Question

An electron in the \(n=4\) state of a 5 nm wide infinite well makes a transition to the ground state, giving off energy in the form of a photon. What is the photon's wavelength?

Short answer

After calculating the energy levels for the initial and final states, the difference between these levels, converting that difference to frequency, and finally converting that frequency to wavelength, the value of wavelength will be obtained as the solution.

Step-by-step solution

Compute the initial and final energy levels

Compute the initial (Ei) and final (Ef) energy levels using the formula for energy in an infinite square well: \(E = n^2 \cdot h^2 / (8 \cdot m \cdot L^2)\). Here, n is the energy state, h is the Planck's constant, m is the mass of an electron, and L is the width of the well. Convert L from nm to m before plugging in the values. For the initial state, n=4 and for the final state (ground state), n=1.

Calculate the energy difference

The energy difference (\( \Delta E\)) is \( \Delta E = E_{i} - E_{f}\). This is the energy of the photon released.

Convert energy difference to frequency

Calculate the frequency of the photon (\( \nu \)) from the energy difference using Planck's relation \( E = h\nu \): \( \nu = \Delta E / h \).

Compute the wavelength

Then, calculate the wavelength using the speed of light wave equation: \( c = \lambda \nu \). Solving for the wavelength: \( \lambda = c / \nu \).

Key concepts

Energy Levels in Infinite Square Well

Energy levels in an infinite square well are quantized, meaning they can only take on specific values. This is different from classical physics where energy is a continuous quantity. In the context of a quantum well, the energy of a particle like an electron is determined by its quantum number, denoted as \( n \).

For an infinite square well, the energy \( E \) is given by the formula: \[ E = \frac{n^2 h^2}{8 m L^2} \] Learn More

• \( n \) is the principal quantum number and indicates the energy state of the electron.
• \( h \) is Planck's constant, an essential parameter in quantum mechanics.
• \( m \) is the mass of the electron.
• \( L \) is the width of the well, affecting the spacing of energy levels.

The lowest energy state, or ground state, is when \( n=1 \). Higher states like \( n=4 \) contain more energy due to the \( n^2 \) dependency. Each energy level corresponds to the possible standing waves that fit inside the well.

Quantum Transition and Photon Emission

A quantum transition occurs when an electron changes energy levels within the infinite square well. As it moves from a higher energy level to a lower one, it must release the excess energy. This energy is emitted as a photon, a particle of light.

In the exercise, the electron transitions from \( n=4 \) to \( n=1 \), the ground state. This specific change is called a quantum transition, as the electron moves from one quantized level to another.

• The energy released during this transition is equal to the difference between the two energy levels: \[ \Delta E = E_{initial} - E_{final} \]

This energy manifesting as a photon connects the principle that energy is conserved, and it adheres to the formula established by Albert Einstein, \( E = hu \), where \( u \) is the frequency of the emitted photon.

Photon Wavelength Calculation

The wavelength of the emitted photon is found by first determining the frequency of the photon from the energy difference using Planck's relation: \[ u = \frac{\Delta E}{h} \]
where \( h \) is Planck's constant. Once the frequency is known, the wavelength \( \lambda \) can be calculated using the speed of light \( c \), with the equation: \[ \lambda = \frac{c}{u} \] Learn More

• The speed of light \( c \) is approximately \( 3 \times 10^8 \) meters per second.
• This relationship explains how the energy of the photon determines both its frequency and wavelength.

In the exercise, performing these calculations will yield the precise wavelength of the photon created by the electron's transition. Photons with different wavelengths constitute the diverse spectrum of electromagnetic waves that we observe.

Understanding Planck's Constant

Planck's constant \( h \) is a fundamental constant in physics that characterizes the mathematical descriptions of the microscopic world. Its value is approximately \( 6.626 \times 10^{-34} \) Joule-seconds.

Planck's constant appears in several critical formulas, bridging the gap between energy and frequency:

• It underpins the energy-frequency relation for photons: \( E = hu \).
• Planck's constant represents the smallest action possible in quantum mechanics, a core idea in the theory.
• It is critical in calculations involving quantum systems like the infinite square well.

Essentially, Planck's constant defines the scale at which quantum effects become significant. Without it, the quantization of energy and understanding of quantum transitions would not be possible.