Two adjacent allowed energies of an electron in a one-dimensional box are 5.4 eV and 9.6 eV.

What is the length of the box?

E = energy in nth state = 5.4 eV

E' = energy in (n + 1) th state = 9.6 eV

L = length of the box

m = mass of electron = 9.1 x 10⁻³¹ kg

h = plank's constant = 6.63 x 10⁻³⁴

Energy in nth state is given as

E = n²h² / (8 m L²) eq-1

Energy in (n+1) th state is given as

E' = (n + 1) ²h² / (8 m L²) eq-2

dividing eq-1 by eq-2

E/E' = n² / (n + 1) ²

inserting the values

5.4 / 9.6 = n² / (n + 1) ²

n = 3

using eq-1

E = n²h² / (8 m L²)

inserting the values

5.4 x 1.6 x 10⁻¹⁹ = (3) ² (6.63 x 10⁻³⁴) ² / (8 (9.1 x 10⁻³¹) L²)

L = 7.9 x 10⁻¹⁰ m