Optionals IAS Mains Physics: Questions 93 - 99 of 306

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Question 93

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The energy levels of a hydrogen atom are given by where Show that

Question 94

Planck Mass, Planck Length, Planck Time, Plank Temperature and Planck Energy
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Appeared in Year: 2015

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Short Answer▾

Establish that:

Question 95

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Appeared in Year: 2009

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Essay▾

The quantum mechanical probability distribution function of an electron of ground state of hydrogen atom is

Using the result deduct that N is proportional to. (20 Marks)

Explanation

Applying integration by parts on the above integral

It is clear that exponential will “beat” any power function as of tends to.

Hence N is proportional to.

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Question 96

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Appeared in Year: 2009

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Using Planck’s radiation formula, deduce Wein’s displacement law. (10 Marks)

Explanation

  • Planck’s formula is given by
  • Differentiating it partially with respect to, we get
  • For maximum value of emission, i. e. for maximum value of must be equal to zero i.e..
  • Putting, the above expression becomes
  • It is obvious from this equation that there must be a root in the neighbourhood of 5. Applying the method of approximation the exact value

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Question 97

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Appeared in Year: 1999

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For a particle confined in a one dimensional potential well of length L the wave function is

And outside

Calculate the expectation values of. (20 Marks)

Explanation

Formula used:

Here we have applied formula of integration by parts

I. e.

Denominator: (calculated previously)

Numerator:

Divided and multiplied by 2 and

Question 98

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Appeared in Year: 2013

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If the forces acting on a particle are conservative, show that the total energy of the particle which is the sum of the kinetic and potential energies is conserved.

Explanation

If the particle is acted upon by the forces which are conservation, that is, if the forces are derivable form a scalar potential energy function in the manner then the total energy of the particle (Kinetic + Potential) is conserved.

Suppose, under the action of such a force, a particle moves form particle will be

If the particle moves distance in

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Question 99

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Appeared in Year: 2013

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Show that a particle of rest mass and total energy E and linear momentum satisfies the relation Where c is the velocity of light in free space.

Explanation

A simple relation between relativistic momentum (and relativistic energy (E) of a particle of rest mass is easily obtained from the expressions for and . For, clearly , so that

Which enables us to determine the velocity of the particle from the values of its momentum and energy since, clearly,

Another important relation between and E may be obt

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