Solve your IGNOU Doubts

Question:

A population of single-celled organisms was grown in a Petri dish over a period of 24 hours. The number of organisms at a given time is recorded in the table below. Fit the data in the table below to an exponential model $y = \alpha \exp(\beta x)$ :

x (time in hours)	0	4	8	12	16	20	24
y	25	36	52	68	85	104	142

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Question:

The rotational constant for the excited electronic state of a particular molecule is 20% smaller as compared to the rotational constant of its ground electronic state. Determine the values for the first 15 lines of the R-branch. Discuss the variation of the separation between successive transition lines of this branch.

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Question:

The HCl molecule shows its first and second overtone transitions at $566.0 \text{ cm}^{-1}$ and $8339.0 \text{ cm}^{-1}$ , respectively. Assuming anharmonic oscillator model, determine the anharmonicity constant, the equilibrium vibrational frequency, zero point energy, and the force constant of this molecule.

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Question:

What are the advantages and drawbacks of the Secant method when compared to the Bisection Method and the Newton-Raphson Method?

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Question:

Write the Hamiltonian of a multi-electron atom. Obtain an expression for the effective potential experienced by an electron of this atom under the central field approximation.

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Question:

Use the Secant method with initial guesses of $x_0 = 1$ and $x_1 = 2$ to calculate the root/s of the equation $f(x) = x^3 - x - 1$ up to three iterations. Calculate the absolute relative approximate error at each stage.

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Question:

Explain why a polynomial is a useful choice for an interpolating function.

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Question:

The series for e^x can be written as:

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
Calculate the truncation error when the first four terms of the series are used to evaluate e^1.5.

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Question:

Explain round-off error and truncation error, with one example of each.

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Question:

Using molecular orbital method, construct all pssible trial wavefunctions (including spin) for a diatomic molecule. Of these functions, which trial function is expected to have the minimum energy? Give justification for your answer.

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Question:

Using Thomas-Fermi method, derive an expression for the total kinetic energy of N electrons in a cubic box of side L, and hence obtain the expression for pressure.

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Question:

The anti-symmetric spin function for a two electron system is given as:
$$\chi_-(1, 2) = \frac{1}{\sqrt{2}}[\alpha(1)\beta(2) - \beta(1)\alpha(2)]$ $
Prove that $\chi_-(1, 2)$ is an eigenfunction of $(\hat{S})^2$ .

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Question:

Consider K meson of mass m_K decaying into two pions of mass $m_\pi$ each, in the laboratory frame. Show that the energy of each pion is equal to $\frac{m_K}{2}$ and the momenta are $p_\pi = \frac{1}{2}\sqrt{m_K^2 - 4m_\pi^2}$ .

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Question:

a) Determine the shift in energy caused by relativistic correction to kinetic energy for $n=2$ and $n=1$ levels of hydrogen atom. Using appropriate selection rules, show which transitions are allowed.

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Question:

Write down the wave function for $|\pi^-\rangle$ . Use the isospin raising operator to derive the wavefunctions for $|\pi^0\rangle$ and $|\pi^+\rangle$ .