Problem 53 - University Physics with Modern Physics

Question

Photon in a Dye Laser. An electron in a long, organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.18 nm. What is the wavelength of the photon emitted when the electron undergoes a transition (a) from the first excited level to the ground level and (b) from the second excited level to the first excited level?

Step-by-Step Solution

Answer

The wavelength for transition (a) is approximately 516 nm, and for transition (b) is approximately 310 nm.

Step 1: Understand the Particle in a Box Model

In this problem, we consider an electron in a box of length \( L = 4.18 \text{ nm} \). The energy levels for a particle in a 1D box are given by \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( n \) is the quantum number, \( h \) is Planck's constant, and \( m \) is the mass of the electron.

Step 2: Calculate Energy Difference for Transition (a)

To find the energy difference \( \Delta E \) between the first excited state \( (n=2) \) and the ground state \( (n=1) \), calculate: \( \Delta E = E_2 - E_1 = \frac{(2^2 - 1^2)h^2}{8mL^2} = \frac{3h^2}{8mL^2} \).

Step 3: Convert Energy Difference to Wavelength for Transition (a)

The energy difference corresponds to the energy of the photon emitted, given by \( \Delta E = \frac{hc}{\lambda} \). Solve for \( \lambda \) to get \( \lambda = \frac{hc}{\Delta E} = \frac{8mL^2c}{3h} \).

Step 4: Calculate Energy Difference for Transition (b)

For the transition from the second excited state to the first excited state \((n=3 \text{ to } n=2)\), the energy difference is \( \Delta E = E_3 - E_2 = \frac{(3^2 - 2^2)h^2}{8mL^2} = \frac{5h^2}{8mL^2} \).

Step 5: Convert Energy Difference to Wavelength for Transition (b)

Again, using the relationship \( \Delta E = \frac{hc}{\lambda} \), solve for \( \lambda \): \( \lambda = \frac{hc}{\Delta E} = \frac{8mL^2c}{5h} \).

Step 6: Final Calculations

Substitute numerical values: \( h = 6.626 \times 10^{-34} \text{ J s} \), \( m = 9.109 \times 10^{-31} \text{ kg} \), \( c = 3 \times 10^8 \text{ m/s} \), and \( L = 4.18 \times 10^{-9} \text{ m} \) into the equations for each transition. Perform the calculations to find the wavelengths.

Key Concepts

Photon Emission

Photon emission occurs when an electron transitions between energy levels in an atom or molecule, releasing energy in the form of a photon. This concept is fundamental to understanding phenomena in quantum mechanics, like the operation of lasers, including dye lasers. In the context of the exercise provided, as the electron in the molecule transitions from one energy level to another, it emits a photon. The energy of this photon is directly correlated to the difference in energy between the two levels.
Photon emission happens as discrete packets of energy, based on the principle that energy levels in atoms and molecules are quantized. This means that electrons can only exist in specific energy states and not in between these states. When an electron moves from a higher energy state to a lower one, the excess energy is emitted as a photon. The specific wavelength of this photon can be calculated using the energy difference, providing valuable insights into the electronic transitions occurring within the molecule.

Quantum Mechanics

Quantum mechanics underlies the framework for "Particle in a Box" problems. In this approach, particles like electrons are described not as classical points but by wavefunctions. These wavefunctions represent the probability of finding a particle in a particular location.
In quantum mechanics, the "Particle in a Box" model serves as a simplified illustration where a particle is confined to move freely within a perfectly rigid and impenetrable boundary. It is a fundamental problem that provides insight into the quantized nature of energy levels. The electron behaves like a wave, and only certain wavelengths \( \lambda \) that fit the box exactly (called standing waves) are allowed. This quantization results in discrete energy levels calculated by the formula:

• \( E_n = \frac{n^2 h^2}{8mL^2} \)

where \( E_n \) is the energy of the level \( n \), \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( L \) is the length of the box. Hence, the understanding of quantum mechanics is crucial to predict the behavior of particles in such constrained systems.

Energy Levels

Energy levels refer to the discrete set of energies that electrons can possess in an atom or molecule. These states are often depicted as a ladder, with each rung representing one quantized energy level. For the "Particle in a Box" model, energy levels arise from the restrictions on the electronic wavefunctions within a confined space.
The specific energy for each level \( n \) is determined by:

• \( E_n = \frac{n^2 h^2}{8mL^2} \)

In this context, energy levels are crucial for understanding photon emission because the difference in these levels gives rise to the particular wavelength of the photon emitted during electronic transitions. Only when electrons transition from higher to lower levels do they emit energy as photons, and this emitted energy corresponds to the difference between the energy levels involved. This model helps visualize what happens as electrons shift between quantized states.

Wavelength Calculation

Calculating the wavelength of a photon involved in electronic transitions is essential for understanding the photon's energy and characteristics. Based on the exercise solution, the calculation commences by establishing the difference in energy levels (\( \Delta E \)). Once \( \Delta E \) is determined, the wavelength \( \lambda \) can be calculated using the formula:

• \( \Delta E = \frac{hc}{\lambda} \)

Here, \( h \) is Planck's constant, and \( c \) is the speed of light. Rearranging the equation gives:

• \( \lambda = \frac{hc}{\Delta E} \)

This relationship shows how the energy of the emitted photon (related to the energy difference) can be directly converted into a measurable wavelength.
Understanding wavelength calculation is pivotal in fields like spectroscopy, where the analysis of wavelengths helps determine the structure and properties of molecules. By calculating the exact wavelengths emitted (as done in the exercise), scientists can gather precise information about the energy levels and behavior of electrons within atoms or molecules.