Solve your IGNOU Doubts

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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Question:

Describe the software techniques used for improving the signal-to-noise ratio of an electronic system.

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Question:

i) Write down the eigenfunctions for the $\psi_{300}$ and $\psi_{310}$ states of the hydrogen atom.
ii) Write the eigenvalues of $\hat{L}^2$ and $\hat{L}_z$ for the states $\psi_{300}$ and $\psi_{310}$ .
iii) Calculate the most probable value of r for the hydrogen atom in the state $\psi_{210}$ .

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Question:

Explain the terms lock range and capture range with reference to PLL. Why is the capture range smaller than the lock range?

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Question:

Explain the working of a shunt regulator using zener diode. What are the limitations of this circuit due to the non-idealities of the zener diode?

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Question:

Draw the circuit of antilogarithmic amplifier using an op-amp and obtain the expression for its output voltage.

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Question:

Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.

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Question:

Explain the working of an op-amp based full wave precision rectifier with the help of appropriate circuit diagram. What are its advantages over a simple diode rectifier?

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Question:

Show that for any stationary state of a symmetric potential well: $\langle x^2 \rangle \ne 0$ .

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Question:

What are CMRR and slew rate of an op-amp? Explain with examples their effects on the performance of an op-amp.

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Question:

A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity $3.0 \times 10^6\text{ ms}^{-1}$ , and then its velocity reduces to $2.0 \times 10^6\text{ ms}^{-1}$ , due to the difference in potential. Calculate the transmitted and reflected currents.