Question

Consider an electron that is confined to the \(x y\) -plane by a twodimensional rectangular infinite potential well. The width of the well is \(w\) in the \(x\) -direction and \(2 w\) in the \(y\) -direction. What is the lowest energy that is shared by more than one distinct state, that is, two different states having the same energy?

Short answer

Answer: The lowest shared energy is \(2\left(\dfrac{h^2}{8m}\right)\dfrac{1}{w^2}\).

Step-by-step solution

Understand the energy formula

For an electron in a two-dimensional infinite potential well, the energy formula is given by: \(E_{n_x, n_y} = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{w^2} + \dfrac{n_y^2}{(2w)^2} \right)\) where \(E_{n_x, n_y}\) is the energy corresponding to energy levels \((n_x, n_y)\), \(h\) is Planck's constant, \(m\) is the mass of the electron, and \(w\) is the width of the well in the \(x\) direction.

List the energy levels

To find the lowest energy shared by more than one distinct state, let's find the energy values for the first few energy levels \((n_x, n_y)\). We start from \((1,1)\) and increase \(n_x\) and \(n_y\) by 1 until we find a shared energy. We can simplify the energy formula by replacing \(\frac{h^2}{8m}\) with constant \(A\) and taking \(w=1\) to analyze the energy levels: \(E_{n_x, n_y} = A \left(\dfrac{n_x^2}{1^2} + \dfrac{n_y^2}{(2)^2} \right) = A \left(n_x^2 + \dfrac{n_y^2}{4} \right)\) Let's list the energy levels \((n_x, n_y)\) and their corresponding energies: \((1,1): E_{11} = A(1^2 + \dfrac{1^2}{4}) = A\left(1 + \dfrac{1}{4}\right) = \dfrac{5}{4}A\) \((1,2): E_{12} = A(1^2 + \dfrac{2^2}{4}) = A\left(1 + 1\right) = 2A\) \((2,1): E_{21} = A(2^2 + \dfrac{1^2}{4}) = A\left(4 + \dfrac{1}{4}\right) = \dfrac{17}{4}A\) \((2,2): E_{22} = A(2^2 + \dfrac{2^2}{4}) = A\left(4 + 1\right) = 5A\)

Identify the lowest shared energy

Analyzing the energy levels listed above, we find that energy levels \((1,2)\) and \((2,1)\) have the same energy (\(E_{12}=E_{21}=2A\)). Since no other shared energy values are found for these energy levels, the lowest shared energy is \(2A\). In terms of the original energy expression: \(E = 2\left(\dfrac{h^2}{8m}\right)\dfrac{1}{w^2}\)