Solve your IGNOU Doubts

Question:

Explain the Lorentz transformation between two inertial frames. Obtain the transformation laws for the electric and magnetic fields.

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Question:

Obtain the radial action and the angular action variables. Hence, express the total action.

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Question:

A particle of mass m moves under a potential $V(r) = -k/r, k > 0$ . The total energy of the system is
$$E = \frac{1}{2}m\dot{r}^2 + \frac{l^2}{2mr^2} - \frac{k}{r}$

where l is the angular momentum.

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Question:

A simple pendulum with a bob of mass m and a string of length l₀ with Hamiltonian
$$H = \frac{p_{\theta}^2}{2ml_0} - mgl_0 \cos \theta .$

Construct Hamilton-Jacobi equation and obtain the characteristic function in integral form. Write the new coordinates P, Q.

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Question:

The Hamiltonian of a system with two degrees of freedom is given as:

$$H(q_1, q_2, p_1, p_2) = \frac{p_1^2}{2} + \frac{p_2^2}{2} + \sin(q_1 + 2q_2).$

Determine the new Hamiltonian K.

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Question:

The Hamiltonian of a system with two degrees of freedom is given as:

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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$ $