# Optionals IAS Mains Physics: Questions 66 - 72 of 306

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## Question 66

### Explanation

Expectation value of any operator is defined as

If the given wave function is normalized then as given in this case wave function is normalized so we need to only find

In spherical polar coordinates (the volume element is given by).

Note:

Put

Using the formula for gamma function

( Contribution comes from the integration of parts)

Again using the f

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## Question 67

Appeared in Year: *2010*

### Describe in Detail

Essay▾A series circuit has an inductance of micro henries, a capacitance of microfarad and a resistance of ohms. Fine the resonant frequency and quality factor of the circuit. (20 Marks)

### Explanation

The resonant frequency of series LCR circuit is given by

Quality factor

## Question 68

Appeared in Year: *2003*

### Write in Short

Short Answer▾Calculate the De Broglie wavelength associated with electrons accelerated through a potential difference of. (10 Marks)

## Question 69

Appeared in Year: *2003*

### Describe in Detail

Essay▾Explain what do you understand by Heisenberg uncertainty principle. Using this principle determine the energy of the ground state of one dimensional simple Harmonic oscillator. (10 Marks)

### Explanation

- Since a particle is represented within the context of quantum mechanics by means of a wave function corresponding to the particle’s wave, and since wave function cannot be localized, then a microscopic particle is somewhat spread over space and unlike classical particles, cannot be localized in space. In addition, we have seen in the double- slit e

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## Question 70

Appeared in Year: *2001*

### Describe in Detail

Essay▾The phase velocity in a material is where k is the propagation constant. Prove that the group velocity will be half of the phase velocity. (20 Marks)

### Explanation

Hence proved.

## Question 71

### Write in Short

Short Answer▾Show that the Fermi velocity is related to the energy of electrons through the relation

## Question 72

Appeared in Year: *2015*

### Write in Short

Short Answer▾The observed vibrational frequency of molecule is. What is the effective force constant of the molecule? (10 Marks)