What's the quantum number for a particle in an infinite square well if the particle's energy is 64 times the ground-state energy?

8

Explanation:

The energy levels is given as

E(n) = n² * h² / ( 8 * m * L²), where

n = 1, 2, 3, 4,... etc

At ground state energy (n = 1), therefore is E(g) = h² / (8 * m * L²).

We can then say that

E(n) = n² * Eg

Therefore, to have E = 64 Eg, we must have n² = 64. And for n² to be equal to 64, we find the square root of n

n = √64

n = 8

Essentially, the needed quantum number is 8