Question

A particle is trapped inside a one-dimensional infinite potential well of width \(19.3 \mathrm{nm}\). The energy difference between the \(n=2\) and the \(n=1\) states is \(2.639 \cdot 10^{-25} \mathrm{~J}\). What is the mass of the particle?

Short answer

Question: Given a particle trapped in a one-dimensional infinite potential well of width 19.3 nm, and the energy difference between the n=1 and n=2 states is \(2.639 \times 10^{-25} \mathrm{J}\), find the mass of the particle. Answer: The mass of the particle is approximately \(9.03 \times 10^{-30} \mathrm{kg}\).

Step-by-step solution

Recall the formula for the energy levels in an infinite potential well

The energy levels of a particle in an infinite potential well can be given by the formula: \(E_n = \dfrac{n^2 \pi^2 \hbar^2}{2ma^2}\) where \(E_n\) is the energy of the level n, n is the quantum number, \(\hbar\) is the reduced Planck's constant, m is the mass of the particle, and a is the width of the potential well.

Calculate the energy difference

We are given the energy difference between the \(n=1\) and \(n=2\) states \(ΔE = 2.639 \cdot 10^{-25} \mathrm{~J}\). Using the energy level formula from Step 1, we can write the energy difference as: \(ΔE = E_2 - E_1 = \dfrac{2^2 \pi^2 \hbar^2}{2ma^2} - \dfrac{1^1 \pi^2 \hbar^2}{2ma^2}\)

Simplify the equation

Now we can simplify the energy difference equation as follows: \(ΔE = \dfrac{\pi^2 \hbar^2}{2ma^2}(2^2 - 1^2)\) \(ΔE = \dfrac{3 \pi^2 \hbar^2}{2ma^2}\)

Solve for the mass of the particle

We can now solve for the mass of the particle by rearranging the equation: \(m = \dfrac{3 \pi^2 \hbar^2}{2 \Delta E a^2}\) We are given the width of the potential well \(a = 19.3 \mathrm{nm}\) and \(ΔE = 2.639 \cdot 10^{-25} \mathrm{~J}\). Also, we know that the reduced Planck's constant \(\hbar = 1.054 \times 10^{-34} \mathrm{Js}\). Plug in the values and calculate the mass: Note: We need to convert the width a from nanometers to meters as the energy and Planck's constant values are in SI units: \(a = 19.3 \mathrm{nm} \times \dfrac{1 \mathrm{m}}{10^9 \mathrm{nm}} = 19.3\times 10^{-9}\mathrm{m}\) \(m = \dfrac{3 (\pi)^2 (1.054 \times 10^{-34} \mathrm{Js})^2}{2 (2.639 \times 10^{-25} \mathrm{J})(19.3 \times 10^{-9} \mathrm{m})^2}\) \(m \approx 9.03 \times 10^{-30} \mathrm{kg}\) The mass of the particle inside the one-dimensional infinite potential well is approximately \(9.03 \times 10^{-30} \mathrm{kg}\).