Question

A system of N identical, non-interacting particles are placed in a finite square well of width L and depth V. The relationship between V and L are such that only 2 bound states exist. What is this relationship? Hint: What is the requirement on E for a bound state? For these two bound states, what is the expected energy of the system as a function of temperature? The result only applies when T is low enough so that the probability of populating a state with E>V is very small. In this range of temperatures, what is the constant volume c(T)? Again for constant volume, what is the change in entropy between two temperatures?

Answer 1

The wavefunction ψ(x) for a particle with energy E in a potential U(x) satisfies the time-independent Schrodinger equation.

Inside the well (−L ≤ x ≤ L), the particle is free. The wavefunction symmetric about x = 0 is ψ(x) = A cos kx where k = r 2mE ~ 2 .

Outside the well (−∞ < x < −L and L < x < ∞), the potential has constant value U > E. The wavefunction symmetric about x = 0 is ψ(x) = Be−α|x| where α = r 2m(U − E) ~ 2 . ψ(x) and its derivative are continuous at x = L: A cos kL = Be−αL

Ak sin kL = Bαe−αL

from which k tan kL = α

or, alternatively, θ sec θ = ±a

where θ = kL

and a = r 2mUL2 ~ 2

are equations for the allowed values of k. The equation with the positive sign yields values of θ in the first quadrant. The equation with the negative sign yields values of θ in the third quadrant. Solving numerically for an electron in a well with U = 5 eV and L = 100 pm yields the ground state energy E = 2.43 eV.