Question

An electron in a one-dimensional box has ground-state energy 2.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

Short answer

The wavelength is approximately 77.6 nm.

Step-by-step solution

Understand the Problem

An electron in a 1D box has quantized energy levels given by the formula \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( n \) is the principal quantum number, \( h \) is Planck's constant, \( m \) is the electron's mass, and \( L \) is the length of the box. We know the ground-state energy \( E_1 = 2.00 \text{ eV} \). We need to find the wavelength of the photon absorbed when the electron transitions from \( n=1 \) to \( n=3 \), which is the second excited state.

Set Up Energy Relation

For the ground state, the energy is \( E_1 = \frac{h^2}{8mL^2} = 2.00 \text{ eV} \). For the second excited state (\( n=3 \)), the energy is \( E_3 = \frac{9h^2}{8mL^2} \). The change in energy \( \Delta E \) when the electron transitions from \( E_1 \) to \( E_3 \) is given by \( \Delta E = E_3 - E_1 \).

Calculate Change in Energy

Substitute the expressions for \( E_1 \) and \( E_3 \) into \( \Delta E = E_3 - E_1 \):\[ \Delta E = \frac{9h^2}{8mL^2} - \frac{h^2}{8mL^2} = \frac{8h^2}{8mL^2} = 8 \times 2.00 \text{ eV} = 16.00 \text{ eV} \]

Use Energy-Photon Wavelength Relation

The energy of the absorbed photon \( \Delta E \) is related to the wavelength \( \lambda \) by the equation: \( \Delta E = \frac{hc}{\lambda} \), where \( c \) is the speed of light. We rearrange to find \( \lambda \):\[ \lambda = \frac{hc}{\Delta E} \]

Calculate the Wavelength

Substitute \( \Delta E = 16.00 \text{ eV} = 16.0 \times 1.602 \times 10^{-19} \text{ J} \), \( h = 6.626 \times 10^{-34} \text{ J s} \), and \( c = 3.00 \times 10^8 \text{ m/s} \) into the wavelength expression:\[ \lambda = \frac{(6.626 \times 10^{-34} \text{ J s})(3.00 \times 10^8 \text{ m/s})}{16.0 \times 1.602 \times 10^{-19} \text{ J}} \approx 77.6 \text{ nm} \]

Key concepts

Particle in a Box

The 'particle in a box' concept is a fundamental idea in quantum mechanics. It simplifies understanding how particles, like electrons, behave in confined spaces. Imagine an electron trapped in a perfectly rigid, one-dimensional box with impenetrable walls. This is an idealized representation that helps us explore quantum behaviors.

In this model, the potential energy of the particle is zero inside the box and infinitely large at the walls. Hence, the particle can't escape. This confinement leads to quantization of energy levels, meaning the particle can only possess specific energy values. Due to the constraints of the box, electrons exhibit wave-like properties. Their behavior is described by a wavefunction, a mathematical representation indicating where the particle is likely to be within the box.

The quantized energy levels of a particle in a 1D box are determined by the formula: \[E_n = \frac{n^2 h^2}{8mL^2}\]where \(n\) is the principal quantum number, \(h\) is Planck's constant, \(m\) is the particle's mass, and \(L\) is the length of the box. The principal quantum number, \(n\), can take positive integer values like 1, 2, 3, etc., representing different energy states. Each state corresponds to a specific electron standing wave within the box.

Energy Levels

In quantum mechanics, energy levels refer to the specific energy states that a system, like an electron in a 'particle in a box', can occupy. These energy levels are quantized, meaning the particle can't have energy values between these levels. Instead, it can only exist in these predefined states.

For an electron in a one-dimensional box, the energy levels are calculated with the equation mentioned earlier, \(E_n = \frac{n^2 h^2}{8mL^2}\). Here:

• \(n\) is the principal quantum number.
• \(h\) is Planck’s constant \((6.626 \times 10^{-34} \text{ Js})\).
• \(m\) is the electron's mass \((9.109 \times 10^{-31} \text{ kg})\).
• \(L\) is the box's length.

The values of \(E_n\) determine how much energy an electron possesses in each state. For instance, the ground state (first energy level) has the lowest energy value. The energy increases as \(n\) increases, meaning higher energy levels are more energetically costly.

Understanding these levels is crucial for analyzing phenomena like photon absorption and electron transitions.

Photon Absorption

Photon absorption is a process whereby an electron or other charged particles absorb energy from photons. A photon is a particle representing a quantum of light or electromagnetic radiation. When an electron in a 'particle in a box' system absorbs a photon, it gains energy. This absorption results in the electron moving from a lower energy level to a higher one.

The difference in energy levels determines the wavelength of the absorbed photon. As seen in the exercise, the energy corresponding to a photon can be expressed using the equation:\[\Delta E = \frac{hc}{\lambda}\]where:

• \(\Delta E\) is the change in energy (in this context, from one quantized level to another).
• \(h\) is Planck's constant.
• \(c\) is the speed of light in a vacuum \((3.00 \times 10^8 \text{ m/s})\).
• \(\lambda\) is the wavelength of the absorbed photon.

In practical applications, when calculating the wavelength of an absorbed photon, one essential step is to find the energy difference \(\Delta E\) between the initial and final states of the electron. This energy difference directly correlates with the energy of the absorbed photon and thus with its wavelength.

Quantum Transitions

Quantum transitions refer to the movement of an electron between different energy levels within a system, such as the 'particle in a box.' These transitions often result from photon absorption. Whenever an electron jumps from a lower energy state to a higher one, it is said to undergo a quantum transition.

For example, if an electron is excited from the ground state (where \(n=1\)) to the second excited state (where \(n=3\)), it means it has absorbed a photon of energy equal to the difference between these two states. This concept is crucial in quantum mechanics, because these transitions underpin the fundamental principles of atomic and molecular physics.

Quantum transitions are not limited to absorption. After the electron absorbs energy and moves to a higher state, it eventually returns to a lower energy level. This decay process involves the emission of a photon, a process known as spontaneous emission. The energies involved during absorption and emission processes decide the wavelengths of absorbed or emitted radiation, crucial for technologies like lasers and even in understanding how stars shine.