A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 * 10-11 m makes a transition from the nX = 2, nY = 2, nZ = 1 state to the nX = 1, nY = 1, nZ = 1 state. What is the wavelength of this photon?

Verified step by step guidance

1. Calculate the energy levels of the initial and final states using the formula for the energy of an electron in a three-dimensional box: E = \frac{h^2}{8mL^2}(n_x^2 + n_y^2 + n_z^2), where h is Planck's constant, m is the mass of the electron, L is the side length of the box, and n_x, n_y, n_z are the quantum numbers for the x, y, and z directions respectively.
2. Substitute the given values and quantum numbers for the initial state (nX = 2, nY = 2, nZ = 1) into the energy formula to find the initial energy E_i.
3. Substitute the given values and quantum numbers for the final state (nX = 1, nY = 1, nZ = 1) into the energy formula to find the final energy E_f.
4. Calculate the energy of the emitted photon, which is the difference between the initial and final energy states: \Delta E = E_i - E_f.
5. Use the energy of the photon to find its wavelength using the formula \lambda = \frac{hc}{\Delta E}, where c is the speed of light in vacuum.

Key Concepts

Quantum States in a Particle in a Box

In quantum mechanics, a particle confined in a three-dimensional box can only occupy specific energy levels, defined by quantum numbers (nX, nY, nZ). The energy levels are quantized, meaning the particle can only exist in these discrete states. The transition between these states results in the emission or absorption of energy, often in the form of a photon.

Energy of a Photon

The energy of a photon is directly related to its frequency and inversely related to its wavelength, described by the equation E = hν = hc/λ, where E is energy, h is Planck's constant, ν is frequency, c is the speed of light, and λ is wavelength. When an electron transitions between energy levels, the energy difference corresponds to the energy of the emitted photon.

Wavelength Calculation

To find the wavelength of the emitted photon, one must first calculate the energy difference between the initial and final quantum states using the formula E = (h²π²)/(2mL²) * (nX² + nY² + nZ²). After determining the energy, the wavelength can be calculated using the photon energy equation, λ = hc/E. This process is essential for understanding the relationship between quantum transitions and electromagnetic radiation.