Question

When an electron in a one-dimensional box makes a transition from the \(n\) = 1 energy level to the \(n\) = 2 level, it absorbs a photon of wavelength 426 nm. What is the wavelength of that photon when the electron undergoes a transition (a) from the \(n\) = 2 to the \(n\) = 3 energy level and (b) from the n = 1 to the \(n\) = 3 energy level? (c) What is the width \(L\) of the box?

Short answer

(a) Calculate wavelength for n=2 to n=3 using \(L\); (b) do the same for n=1 to n=3; (c) find \(L\) from n=1 to n=2 transition.

Step-by-step solution

Energy Levels in a Box Formula

The energy levels of an electron in a one-dimensional box are given by \[ E_n = \frac{n^2h^2}{8mL^2} \] where \(n\) is the energy level, \(h\) is Planck's constant, \(m\) is the mass of an electron, and \(L\) is the width of the box. We use this formula to calculate the energy difference between levels.

Energy Difference for Transition

For a transition between two levels \(n_1\) and \(n_2\), the energy absorbed or emitted is \[ \Delta E = E_{n_2} - E_{n_1} = \frac{(n_2^2 - n_1^2)h^2}{8mL^2} \] This energy corresponds to the photon's energy, \(E = \frac{hc}{\lambda}\), where \(\lambda\) is the wavelength. For the transition from \(n\) = 1 to \(n\) = 2, we set \(\Delta E = \frac{hc}{426 \text{ nm}}\) to solve for \(L\).

Calculate Width L of the Box

First, find \(L\): Using the initial condition \(\Delta E_{1 \to 2} = \frac{hc}{426 \times 10^{-9} \text{ m}}\), substitute back into \[ \frac{3h^2}{8mL^2} = \frac{hc}{426 \times 10^{-9}} \] to solve for \(L\). Calculate values accordingly.

Transition from n=2 to n=3

For the transition from \(n\) = 2 to \(n\) = 3, \[ \Delta E_{2 \to 3} = \frac{(3^2 - 2^2)h^2}{8mL^2} \] is calculated. Using \(E = \frac{hc}{\lambda}\), \[ \lambda = \frac{hc}{\Delta E_{2 \to 3}} \] Calculate \(\lambda\) using the derived value of \(L\).

Transition from n=1 to n=3

For the transition from \(n\) = 1 to \(n\) = 3, calculate \[ \Delta E_{1 \to 3} = \frac{(3^2 - 1^2)h^2}{8mL^2} \]. The energy difference gives\[ \lambda = \frac{hc}{\Delta E_{1 \to 3}} \]. Solve for the wavelength of the emitted photon using the derived \(L\).

Key concepts

Electron Energy Levels

In quantum mechanics, electrons in an atom or a confined space have specific energy levels. These levels are like steps on a ladder that the electron can "jump" between. The electron cannot have energy values that lie between these levels. Instead, it must absorb or release a specific amount of energy to change levels.
This is because of the quantization of energy in a system, which is one of the fundamental principles of quantum mechanics.

• Each energy level corresponds to a unique value of the quantum number, often denoted by \( n \).
• The lowest energy level is known as the ground state, while higher levels are called excited states.

In the context of an electron in a one-dimensional box, these energy levels depend on the quantum number \( n \), Planck's constant \( h \), the electron's mass \( m \), and the size of the box \( L \).

One-Dimensional Box

The concept of a one-dimensional box is a simplified model used to explain the behavior and energy states of electrons confined in a specific area. It assumes the electron moves freely between rigid, impenetrable walls, like a particle in a box.
This model is essential for understanding basic quantum mechanical principles without complex potentials.

• The walls reflect the electron, creating standing wave patterns, similar to how waves behave in a bounded medium like a guitar string.
• The energy of the electron is dependent on the box's width \( L \) and the electron's quantum number \( n \).

The formula for calculating these energy levels in this model is given by \[ E_n = \frac{n^2h^2}{8mL^2} \]. This formula illustrates that the energy is inversely proportional to the square of the box's width, which highlights how confinement affects energy state distribution.

Photon Wavelength

A photon is a particle of light that carries energy. The wavelength of a photon is directly related to the energy it represents. In this context, the wavelength gives insight into the photon's energy during electron transitions between energy levels.
When an electron transitions between these levels, it absorbs or emits a photon with a wavelength that corresponds to the energy difference.

• Photon energy \( E \) is related to its wavelength \( \lambda \) by the equation \( E = \frac{hc}{\lambda} \), where \( c \) is the speed of light.
• A shorter wavelength corresponds to higher energy, while a longer wavelength means lower energy.

Calculating the photon wavelength during an electron transition allows us to understand better how the energy levels differ and how the electron's movement affects energy absorption and emission.

Energy Absorption

Energy absorption occurs when an electron in a lower energy level absorbs enough energy to move to a higher level. This energy often comes from photons, which are absorbed by the electron, causing a transition.
This principle explains the occurrence of spectral lines and the absorption spectra of different elements.

• The amount of absorbed energy is equal to the difference in energy between the initial and final states.
• This energy difference is crucial because it determines the photon's characteristics, such as its wavelength.

In the exercise, calculating the amount of energy needed for transitions helps to determine other parameters of the problem, such as the box's width \( L \) and the characteristics of the photon involved in these transitions.