JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Modern Physics

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge

Z e

are defined by their principal quantum number

n

, where

n ≫ 1

. Which of the following statement(s) is(are) true?
(A) Relative change in the radii of two consecutive orbitals does not depend on

Z

(B) Relative change in the radii of two consecutive orbitals varies as

1/ n

1/ n^{3}

(D) Relative change in the angular momenta of two consecutive orbitals varies as

1/ n

(A)Relative change in the radii of two consecutive orbitals does not depend on $Z$ ✓ Correct
(B)Relative change in the radii of two consecutive orbitals varies as $1/ n$ ✓ Correct
(C)Relative change in the energy of two consecutive orbitals varies as $1/ n^{3}$
(D)Relative change in the angular momenta of two consecutive orbitals varies as $1/ n$ ✓ Correct

Answer: (A), (B), (D)

Solution

For hydrogen-like atoms:

Radius:

r_{n} = \frac{n ^{2}}{Z} a_{0}

, so

Δ r_{n} = \frac{2 n a _{0}}{Z}

and

\frac{Δ r _{n}}{r _{n}} = \frac{2}{n}

(independent of

Z

, varies as

1/ n

). (A) and (B) are correct.

Energy:

E_{n} = \frac{- 13.6 Z ^{2}}{n ^{2}}

eV, so

Δ E_{n} = \frac{13.6 \times 2 Z ^{2}}{n ^{3}}

and

\frac{Δ E _{n}}{E _{n}} = - \frac{2}{n}

(varies as

1/ n

, not

1/ n^{3}

). (C) is incorrect.

Angular momentum:

L_{n} = \frac{nh}{2 π}

, so

Δ L_{n} = \frac{h}{2 π}

and

\frac{Δ L _{n}}{L _{n}} = \frac{1}{n}

. (D) is correct.

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · One or more correct options