Question

What is the energy of a transition capable of producing light of wavelength \(10.6 \mu \mathrm{m}\) ? (This is the wavelength of light associated with a commonly available infrared laser.)

Short answer

Answer: The energy of the transition is 1.88 x 10^-19 J.

Step-by-step solution

Convert the wavelength to meters

Given the wavelength as \(10.6\mu m\) (micrometers), we can convert it to meters using the conversion factor: \(1\mu m = 1\times10^{-6}m\). So, \(10.6\mu m = 10.6\times10^{-6}m = 1.06\times10^{-5}m\).

Use the formula to calculate the energy

The formula that relates energy (\(E\)), wavelength (\(\lambda\)), and the speed of light (\(c\)) is given by: \(E = \dfrac{hc}{\lambda}\), where \(h\) is the Planck's constant (\(6.63\times10^{-34} Js\)) and \(c\) is the speed of light (\(3\times10^{8}m/s\)). Plug in the values for \(h, c, \lambda\) to calculate the energy \(E\): \(E = \dfrac{(6.63\times10^{-34}Js)(3\times10^{8}m/s)}{1.06\times10^{-5}m}\)

Evaluate the expression

Calculate the given expression for the energy \(E\): \(E = \dfrac{(6.63\times10^{-34}Js)(3\times10^{8}m/s)}{1.06\times10^{-5}m} = 1.88\times10^{-19}J\) The energy of the transition capable of producing light of wavelength \(10.6\mu m\) is \(1.88\times10^{-19}J\).

Key concepts

Wavelength to Energy Conversion

Understanding the transition from wavelength to energy is crucial when dealing with spectral analysis and quantum physics. In the visible light spectrum, different colors are associated with specific wavelengths, and similarly, different energies correspond to those wavelengths. The conversion process relies on the inverse relationship between energy and wavelength. As such, radiations with shorter wavelengths, like gamma rays, have higher energy, while those with longer wavelengths, like radio waves, have lower energy.

In the context of the given problem, where we are examining infrared light, this wavelength is considerably longer than visible light, placing it on the low-energy end of the electromagnetic spectrum. However, despite its lower energy, this part of the spectrum has practical applications, including the use of lasers in various technologies. To convert the wavelength to energy, we use the formula: \[\begin{equation} E = \frac{h c}{\lambda} \end{equation}\], where 'E' represents the energy in joules (J), 'h' is Planck's constant, 'c' is the speed of light, and '\lambda' is the wavelength in meters.

By plugging the values into this equation, we can derive the energy associated with a photon with a given wavelength. This process not only allows for understanding light's behavior but also for designing equipment that can use this energy, such as infrared lasers that emit light at the specified wavelength.

Infrared Laser Physics

Infrared lasers operate in a part of the electromagnetic spectrum that is not visible to the human eye. These lasers emit light with wavelengths longer than visible light and are widely used in various applications like remote controls, fiber-optic communication, and medical procedures. The physics behind their operation is grounded in the principles of stimulated emission and quantum transitions.

Lasers produce coherent light, meaning the light waves are in phase and have the same frequency, a phenomenon that can be precisely controlled. Leveraging the wavelength-energy relationship, it is possible to design infrared lasers to emit light at specific wavelengths, such as the 10.6 micrometer mark, which is the focal point of our exercise. This specific wavelength falls into the category of the mid-infrared range and is harnessed for its properties, including the ability to be absorbed by many materials, making it useful for cutting and heating applications.

Understanding the energy transition associated with this wavelength helps in designing and optimizing laser systems that are both safe and effective. In research and industrial settings, determining the energy of these transitions allows for a nuanced manipulation of the laser's interaction with various materials.

Planck's Constant

A pivotal figure in quantum mechanics, Planck's constant (represented as 'h') is a fundamental physical constant that describes the size of the quantum of action. Its value is approximately \[\begin{equation} h = 6.63 \times 10^{-34} \text{Js} \end{equation}\], which means that it expresses the proportional relationship between the frequency of a photon and its energy. This constant is at the heart of the quantum theory and introduces the concept of quantization in the energy levels of atoms and molecules.

What Planck's constant effectively communicates is that energy can only be exchanged in discrete amounts, or 'quanta', rather than flowing continuously. This discreteness is crucial in understanding phenomena such as the photoelectric effect, atomic emission spectra, and the operation of devices such as LEDs and lasers mentioned in our original exercise. The introduction of Planck's constant bridged the gap between classical physics and quantum mechanics, providing a mathematical basis for the wave-particle duality of light.

For students and scientists alike, comprehending Planck's constant is vital to any discussion of energy at the quantum level, and hence it plays a central role in calculating the energy associated with the transition of electrons that can emit specific wavelengths of light, such as the infrared wavelengths used in laser technology.