A photon is emitted when an electron in a three-dimensional cubical box of side length $8.00\times {10}^{-11}$ m makes a transition from the $n_X=2$, $n_Y=2$, $n_Z=1$ state to the $n_X=1$, $n_Y=1$, $n_Z=1$ state. What is the wavelength of this photon?

Verified step by step guidance

1. Step 1: Understand the problem. The electron transitions between two quantum states in a three-dimensional cubical box. The energy difference between these states determines the energy of the emitted photon, which can be used to calculate its wavelength using the formula \( \lambda = \frac{hc}{E} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( E \) is the energy difference.
2. Step 2: Write the expression for the energy levels in a three-dimensional cubical box. The energy of a quantum state is given by \( E = \frac{h^2}{8mL^2} (n_X^2 + n_Y^2 + n_Z^2) \), 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 respective dimensions.
3. Step 3: Calculate the energy for the initial state \( (n_X = 2, n_Y = 2, n_Z = 1) \) using the formula \( E_{initial} = \frac{h^2}{8mL^2} (2^2 + 2^2 + 1^2) \). Substitute the values of \( h \), \( m \), and \( L \) into the equation.
4. Step 4: Calculate the energy for the final state \( (n_X = 1, n_Y = 1, n_Z = 1) \) using the formula \( E_{final} = \frac{h^2}{8mL^2} (1^2 + 1^2 + 1^2) \). Again, substitute the values of \( h \), \( m \), and \( L \) into the equation.
5. Step 5: Find the energy difference \( \Delta E = E_{initial} - E_{final} \). Use this energy difference to calculate the wavelength of the photon using \( \lambda = \frac{hc}{\Delta E} \). Substitute the values of \( h \), \( c \), and \( \Delta E \) into the equation to determine the wavelength.

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 associated with each state is quantized, meaning that transitions between these states result in the emission or absorption of photons, with energy differences corresponding to the energy of the emitted or absorbed light.

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, allowing us to calculate its wavelength.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately 6.626 x 10^-34 J·s. This constant is crucial for calculations involving the energy of photons emitted during electronic transitions, as it provides the proportionality factor needed to convert frequency to energy.