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QUESTION: 1

A certain excited state of hydrogen atom is known to have a life of 2.5 × 10^{–14} * s*. The minimum error, with which the energy of the excited state can be measured is.

Solution:

The energy time uncertainty relation is

The correct answer is: 0.026 **eV**

QUESTION: 2

Which of the following is the *eigenfunction* of the linear momentum operator with a positive *eigenvalue *

Solution:

Compare this with

is the eigenvalue

The correct answer is: e^{ikx}

QUESTION: 3

An eigenfunction of an operator The corresponding eigenvalue is.

Solution:

The correct answer is: α^{2}

QUESTION: 4

The ground state energy of an electron confined to a box 1**Å** wide is.

Solution:

a = 1 * 10^{-10}m m = 9.1 * 10^{-31}kg

Energy of electron in ground state

E_{1} = h^{2}/8ma^{2}

⇒ [(6.625 * 10^{-34})^{2}]/(8 * 9.1 * 10^{-31} * 1 * 10^{-10})

E_{1} = 6.016 * 10^{-18} J

QUESTION: 5

Which of the following values of * A* will properly normalize the wave function of a rigid dumbbell rotating about its centre having φ dependence as ψ(φ) = Ae

Solution:

For normalization

The correct answer is:

QUESTION: 6

A particle of mass *m* has the wave function ψ(x, t) = e^{iωt} (α cos kx + β sin kx) where α, β are complex constants and ω and *k* are real constants. The probability current density is equal to :

Solution:

Probability current density

subtract (2) and (1)

QUESTION: 7

If de-Broglie wavelength of an electron is 7.3Å. Then the velocity of electron is. Given, m_{e} = 9.1 × 10^{–31} kg and h = 6.6 × 10^{–34} Js.

Solution:

Λ = h/mv

v = h/mΛ

v = (6.6*10^{-34})/(9.1 × 10^{–31 }* 7.3 * 10^{-10})

= 9.93 × 10^{4} **m/s**

The correct answer is: 9.93 × 10^{4} **m/s**

QUESTION: 8

Which of the following is correct set of *eigenvalues* of the Hermitian matrix

Solution:

The correct answer is: 1, 3

QUESTION: 9

If ψ is a normalized solution of the Schrödinger equation and *Q* is the operator corresponding to a physical quantity ** x**, then the quantity may be integrated in order to obtain :

Solution:

The correct answer is: mean value of *x*

QUESTION: 10

A particle limited to the ** x**–axis has the wave frequency ψ = ax between

Solution:

The correct answer is: *a*^{2}/4

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