not prove it experimentally. ( )
- Thomson
- Davisson and Germer
- de Broglie
- Schrödinger

- The interplanar spacing of gold foil obtained by G.P. Thomson's method agree very well with that obtained by ___________ method. ( )
- interference
- X–ray
- diffraction
- none

- The thickness of gold foil used in G.P. Thomson experiment was: ( )
- 10
^{-3}m - 10
^{-5}m - 10
^{-7}m - 10
^{-8}m

- 10
- The original aim of Davisson and Germer experiment was to find the ___________ by a metal target. ( )
- intensity of scattered electrons
- electron diffraction
- to find interplanar spacing
- none

- The de Broglie wavelength of electrons obtained from Davisson and Germer experiment is: ( )
- 0.0165 nm
- 0.165 nm
- 1.65 nm
- 16.5 nm

- Schrödinger's wave equation was derived based on ___________idea of matter waves. ( )
- de Broglie's
- Schrödinger's
- Thomson's
- Newton's

- If
*ψ*(*x, y, z, t*) represent wave function associated with a moving particle, then |*ψ*(*x, y, z, t*)|^{2}represents: ( )- intensity
- amplitude
- probability density
- none

- If
*E*_{1}is the ground state energy of a particle, then the increase in energy from*n*th energy level to next higher level is: ( ) - The normalized wave function of a particle in a one-dimensional potential well of width ‘
*L*’ is: ( ) - The most probable position of a particle in one-dimensional potential well of width ‘L’ in the first quantum state is: ( )

#### Answers

1. d | 2. a | 3. c |

4. a | 5. b | 6. b |

7. b | 8. b | 9. a |

10. b | 11. b | 12. c |

13. b | 14. a | 15. c |

16. c | 17. a | 18. a |

19. b | 20. a | 21. b |

22. a | 23. d | 24. a |

25. b | 26. c | 27. b |

28. c | 29. d | 30. b |

31. a | 32. b | 33. c |

34. b | 35. a | 36. d |

37. d | 38. a | 39. c |

40. a | 41. b | 42. c |

43. b | 44. d | 45. a |

46. b | 47. a | 48. c |

49. a | 50. d | 51. c |

#### Review Questions

- Derive an expression for the distribution of identical distinguishable particles based on classical concepts?
- Derive Maxwell–Boltzmann distribution expression?
- Derive Fermi–Dirac distribution equation for identical indistinguishable particles.
- Derive Bose–Einstein distribution for bosons.
- Distinguish between Maxwell–Boltzmann distribution, Fermi–Dirac distribution and Bose–Einstein distribution.
- Write short notes on (i) electron gas (ii) photon gas and (iii) Fermi energy.
- Derive an expression for density of available electron states between the energies
*E*and*E + dE.* - Explain black body radiation.
- What are matter waves? Explain their properties.
(Set-1–Sept. 2006), (Set-1–May 2004), (Set-1–May 2003), (Set-4–Nov. 2003)

- Derive the expression for de Broglie wavelength.
(Set-1–Sept. 2006), (Set-1–May 2004), (Set-1–May 2003), (Set-4–Nov. 2003)

- Explain the concept of matter waves.
(Set-1, Set-3–May 2006), (Set-2, Set-3–June 2005), (Set-2–May 2004), (Set-4–May 2003)

- Describe Davisson and Gemer's experiment and explain how it enabled the verification of wave nature of matter.
(Set-1, Set-3–May 2006), (Set-2, Set-3–June 2005), (Set-2–May 2004), (Set-4–May 2003)

- Derive time-independent Schrödinger's wave equation for a free particle.
(Set-2–Sept. 2006), (Set-4–June 2005), (Set-1–Nov. 2004), (Set-2–Nov. 2004), (Set-2, Set-3–May 2003)

- Explain the physical significance of wave function.
(Set-1, Set-3–Sept. 2008), (Set-4–June 2005), (Set-2, Set-3–Nov. 2004), (Set-3-–May 2003), (Set-1–Nov. 2003)

- Show that the wavelength ‘λ’ associated with an electron of mass ‘
*m’*and kinetic energy*‘E’*is given by(Set-2–Sept. 2006), (Set-3–Nov. 2004), (Set-2–May 2003)

- Assuming the time-independent Schrödinger's wave equation, discuss the solution for a particle in one-dimensional potential well of infinite height.
(Set-3–Nov. 2004), (Set-1–Nov.2003)

- Describe in detail, with a neat diagram, Davison and Germer experiment to show that particles behave like waves.
(Set-4–Sept. 2008), (Set-4–Nov. 2004), (Set-3–Nov. 2003)

- Describe an experiment to establish the wave nature of electrons.
(Set-4–May 2007), (Set-1–June 2005), (Set-4–May 2004)

- Explain the difference between a matter wave and an electromagnetic wave.
(Set-4–May 2007), (Set-1–June 2005), (Set-4–May 2004)

- Show that the wavelength of an electron accelerated by a potential difference
*V*volts is for non-relativistic case.(Set-4–May 2007), (Set-1–June 2005), (Set-4–May 2004)

- Apply Schrödinger's equation to the case of a particle in a box and show that the energies of the particle are quantized.
(Set-2–Nov. 2003

- Explain de Broglie hypothesis.
(Set-1–May 2008), (Set-3–May 2004

- Explain G.P. Thomson's experiment in support of this hypothesis.
(Set-3-–May 2004)

- Starting with the plane wave equation associated with a moving particle, formulate the time-independent Schrödinger's wave equation.
(Set-2–Nov. 2003)

- Explain in detail the properties of matter waves
(Set-2–May 2008)

- Describe G.P. Thomson's experiment in support of de Broglie hypothesis.
(Set-1–May 2008)

- Describe Davisson and Germer experiment to verify the wave nature of matter.
(Set-2–May 2008)

- Distinguish between a matter wave and an electromagnetic wave.
(Set-4–May 2008)

- Describe G.P. Thomson's experiment to study electron diffraction.
(Set-4–May 2008)

- Discuss the de Broglie hypothesis of duality of matter particles.
(Set-4–Sept. 2008), (Set-3–May 2008)

- Describe G.P. Thomson's experiment to verify the wave nature of matter.
(Set-3–May 2008)

- Discuss, in detail, the physical significance of wave function.
(Set-2–Sept. 2008)

- Show that the energies of a particle in a 3-dimensional potential box, are quantized.
(Set-2–Sept. 2008)

- Deduce an expression for energy of an electron confined to a potential box of width ‘
*x*’(Set-1–Sept. 2008)

- Derive 3-dimensional, time independent Schrödinger wave equation for an electron.
(Set-1–Sept. 2008)

- Derive one-dimensional, time independent Schrödinger wave equation for an electron.
(Set-3–Sept. 2008)

- What is de Broglie's hypothesis? Describe any one experiment by which the hypothesis was verified.
- Show that the energies of a particle in a potential box are quantized.
- Explain the concept of wave-particle duality and obtain an expression for the wavelength of matter waves.
- Discuss the de Broglie's hypothesis of duality of material particles. Give in detail the experiment of Davisson and Germer in support of the hypothesis.
- What are matter waves? Obtain an expression for the wavelength of matter waves.
- Explain in detail the Davisson and Germer's experiment to prove the existence of matter waves.
- Explain the dual nature of light. Describe G.P. Thomson's experiment to verify the dual nature of matter.
- Obtain eigen values of energy, normalized wave functions and probability functions for a particle in one-dimensional potential box of side ‘
*L*’. - Derive the Schrödinger's time-independent wave equation of an electron and write the significance of orthonormality condition of wave function.
- Give the graphical presentation for the probability of metallic electron in its second allowed state as a function of length of potential box.
- Show that for a quantum particle confined to an infinite deep potential box with finite length, the energy levels are quantized.
- Write the time-independent Schrödinger's wave equation of electron and write the physical interprelation of
*ψ*. - With suitable picturization of potential well and imposed boundary conditions, derive the Schrödinger's equation for metallic electron and prove that energy levels are unequally spaced.