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