Question

An electron made a transition between allowed states, emitting a photon. Which of the following physical constants is (are) needed to calculate the energy of the photon from the measured wavelength? a) the Plank constant, \(h\) b) the basic electric charge, \(e\) c) the speed of light in vacuum, \(c\) d) the Stefan-Boltzmann constant, \(\sigma\)

Short answer

Answer: The required physical constants for calculating the energy of the photon are a) the Planck constant, \(h\) and c) the speed of light in vacuum, \(c\).

Step-by-step solution

1. Determine the relation between the energy of a photon and its wavelength

The energy of a photon can be expressed using the Planck-Einstein relation as follows: $$ E = h \cdot \frac{c}{\lambda} $$ Where: - \(E\) is the energy of the photon - \(h\) is the Planck constant - \(c\) is the speed of light in vacuum - \(\lambda\) is the wavelength of the photon

2. Identify the physical constants in the equation

From the equation above, we can see that there are two physical constants involved: 1. Planck constant, \(h\) 2. The speed of light in vacuum, \(c\)

3. Compare the constants in the equation to the given choices

Comparing the constants in the equation to the given choices, we can conclude that: a) The Planck constant, \(h\) is needed. (Correct) b) The basic electric charge, \(e\) is not needed. (Incorrect) c) The speed of light in vacuum, \(c\) is needed. (Correct) d) The Stefan-Boltzmann constant, \(\sigma\) is not needed. (Incorrect) So, the correct answer is a) the Planck constant, \(h\) and c) the speed of light in vacuum, \(c\).

Key concepts

Planck Constant

The Planck constant, denoted as \(h\), is a fundamental quantity in quantum mechanics, playing a pivotal role in the quantization of energy. It was first discovered by Max Planck in 1900 and has since been a cornerstone in the understanding of atomic and subatomic processes. The value of the Planck constant is approximately \(6.62607015 \times 10^{-34}\text{ Js}\).

This constant is especially crucial when dealing with the interaction between matter and electromagnetic radiation, such as when a photon is emitted by an electron transitioning between states. In the context of photon energy calculation, the Planck constant represents the proportionality between the energy of the photon and its frequency. The smaller the value of \(h\), the less energy each quantized 'packet' or photon will carry when related to a specific frequency. It's one of the key constants required to calculate the energy of a photon using the Planck-Einstein relation.

Speed of Light in Vacuum

The speed of light in a vacuum, represented by the symbol \(c\), is another fundamental physical constant and is central to the fields of physics and astronomy. The speed of light is precisely \(299,792,458 \text{ meters per second (m/s)}\). In the vacuum of space, this speed does not change and acts as the universal speed limit for the propagation of light and all electromagnetic radiation.

Understanding the concept of the speed of light is critical when calculating the energy of a photon because it provides a direct relationship between the wavelength of light and its frequency. Since these two properties are inversely proportional, knowing the speed of light allows us to understand how the energy of a photon is influenced by its wavelength, which is vital in exercises such as determining the energy output of an electron transitioning between energy levels.

Planck-Einstein Relation

The Planck-Einstein relation is a formula that establishes a direct link between the energy of a photon and its frequency. It is expressed as \(E = h u\), where \(E\) denotes the energy of the photon, \(h\) is the Planck constant, and \(u\) represents the frequency. Alternatively, because frequency \(u\) can be described as the speed of light \(c\) divided by the wavelength \(\text{ambda}\) (i.e., \(u = \frac{c}{\text{ambda}}\)), the formula can also be written as:
\[E = h \frac{c}{\text{ambda}}\]
This equation is fundamental when calculating the energy of photons, as it connects the quantized nature of light at the subatomic level to measurable properties like frequency and wavelength. The Planck-Einstein relation is the reason we need both the Planck constant and the speed of light in vacuum to determine the energy of a photon.

Wavelength of a Photon

The wavelength of a photon, commonly represented by the Greek letter \(\text{ambda}\), is a measure of the distance between repeating units of a wave pattern. It is an essential property of photons and directly affects their energy levels. In terms of light, wavelength determines the color we observe, with longer wavelengths corresponding to red and shorter wavelengths to violet.

For calculating energy, the wavelength is inversely proportional to the frequency; as the wavelength increases, the frequency decreases, and vice versa. This relationship between wavelength and energy is described by the Planck-Einstein relation, which indicates that a photon with a shorter wavelength will have higher energy due to the inverse proportionality defined by \(c = \text{ambda} u\). Knowledge of the photon's wavelength is necessary to calculate its energy, as the two are inversely related.