Identical particles are trapped in one-dimensional wells with infinite potential energy at the walls. The widths L of the traps and the quantum numbers n of the particles are Rank them according to the kinetic energies of the particles, least to greatest.
A) 1, 2, 3, 4
B) 4, 3, 2, 1
C) 1 and 3 tied, then 2, then 4
D) 4, then 1 and 3 tied, then 2
E) 2, then 1 and 3 tied, then 4

Question

Identical particles are trapped in one-dimensional wells with infinite potential energy at the walls. The widths L of the traps and the quantum numbers n of the particles are  Rank them according to the kinetic energies of the particles, least to greatest.
A) 1, 2, 3, 4 
B) 4, 3, 2, 1 
C) 1 and 3 tied, then 2, then 4 
D) 4, then 1 and 3 tied, then 2 
E) 2, then 1 and 3 tied, then 4

Quizplus · Accepted Answer

The kinetic energy of a particle in a one-dimensional box is given by $E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$. As the width of the traps is the same for all particles, the kinetic energy is proportional to $n^2$. Thus, particle 4 with $n=4$ has the greatest kinetic energy, followed by particles 1 and 3 with $n=1$ and $n=3$ tied for second, and particle 2 with $n=2$ having the least kinetic energy. Therefore, option D is the correct answer.

Identical Particles Are Trapped in One-Dimensional Wells with Infinite Potential