Question

An electron is confined in a one-dimensional infinite potential well of \(1.0 \mathrm{nm}\). Calculate the energy difference between a) the second excited state and the ground state, and b) the wavelength of light emitted by this radiative transition.

Short answer

#Answer# The wavelength of the light emitted during the transition between the second excited state and the ground state of an electron confined in a one-dimensional infinite potential well is approximately: \(λ ≈ 382.7\mathrm{nm}\)

Step-by-step solution

Calculate the energy levels for the ground state and second excited state

The energy levels of an electron in a one-dimensional infinite potential well are given by the formula: \(E_n = \dfrac{n^2h^2}{8mL^2}\) where \(E_n\) is the energy of the nth level, n is an integer denoting the energy level, h is the Planck's constant, m is the mass of the electron, and L is the width of the well. For the ground state (n = 1) and the second excited state (n = 3), we have : \(E_1 = \dfrac{1^2h^2}{8mL^2}\) and \(E_3 = \dfrac{3^2h^2}{8mL^2}\)

Determine the energy difference between the two states

The energy difference between the second excited state (n = 3) and the ground state (n = 1) is given by: \(ΔE = E_3 - E_1 = \dfrac{h^2(3^2 - 1^2)}{8mL^2}\) Using the given well width \(L = 1.0\mathrm{nm}\) and substituting the values of the mass of the electron \(m = 9.10938356 × 10^{-31}\mathrm{kg}\) and Planck's constant \(h = 6.62607004 × 10^{-34}\mathrm{Js}\), we can calculate the energy difference. \(ΔE = \dfrac{(6.62607004 \times 10^{-34}\mathrm{Js})^2(3^2 - 1^2)}{8(9.10938356 \times 10^{-31}\mathrm{kg})(1.0 \times 10^{-9}\mathrm{m})^2}\)

Calculate the wavelength of the emitted light

According to the energy conservation principle, the energy difference is equal to the energy of the emitted photon, given by: \(ΔE = h\nu = h \dfrac{c}{λ}\) where ν is the frequency of the emitted light, c is the speed of light, and λ is the wavelength we need to calculate. Rearranging for λ: \(λ = \dfrac{hc}{ΔE}\) Now, we can substitute the values of the energy difference, Planck's constant, and the speed of light \(c = 3 \times 10^8\mathrm{m/s}\) to calculate the wavelength of the emitted light. \(λ = \dfrac{(6.62607004 \times 10^{-34}\mathrm{Js})(3 \times 10^8\mathrm{m/s})}{ΔE}\) Using the previously calculated energy difference and simplifying will give us the wavelength of the emitted light.

Key concepts

Infinite Potential Well

The concept of an infinite potential well is a fundamental idea in quantum mechanics, illustrating how particles like electrons behave when confined to extremely tiny regions of space. Imagine a particle trapped inside a box with impenetrable walls. In the real world, no wall can be perfectly impenetrable, but this idealization simplifies understanding of quantum behavior.

An infinite potential well is defined by walls with infinite energy, meaning that a particle within the well cannot escape—its potential energy outside the well is considered infinitely large. Within the well, the particle exhibits wave-like properties and can only occupy certain quantized energy levels. With this model, we can describe phenomena that occur at the nanometer scale, especially those pertaining to electrons in atoms or semiconductor materials.

Energy Levels of Electrons

The energy levels of an electron confined within an infinite potential well can only take specific discrete values. Quantum mechanics dictates that these levels are related to the size of the well and other physical constants. The formula \(E_n = \frac{n^2h^2}{8mL^2}\) shows that energy levels depend on an integer, n, which can only be a positive whole number—these are known as quantum numbers.

The ground state corresponds to \(n=1\), and as n increases, the electron occupies higher energy levels, or 'excited states.’ These levels get progressively closer together as the energy increases, meaning the electron has a wealth of ways it could gain or lose energy through interactions, such as absorbing or emitting photons.

Planck's Constant

Planck's constant, symbolized by \(h\), is a crucial quantity in quantum physics. Max Planck introduced the constant in 1900, helping spark the development of quantum mechanics. It characterizes the quantization of energy, momentum, and angular momentum in the microscopic world.

With the value of approximately \(6.626 \times 10^{-34} \text{ J s}\), Planck's constant establishes the scale at which quantum effects become noticeable. It sets the proportionality between the energy \(E\) of a photon and its frequency \(u\), described by the famous equation \(E = hu\). The constant is also influential in calculating energy levels within quantum wells, as seen in the given exercise, and is critical in understanding the nature of the quantized energy transitions in atomic and molecular systems.

Radiative Transition

A radiative transition occurs when an electron moves between energy levels in an atom, molecule, or quantum well, either absorbing or emitting a photon in the process. This transition is governed by the conservation of energy: the energy difference between the initial and final states must equal the energy of the photon involved.

In the case of emission, an electron drops from a higher energy state to a lower one, and a photon carrying the energy difference is emitted. This photon's energy (\(E = hu\)) can be observed as light with a specific color or wavelength. Understanding these transitions is key to technologies like lasers, fluorescence microscopy, and even the analysis of starlight to determine the composition of distant stars.

Wavelength Calculation

In the context of a radiative transition, the emitted photon has a specific wavelength corresponding to the energy difference between the electron's initial and final states. To calculate the wavelength \(\lambda\) of this photon, we use the equation \(\lambda = \frac{hc}{\Delta E}\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\Delta E\) is the energy difference.

By rearranging the equation for frequency \(E = hu\) to solve for wavelength, we understand why Planck's constant is integral for linking the energy of a photon to its wavelength. This calculation is significant in many areas of physics, chemistry, and engineering, for example in determining the color of light emitted by LEDs or the energy required to promote electrons during photosynthesis in plants.