Quantum Mechanics I [Optionals IAS Mains Physics]: Questions 1 - 6 of 36

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Question 1

Quantum Mechanics I
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Appeared in Year: 2011 (IFS)

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Essay▾

On the basis of uncertainty principle calculate the size of Hydrogen atom.

(Paper-2) (Section - A)

Explanation

  • From a hydrogen atom, it is not possible to predict exact the position of the electron or the momentum of the electron. Every time the electron is at somewhere but it has amplitude to in different places so there is a probability of it being found in different places.
  • These places cannot all be at the nucleus; we shall suppose there is a spread in p…

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Question 2

Appeared in Year: 2011 (IFS)

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Essay▾

Solve the Schrodinger equation for a potential step function given by,

and calculate the reflection and transmission coefficients. Show that for there is a finite probability of finding the particle in a classically forbidden region.

Explanation

  • Consider . The time independent Schrodinger equation in one dimension is given as,

  • And according to question, the potential step function is given as,

  • Let the region of negative is denoted by and region of positive is denoted by . Also corresponding wave functions and , respectively.
  • From given conditions, Schrodinger equation are,

Where,

And,

Where…

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Question 3

Appeared in Year: 2011 (IFS)

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Essay▾

For a quantum mechanical system prove that all energy eigen – values are real and if , then the corresponding eigen functions are orthogonal.

Explanation

  • The eigenvalue equation, of the sets of energies and wave functions obtained by any quantum mechanics problems is given as,

    For another value of the quantum number, we can write

  • Multiply equation (1) by and the complex conjugate of equation (2) by . Then subtract the two expressions and integrate over .

  • The left hand side of equation (3) becomes zero…

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Question 4

Appeared in Year: 2011

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Essay▾

Solve the Schrodinger equation for a particle of mass m in an infinite rectangular well defined by the potential

Obtain the normalized Eigen function and the corresponding Eigen values. (25 Marks)

Explanation

The One-Dimensional Time Independent Schrodinger Equation
  • The one-dimensional time independent Schrodinger equation of the particle inside the well where , is

    … eq. (1)

  • As the potential energy is infinite outside the well, i.e. for , the probability of finding the particle outside the well is zero. Therefore any wave function should vanish outside …

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Question 5

Quantum Mechanics I

Appeared in Year: 2011

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Essay▾

Calculate (15 Marks)

Explanation

For the previous problem, let as find

where As the given wave function is normalized

Question 6

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Appeared in Year: 2007

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Essay▾

Solve the Schrodinger equation for a linear harmonic oscillator obtain the eigen values and the corresponding Eigen functions. (40 Marks)

Explanation

  • The Schrodinger equation for harmonic oscillator is

    … eq. (1)

  • And solve it directly by the series method. Let us introduce a dimensionless variable

    … eq. (2)

  • In terms of , the Schrodinger equation reads

    … eq. (3)

  • Where k is the energy, in units of

    … eq. (4)

  • Our problem is to solve equation (3) and in the process obtain the “allowed” values of k (and h…

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