Consider a particle in a square well. It has mass m and the length of the box is I. Consider the following situations (1) the particle of mass m in the ground state (2) the particle of mass m in the first excited state (3) a particle of mass m in the ground state (4) a particle of mass m in a box of length 2 (5) a particle of mass m in a box of length 21 in the first excited state Which is the proper order from lowest energy to highest energy? (Use". > to reflect the relative values; for example (1) (3) く (4) 5). 2) Select the correct answer O 1) (2) (3) (4) (5)

The energy of the particle in a square power well is given by the Schrödinger equation, with

E₁ = RA (h² / 8m L²) n²

With h the Planck constant the mass of the particle, L the length of the box and n an integer starting 1

We find the energy for each situation presented

1) For the fundamental state n = 1

E₁ = RA (h / 8m L²)

2) the first excited state corresponds to n = 2

E₂ = RA (h / 8m L²) 2²

E₂ = E₁ 4

3) E₁ = RA (h / 8m L²)

4) The length of the box is L = 2L

E' = RA (h' / 8m (2L) ²)

E' = RA (h' / 8m L²) Ra ¼

E' = E₁ / 2

5) L = 2l first excited state

n = 2

E '' = RA (h / 8m (2L) ²2) 2²

E '' = RA (h ' / 8m (L) ²) 4/2

E '' = E₁ 2

All energies are in relation to the fundamental state (E1), milking from least to greatest

E1 / 2
4 <3 = 1 <5 <2