Q1 (a) What do you understand by wave packet? Explain the difference between phase velocity and group velocity.

Show that the de Broglie group

velocity associated with the wave packet is equal to the velocity of the particle.
A1(a) A wave packet is a wave that is localized in space and time. It is composed of a superposition of many waves with
different frequencies and wavelengths, and its shape changes as it propagates through space. Wave packets are
commonly used to describe the behavior of quantum particles, such as electrons and photons.

Phase velocity is the velocity at which the phase of a wave propagates. It is given by the ratio of the wavelength to the
period of the wave.

Group velocity, on the other hand, is the velocity at which the envelope of the wave packet propagates. It is the velocity at
which the energy or information is transmitted by the wave.

Now we need to use the de Broglie wavelength given by λ = h/p.

For a wave packet, we can define the group velocity as v_g = dω/dk. Using the de Broglie relation, we can write k = p/h,
and ω = E/h. Therefore, we have:

v_g = dω/dk = (dE/dp) / (h/p) = p/(h/m) = p/m = v

Hence, we have shown that the de Broglie group velocity associated with the wave packet is equal to the velocity of the
particle.

(b)Explain de-Broglie’s hypothesis. Describe an experiment for the confirmation of wave nature of electrons.

A1(b) De Broglie's hypothesis states that all matter, including particles like electrons and protons, exhibit wave-like
properties. The hypothesis proposes that for any particle with momentum p, there is an associated wavelength λ given by
λ = h/p.

The confirmation of the wave nature of electrons was provided by the famous double-slit experiment.

The experiment consisted of firing a beam of electrons at a thin crystal of nickel, which acted as a diffraction grating. The
nickel crystal was oriented in such a way that its atoms were regularly spaced, creating a series of parallel slits. The
electrons diffracted as they passed through the slits, creating an interference pattern on a detector screen placed behind
the crystal.

The resulting pattern showed bright and dark fringes. This demonstrated that electrons exhibit wave-like properties, and
the diffraction of electrons was consistent with the de Broglie hypothesis.

This provided strong evidence for the wave-like behavior of electrons, confirming de Broglie's hypothesis that all matter
exhibits wave-particle duality.

(c)Find the phase and group velocities of an electron whose de-Broglie wavelength is 1.2 Ǻ.

The de Broglie wavelength of an electron is given by:

λ = h/p

p = h/λ (eqn 1)

v(phase) = λf

where f is the frequency of the electron. Since the electron is not a classical wave, it does not have a well-defined
frequency. However, we can use the relation between the energy and momentum of a particle to calculate the phase
velocity:

E = p^2/2m

p = sqrt(2mE)

Putting this in eqn 1

λ = h/sqrt(2mE)

E = h^2/(2mλ^2)
λ = 1.2 Å, m = 9.10938356 × 10^-31 kg, h = 6.62607015 × 10^-34 J s

E = (6.62607015 × 10^-34 J s)^2 / (2 × 9.10938356 × 10^-31 kg × (1.2 × 10^-10 m)^2) = 2.761 × 10^-18 J

The phase velocity of the electron is then:

v_phase = sqrt(2E/m) = sqrt(2 × 2.761 × 10^-18 J / 9.10938356 × 10^-31 kg) = 6.24 × 10^5 m/s

To calculate the group velocity of the electron, we need to know the dispersion relation for electrons in a material. For free
electrons, the dispersion relation is:

E = p^2/2m

which is the same as the equation we used to calculate the phase velocity.

Therefore, for free electrons, the group velocity is equal to the phase velocity:

v_group = v_phase = 6.24 × 10^5 m/s

Q2 (a) State Heisenberg’s uncertainty principle. Give its experimental verification and use it to prove the non- existence of electrons in the
nucleus.
Heisenberg's uncertainty principle states that it is impossible to simultaneously measure the position and momentum of a
particle with perfect accuracy.

Δx*Δp ≥ h/4π

This principle has been verified by measuring the position and momentum of electrons in an atom using techniques like
electron microscopy and spectroscopy. The principle has also been demonstrated using a thought experiment involving
the diffraction of electrons through a pair of narrow slits, where the uncertainty in the position of the electron is related to
the width of the slit and the uncertainty in its momentum is related to the diffraction pattern.

The uncertainty principle can be used to prove the non-existence of electrons in the nucleus of an atom. If an electron
were to be located within the nucleus, its position would be confined to a very small region, resulting in a very small
uncertainty in its position. However, the momentum of the electron would be extremely high due to the strong nuclear
forces, resulting in a very large uncertainty in momentum. This violates the uncertainty principle, which states that the
product of the uncertainties in position and momentum cannot be smaller than a certain minimum value. Therefore, it is
impossible for electrons to exist within the nucleus of an atom.

(b) Find the smallest possible uncertainty in position of the electron moving with velocity 3×107m/s.

Δx*Δp ≥ h/4π

For an electron moving with velocity 3×10^7 m/s, we can calculate the momentum using the relativistic expression:

p = γmv

For electrons, we can use γ = 1, since their velocities are much less than the speed of light.

Therefore:

p = mv = 9.10938356 × 10^-31 kg × 3 × 10^7 m/s = 2.732815068 × 10^-23 kg m/s

Solving for the uncertainty in position, Δx, we get:

Δx ≥ h/4πΔp

Δx ≥ (6.62607015 × 10^-34 J s)/(4π × 2.732815068 × 10^-23 kg m/s) ≈ 2.4 × 10^-11 m

Therefore, the smallest possible uncertainty in the position of an electron moving with velocity 3×10^7 m/s is
approximately 2.4 × 10^-11 m.

Q3. (a) Derive Schrodinger time dependent wave equation for non-relativistic particle. Give physical interpretation of Hamilton
operator H.
The Schrödinger time-dependent wave equation for a non-relativistic particle can be derived starting from the classical
Hamiltonian for a particle in a potential energy field:
H = (p^2 / 2m) + V(x)

To obtain the quantum mechanical equivalent of this equation, we replace the classical variables p and x with their
corresponding operators in the position and momentum representations, respectively:

p → (-iħ ∂/∂x)

x→x

where ħ is the reduced Planck constant. The Hamiltonian operator H becomes:

H = (-ħ^2 / 2m) (∂^2 / ∂x^2) + V(x)

The wave function Ψ(x,t) of the particle in the position representation can be written as:

Ψ(x,t) = A(x)exp(i(kx - ωt))

Substituting this expression for Ψ into the time-dependent Schrödinger equation:

iħ ∂Ψ/∂t = HΨ

(∂A/∂t)exp(i(kx - ωt)) + A(x)(-iħk^2/2m)exp(i(kx - ωt)) + V(x)A(x)exp(i(kx - ωt)) = iħ(∂A/∂t)exp(i(kx - ωt))

Dividing by A(x)exp(i(kx - ωt)), we obtain:

iħ (∂A/∂t) = (-ħ^2 / 2m) (∂^2 A / ∂x^2) + V(x)A(x)

This is the time-dependent Schrödinger wave equation for a non-relativistic particle.

The Hamilton operator H represents the total energy of the system and contains information about the kinetic and
potential energy of the particle. The first term (-ħ^2 / 2m) (∂^2 / ∂x^2) represents the kinetic energy operator and the
second term V(x) represents the potential energy operator. The Hamiltonian operator describes the dynamics of the
quantum system.
 (b) Set up the Schrodinger equation for a free quantum particle. Write two examples of functions which are not acceptable as wave
functions. Write another two which are acceptable.
The Schrödinger equation for a free quantum particle is given by:

iħ ∂Ψ/∂t = (-ħ^2 / 2m) ∂^2Ψ/∂x^2

Here are two examples of functions that are not acceptable as wave functions:

1. A function that is not square-integrable, such as:

Ψ(x) = sin(x)

This function oscillates infinitely and does not decay to zero at infinity, so its integral over all space is not finite.

2. A function that is not single-valued, such as:

Ψ(x) = √x

This function has two values for x < 0 and x > 0, which violates the requirement that the wave function be single-valued.

On the other hand, here are two examples of functions that are acceptable as wave functions:

1. A Gaussian wave packet, such as:

Ψ(x) = Ae^(-x^2/2σ^2)exp(ikx)

2. A plane wave, such as:

Ψ(x) = exp(ikx)

where k is the wave number.

These functions are square-integrable, single-valued, and continuous, and can be normalized to satisfy the probability
interpretation of the wave function.

Q4. (a) Set up the Schrodinger equation for a particle in an infinite well/box (one-dimensional) Solve it for eigen values and Eigen
functions and plot the first three Eigen functions Ψ1, Ψ2, and Ψ3.
The Schrödinger equation for a particle in a one-dimensional infinite well of width L is given by:

−(ħ^2 / 2m) (∂^2Ψ/∂x^2) = EΨ

The infinite well potential is zero inside the well (0 < x < L) and infinite outside the well. This means that the wave function
must be zero at the boundaries of the well:

Ψ(0) = Ψ(L) = 0

To solve this equation, we assume that the wave function has the form:

Ψ(x) = A sin(kx) + B cos(kx)

where A and B are constants, and k is a constant to be determined.

Applying the boundary conditions Ψ(0) = Ψ(L) = 0, we get:

B = 0 and kL = nπ, where n is an integer.

Thus, the allowed values of k are:

k_n = nπ/L

and the wave function is:

Ψ_n(x) = A_n sin(k_nx)

where A_n is a normalization constant.

The energy of the particle can be obtained from the Schrödinger equation:

E(n) = (ħ^2π^2n^2)/(2mL^2)

Thus, the eigenvalues are quantized and depend only on the quantum number n. The corresponding eigen functions are
given by:

Ψ(n(x)) = √(2/L) sin(nπx/L)

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

Here are the plots of the first three eigenfunctions Ψ1, Ψ2, and Ψ3:

(YAHAN PAR THODA SA SPACE CHOD DENA)

The energy levels are equally spaced, with E_n+1 − E_n = ħ^2π^2/(2mL^2). This result shows that the energy of the
particle is quantized due to the boundary conditions imposed by the infinite well potential.

(b)Find the probability that a particle trapped in a box ‘l’ wide can be found between 0.30 l and 0.65l for the first excited state.

For a particle trapped in a box of width L, the probability density function is given by:

|Ψ_n(x)|^2 = (2/L)sin^2(nπx/L)

where n is the quantum number.

The probability of finding the particle between x_1 and x_2 is given by the integral:

P(x_1 ≤ x ≤ x_2) = ∫x_1^x_2 |Ψ_n(x)|^2 dx

For the first excited state (n=2), the probability of finding the particle between 0.30L and 0.65L is:

P(0.30L ≤ x ≤ 0.65L) = ∫0.30L^0.65L |Ψ_2(x)|^2 dx

= ∫0.30L^0.65L (2/L)sin^2(2πx/L) dx

= (2/L)[(0.35L/2) - (0.30L/2) - (1/4π)cos(4π(0.65L)/L) + (1/4π)cos(4π(0.30L)/L)]

= 0.147

Therefore, the probability of finding the particle between 0.30L and 0.65L for the first excited state is 0.147 or 14.7%

(c)A particle of mass m is moving in an infinitesimal deep square well potential extending from x=0and x=l. Show that <E> =
n2h2/8ml2 and <px> = 0.

The Schrödinger equation for a particle in an infinitesimal square well potential is:

−(ħ^2 / 2m) (∂^2Ψ/∂x^2) = EΨ

Inside the well, the potential is zero, so the Schrödinger equation reduces to:

−(ħ^2 / 2m) (∂^2Ψ/∂x^2) = EΨ

with the boundary conditions Ψ(0) = Ψ(l) = 0.

The solutions to this equation are:

Ψ_n(x) = √(2/l)sin(nπx/l)

where n = 1, 2, 3, ... is an integer.

The energy levels are:

E_n = (n^2π^2ħ^2)/(2ml^2)

The expectation value of the energy <E> is:

<E> = ∫Ψ*HΨ dx

where H is the Hamiltonian operator.

Using the wave function Ψ_n(x), we get:

<E> = ∫Ψ_n*HΨ_n dx

= E_n ∫|Ψ_n(x)|^2 dx

= E_n

Therefore, the expectation value of the energy is equal to the energy of the nth energy level.

The expectation value of momentum <p_x> is:

<p_x> = ∫Ψ*p_xΨ dx

where p_x is the momentum operator.

Using the wave function Ψ_n(x), we get:

<p_x> = −iħ ∫Ψ_n*(∂/∂x)Ψ_n dx

= −iħ [(2/l)nπcos(nπx/l)]_0^l

=0

Therefore, the expectation value of momentum is zero.

Hence, <E> = n^2π^2ħ^2/(2ml^2) and <p_x> = 0 for a particle of mass m moving in an infinitesimal deep square well
potential extending from x=0 and x=l
Q5. (a) What is potential barrier and tunnel effect? Calculate the transmission probability for rectangular barrier for the condition of
E<Vo, where E is energy of particle and Vo is barrier height.
A potential barrier is a region in space where the potential energy of a particle is higher than the energy of the particle.

The tunnel effect is the phenomenon in quantum mechanics where a particle can pass through a potential barrier even
though its energy is less than the height of the barrier. This is possible because the wave function of the particle can
extend into the region beyond the barrier.

For a rectangular potential barrier with height V_o and width a, the transmission probability T is given by:

T = (4k_1^2k_2^2)/(4k_1^2k_2^2 + (k_1^2 + k_2^2)^2sinh^2(ka))

where k_1 = sqrt(2mE)/ħ, k_2 = sqrt(2m(E - V_o))/ħ

For the condition of E < V_o, we have k_2 = iκ, where κ = sqrt(2m(V_o - E))/ħ is imaginary. Substituting this into the above
equation and simplifying, we get:

T = (16k_1^2κ^2)/(4k_1^2κ^2 + (k_1^2 - κ^2)^2sin^2(ka))

This equation shows that there is a non-zero probability of the particle tunneling through the barrier even when its energy
is less than the barrier height.

(b) Electrons with energies 1 eV and 2 eV are incident on a barrier 10eV high and 0.5 nm wide. Find (a) the respective transmission
probabilities (b) how are these affected if the barrier is doubled in width?
The transmission probability of an electron with energy E through a rectangular potential barrier of height V_o and width a
is given by:

T = (4k_1^2k_2^2)/(4k_1^2k_2^2 + (k_1^2 + k_2^2)^2sinh^2(ka))

where k_1 = sqrt(2mE)/ħ, k_2 = sqrt(2m(E - V_o))/ħ

(a) For an electron with energy 1 eV, we have E = 1.602 × 10^-19 J

k_1 = sqrt(2mE)/ħ = 1.374 × 10^10 m^-1 k_2 = sqrt(2m(E - V_o))/ħ = 1.301 × 10^10 m^-1

Similarly, for an electron with energy 2 eV, we have E = 3.204 × 10^-19 J

k_1 = sqrt(2mE)/ħ = 1.947 × 10^10 m^-1 k_2 = sqrt(2m(E - V_o))/ħ = 1.708 × 10^10 m^-1

Substituting these values into the transmission probability equation, we get:

T(1 eV) = 0.057

T(2 eV) = 0.259

Therefore, the respective transmission probabilities are 0.057 and 0.259 for electrons with energies 1 eV and 2 eV incident
on the barrier.

(b) If the width of the barrier is doubled to 1 nm, the value of ka in the transmission probability equation will double. This
will cause the sinh^2 term in the denominator to become much larger than 1, making the transmission probability very
small. Therefore, increasing the width of the barrier will decrease the transmission probability for both electrons.

Q6. Set up the Schrodinger equation for the case of Harmonic oscillator. Write its energy Eigen values and explain zero-point energy.
The Schrodinger equation for the one-dimensional harmonic oscillator is given by:

(-ħ^2/2m) (d^2/dx^2)Ψ(x) + (1/2) kx^2 Ψ(x) = EΨ(x)

To solve this equation, we make the substitution:

Ψ(x) = u(x) exp(-x^2/2l^2)

where u(x) is a polynomial and l is a characteristic length scale.

This leads to the Hermite differential equation, whose solutions are the Hermite polynomials H_n(x). The energy eigen
values of the harmonic oscillator are given by:
 E_n = (n + 1/2) ħω

where ω = sqrt(k/m) is the angular frequency of the oscillator.

The zero-point energy of the harmonic oscillator is the minimum energy that the system can have, corresponding to the
ground state with n = 0. In the case of the harmonic oscillator, the zero-point energy arises because even in the ground
state, the particle has a non-zero uncertainty in its position and momentum. This leads to a minimum energy of E_0 = 1/2
ħω, which is the zero-point energy.

Q7. Write down the principle and working of scanning electron microscope and transmission electron microscope.

Scanning Electron Microscope (SEM): The scanning electron microscope (SEM) is an imaging tool that uses a focused
beam of high-energy electrons to create a detailed image of the surface of a sample. The principle behind the SEM is that
when a beam of electrons is scanned across a sample, the electrons interact with the atoms in the sample and produce
signals that can be detected and used to create an image.

The working of SEM involves the following steps:

1. A sample is prepared by coating it with a thin layer of conductive material, such as gold, to prevent charging and to
enhance the image contrast.
2. The sample is placed in the vacuum chamber of the SEM, which prevents the scattering of electrons by air molecules.
3. A beam of high-energy electrons is generated by an electron gun and focused onto the sample by a series of
electromagnetic lenses.
4. The electron beam is scanned across the sample using a set of electromagnetic coils.
5. As the electron beam interacts with the atoms in the sample, it produces signals such as secondary electrons,
backscattered electrons, and X-rays.
6. These signals are detected by specialized detectors and used to create an image of the sample surface.

Transmission Electron Microscope (TEM): The transmission electron microscope (TEM) is an imaging tool that uses a beam
of high-energy electrons to transmit through a thin sample to create a detailed image of its internal structure. The
 principle behind the TEM is similar to that of the SEM, except that the electrons are transmitted through the sample rather
than being scattered by its surface.

The working of TEM involves the following steps:

1. A sample is prepared by cutting it into a thin section, typically less than 100 nm thick, using specialized tools such as a
microtome.
2. The sample is placed on a thin film of support material, such as carbon or copper, to hold it in place and to enhance image
contrast.
3. The sample is placed in the vacuum chamber of the TEM, which has a high vacuum to prevent the scattering of electrons
by air molecules.
4. A beam of high-energy electrons is generated by an electron gun and focused onto the sample by a series of
electromagnetic lenses.
5. The electron beam is transmitted through the sample and interacts with the atoms in its path, producing signals such as
scattered electrons, diffracted electrons, and X-rays.
6. These signals are detected by specialized detectors and used to create an image of the sample's internal structure.