Solve your IGNOU Doubts

Question:

Define and explain the gyromagnetic ratio and Landé g-factor, stating their physical importance.

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Question:

Derive an expression for the magnetic moment associated with orbital motion of an electron in an atom.

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Question:

Obtain an expression for the magnetic vector potential, $\vec{A}$ due to a circular current loop of radius a carrying steady current I, at a point on its axis at distance x from the centre. Discuss the limiting case for $x \gg a$ .

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Question:

Define the electric displacement vector $\vec{D}$ . Starting from Maxwell's equations, deduce Gauss's law in dielectric media. Explain the physical significance of bound and free charges.

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Question:

calculate the force experienced by the charge q.

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Question:

Show that in the terms of the $|\uparrow\rangle$ and $|\downarrow\rangle$ basis vectors defined by
$\hat{S}_z|\uparrow\rangle = \frac{\hbar}{2}|\uparrow\rangle$ ; $\hat{S}_z|\downarrow\rangle = -\frac{\hbar}{2}|\downarrow\rangle$ , we can write: $\hat{S}_x = \frac{\hbar}{2} \left[ |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow| \right]$

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Question:

i) Write down the angular momentum states $|j, m_j\rangle$ and calculate the matrix elements of $\hat{J}_x, \hat{J}_y$ and $\hat{J}_z$ for $j = 1/2$ .
ii) For the angular momentum state $|3,1\rangle$ show that $J_+|3,1\rangle = \hbar\sqrt{10}|3,2\rangle$ and $J_-|3,1\rangle = 2\hbar\sqrt{3}|3,0\rangle$ .

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Question:

determine the electrostatic potential in the region above the plane,

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Question:

An infinite plane conducting surface is held at zero potential. A point charge q is placed at a distance d above the plane. Using the method of images,

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Question:

Obtain Poisson's equation for electrostatics using the result derived in (a) above. Write down the general solution of Poisson's equation and explain its physical significance for a given charge distribution.

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Question:

For the simple harmonic oscillator
i) Show that $[\hat{H}, \hat{a}] = -\hbar\omega\hat{a}$ .
ii) Calculate the matrix element $\langle n|\hat{x}|n + 1\rangle$

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Question:

Starting from Coulomb's law, derive Gauss's law in differential form.

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Question:

Derive the Heisenberg equations of motion for $\hat{S}_x$ and $\hat{S}_y$ for the Hamiltonian $\hat{H} = \omega\hat{S}_z$ .

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Question:

A uniformly charged solid cylinder of radius a and length L carries a total charge Q. Assuming the observation point lies at a distance x from the centre of the cylinder along its axis and that $L \gg a$ , derive an expression for the electric field on the axis of the cylinder. Discuss the limiting case when $x \gg L$ .

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Question:

$|1\rangle$ and $|2\rangle$ are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator $\hat{A}$ in this basis is given by the spectral representation: $\hat{A} = 3|1\rangle\langle1| - 1|2\rangle\langle2|$ . For a normalized state given by

$$|\psi\rangle = \frac{1}{2}|1\rangle + \frac{\sqrt{3}}{2}|2\rangle, \text{ calculate } \langle\hat{A}\rangle \text{ and the root mean square deviation in } \hat{A} .$

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Question:

The spectral representation of an operator $\hat{\Omega}$ in a two-dimensional orthonormal basis $|\phi_1\rangle,|\phi_2\rangle$ is

$$\hat{\Omega} = 3|\phi_1\rangle\langle\phi_1| + \sqrt{2}|\phi_1\rangle\langle\phi_2| + \sqrt{2}|\phi_2\rangle\langle\phi_1| + 2|\phi_2\rangle\langle\phi_2|$

Determine the matrix elements of $\Omega$ .

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Question:

If an operator $\hat{O}$ is Hermitian and an operator $\hat{U}$ is unitary, show that the operator $\hat{U}\hat{O}\hat{U}^{-1}$ is Hermitian.

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Question:

Write an Assembly language program for microcontroller 8051 to compute the average of 6 bytes data stored from RAM location 45H and store the result at RAM location 30H.

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Question:

Consider the following state vectors: $|\psi\rangle = \begin{pmatrix} 1 \\ i \\ -1 \end{pmatrix}; |\phi\rangle = \begin{pmatrix} 2 \\ 1-i \\ i \end{pmatrix}$ .

Calculate (i) the norm of $|\psi\rangle$ and (ii) the inner product $\langle\psi|\phi\rangle$ .

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Question:

Output of a 4-bit DAC for digital input of 1001 is 4.5 V. Determine its step size and percentage resolution. What will be the output for digital input of 1110?

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