What Are the Quantum Numbers and Properties of an Electron in a 3D Box?

[EugP]

Homework Statement

An electron is confined in a box where a = 60nm, b = 20nm, c = 35nm. For each of the first four quantum levels above the ground state give:

a) The quantum numbers
b) The speed of an electron trapped at that level
c) The wavelength of light needed to excite an electron from the ground state to that level.

Homework Equations

$$E_{n_1n_2n_3}=\frac{h^2}{8m}(\frac{n_1^2}{a^2}+\frac{n_2^2}{b^2}+\frac{n_3^2}{c^2})$$

The Attempt at a Solution

I don't know where to start with this problem. Can someone give me a hint how to begin? It would be greatly appreciated.

 

[nate808]

To begin solving this problem, we can start by using the given equation for the energy levels of an electron in a 3D box. This equation relates the quantum numbers (n1, n2, n3) to the energy level of the electron. From this equation, we can determine the quantum numbers for the first four energy levels above the ground state by plugging in the given values for a, b, and c. The quantum numbers will be different for each energy level, so you will need to solve for each one individually.

To find the speed of an electron trapped at each energy level, we can use the equation for the kinetic energy of an electron, which is equal to 1/2mv^2. The mass of an electron is a known constant, so we can use this equation to solve for the speed of the electron at each energy level by plugging in the calculated energy levels from the previous step.

Lastly, to find the wavelength of light needed to excite an electron from the ground state to each energy level, we can use the equation for the energy of a photon, which is given by E=hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of light. We can use the energy levels calculated in the first step to solve for the wavelength of light needed for each energy level.

I hope this helps you get started on solving the problem. Let me know if you have any further questions or need clarification on any of the steps. Good luck!

 

[Blueshift5]

Sure, let's break down the problem step by step. First, we need to understand the given information. We know that an electron is confined in a box with dimensions a = 60nm, b = 20nm, and c = 35nm. This means that the electron is limited in its movement within these dimensions.

a) The quantum numbers represent the energy levels that the electron can occupy within the box. In this case, we are looking at the first four levels above the ground state. The ground state is represented by the quantum numbers n1 = 1, n2 = 1, and n3 = 1. For the first excited state, n1 = 2, n2 = 1, and n3 = 1. For the second excited state, n1 = 1, n2 = 2, and n3 = 1. For the third excited state, n1 = 1, n2 = 1, and n3 = 2.

b) The speed of an electron trapped at a certain energy level can be calculated using the formula v = √(2E/m), where E is the energy level and m is the mass of the electron. Using the given equation, we can calculate the energy levels for each quantum state and then plug them into the formula to find the corresponding speed.

c) To calculate the wavelength of light needed to excite an electron from the ground state to a certain energy level, we can use the equation E = hc/λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. We can use the energy levels calculated in part b) to find the energy of the photon needed to excite the electron, and then use that to solve for the wavelength.

I hope this helps you get started on the problem. Remember to always break down the information given and use the appropriate equations to solve for the desired values.