Solve your IGNOU Doubts

Question:

Obtain the generating function F₁(q, Q). Using F₁(q, Q), obtain F₂(q, P).

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Question:

Determine the Hamitonian and the value of $\alpha$ so that the transformation is canonical.

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Question:

The transformation of a 1D harmonic oscillator is given as:

$$Q = \frac{\alpha}{q} \text{ and } P = \alpha pq^2$

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Question:

If $G = J_y = zp_x - xp_z$ , using Poisson bracket show that G generates a roation in X-Z plane.

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Question:

Write down the expression for the Routhian for a bead of mass m sliding due to gravity on an elliptical wire for which the Lagrangian is given as:

$$L = \frac{1}{2}m(a^2 \sin^2 \theta + b^2 \cos^2 \theta)\dot{\theta}^2 - mgb \sin \theta$

Hence derive the equation of motion.

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Question:

The Lagrangian of a relativistic particle with charge e is given as:

$$L = -mc^2 \sqrt{1 - \left( \frac{v}{c} \right)^2} - e\phi + e\vec{A} \cdot \vec{v}$

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Question:

The Lagrangian of a bead of m moving without friction along a wire bent in the shape of a parabola in the X-Y pane with $y = x^2$ is given by:

$$L = \frac{1}{2}m(1 + 4x^2)\dot{x}^2 - mgx^2$

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Question:

Determine the Hamiltonian and the Hamilton's equation of motion:

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Question:

The Lagrangian for a harmonic oscillator if given as:

$$L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) + \frac{1}{2}m\omega^2(x^2 + y^2)$
The transformed coordinate under rotation about the Z-axis is given as:

$$x' = x \sin \theta + y \cos \theta$
Determine the corresponding y' so that the transformation leaves the Lagrangian invariant.

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Question:

Obtain the Larmor formula for the total radiated power of a non-relativistic accelerating charge and explain its significance.

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Question:

Derive the current conservation equation from the Dirac equation.

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Question:

Starting from the scalar and vector potentials, derive the expression for the electromagnetic fields produced by a small oscillating electric dipole.

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Question:

Explain how the expression for the energy levels obtained by solving Klein Gordon equation for a Coulomb field differs from the results derived from Schrödinger equation. Why is this solution not able to explain the fine-structure splitting of the energy levels?

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Question:

Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: $V(r) = V_0 \frac{a}{r} \exp\left( -\frac{r}{a} \right)$ . You may use:
$$\int_0^\infty e^{-\alpha x} \sin(\beta x) dx = \frac{\beta}{\alpha^2 + \beta^2} \text{ for } \alpha > 0$ $

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Question:

Describe the principle of guided wave propagation in a rectangular waveguide. Derive the expression for the cut-off frequency of $\text{TE}_{mn}$ modes in terms of waveguide dimensions.

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Question:

A charged particle of mass m and charge q, is confined to a one-dimensional box of side L with $0 \leq x \leq L$ . At t > 0, an electric field $\vec{E} = E_0 e^{-\alpha t} \hat{i}$ acts on the particle where $\alpha$ is a constant. If the particle is in the ground state when t < 0, calculate the probability that it will be in the first excited state for t > 0.

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Question:

Consider the two state problem in which the unperturbed Hamiltonian $\hat{H}_0$ has just two eigenkets, $|1\rangle$ and $|2\rangle$ with: $\hat{H}_0|1\rangle = E_1|1\rangle$ ; $\hat{H}_0|2\rangle = E_2|2\rangle$ , and E₂ > E₁. The system is subjected to a time-dependent perturbation: $\hat{V}(t) = V_0 \cos(\omega t) \left[ |1\rangle\langle2| + |2\rangle\langle1| \right]$ .
Calculate the probability for the system to be in the state $|2\rangle$ at time t, given that it is in the state $|1\rangle$ at $t = 0$ .

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Question:

Show that if the electric field of the incident wave is normal to the plane of incidence, the electric fields of the reflected and transmitted waves are also normal to the plane of incidence.

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Question:

Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:

$V(x) = \begin{cases} a|x| & x \geq 0 \\ 2a|x| & x < 0 \end{cases}$

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Question:

A plane electromagnetic wave of frequency $1 \text{ GHz}$ is normally incident on a medium with relative permittivity $\varepsilon_r = 4$ . Calculate the wavelength of the wave in the medium.