Question

Eye Surgery The argon fluoride (ArF) laser used in some refractive eye surgeries emits electromagnetic radiation of 193.3 nm wavelength. What is the frequency of the ArF laser’s radiation? What is the energy of a single quantum of the radiation?

Short answer

The frequency of the ArF laser's radiation is approximately \(1.55 × 10^{15}\, Hz\), and the energy of a single quantum of the radiation is approximately \(1.03 × 10^{-18}\, J\).

Step-by-step solution

Calculate the frequency

First, we need to calculate the frequency of ArF laser radiation. We are given the wavelength (λ) as 193.3 nm, but we need to convert it to meters for our calculations. 1 nanometer (nm) is equal to \(1 × 10^{-9}\) meters (m): \(λ = 193.3\, nm × 1 × 10^{-9}\, \frac{m}{nm} = 193.3 × 10^{-9}\, m\) Now, we can use the formula \(ν = \frac{c}{λ}\) to find the frequency (ν). The speed of light (c) is approximately equal to \(3.0 × 10^{8}\, m/s\). \(ν = \frac{3.0 × 10^{8} m/s}{193.3 × 10^{-9} m} = 1.55 × 10^{15}\, Hz\) So, the frequency of the ArF laser's radiation is approximately \(1.55 × 10^{15}\, Hz\).

Calculate the energy of a single quantum

Now that we have the frequency (ν), we can use the formula \(E = hν\) to find the energy of a single quantum. Planck's constant (h) is approximately equal to \(6.63 × 10^{-34}\, Js\). \(E = (6.63 × 10^{-34}Js)(1.55 × 10^{15}Hz) = 1.03 × 10^{-18} J\) So, the energy of a single quantum of the ArF laser's radiation is approximately \(1.03 × 10^{-18}\, J\).

Key concepts

Electromagnetic Radiation

To understand the behavior of ArF laser radiation, as used in refractive eye surgeries, it's essential to grasp the concept of electromagnetic radiation. Electromagnetic radiation is a form of energy that is all around us and takes the form of waves. These waves are generated by the vibration of electric and magnetic fields and can travel through a vacuum at the speed of light. Electromagnetic radiation encompasses a wide range of wavelengths and frequencies, from very long radio waves to very short gamma rays. The laser used in the aforementioned eye surgery emits radiation at a particular wavelength, in this case, 193.3 nm (nanometers), which lies within the ultraviolet portion of the electromagnetic spectrum.

The properties of electromagnetic waves are essential in many technologies, not only in medical applications like laser surgery but also in wireless communications and many forms of imaging. The wavelength of any electromagnetic radiation determines its color in the visible spectrum and its properties in the non-visible spectrum, which in turn dictates how it interacts with matter and its potential applications.

Quantum Energy

Diving deeper into the nature of the ArF laser radiation requires an understanding of quantum energy. At the quantum level, energy is not continuous, but rather it comes in discrete packets known as quanta. When dealing with light, these packets of energy are referred to as photons.

The energy of a single photon is directly related to the frequency of the light, with higher frequency light having more energetic photons. The frequency is the number of wave cycles that pass a given point per second. The relationship between energy and frequency is given by the equation E = hν, where E is the energy of a photon, h is Planck's constant, and ν (nu) is the frequency of the light. Therefore, when the frequency of the ArF laser radiation was calculated as roughly 1.55 x 10¹⁵ Hz, it allowed for the subsequent calculation of the energy of an individual photon emitted from the laser.

Planck's Constant

At the heart of understanding the energy of electromagnetic radiation is Planck's constant (h). This fundamental constant has the value of approximately 6.63 x 10^-34 joule-seconds (Js) and plays a pivotal role in quantum mechanics. It represents the proportionality constant between the energy (E) of a photon and the frequency (ν) of its associated electromagnetic wave.

Importance in Quantum Physics

Planck's constant is crucial because it highlights the quantized nature of energy at microscopic scales. It is utilized when calculating the energy of photons, determining the emission spectra of atoms, and describing the behaviors of particles at the quantum level. The exact value of Planck's constant is carefully defined and measured because it allows for precise calculations in many areas of physics, particularly in understanding the energy of electromagnetic radiation, like that from an ArF laser.

Speed of Light

The speed of light (c) in a vacuum is approximately 3.0 x 10⁸ meters per second (m/s) and is another fundamental constant of the universe. This constant is crucial when dealing with electromagnetic radiation because all forms of light and such radiation travel at this speed in a vacuum.

The speed of light factors into numerous equations and principles in physics, one of which is used to calculate the frequency of the ArF laser radiation. In the eye surgery example, the equation ν = c/λ, which gives the frequency (ν) of light as the speed of light (c) divided by its wavelength (λ), is derived from the universal nature of c. Calculating the frequency was the first step in determining the energy of the laser's radiation. Understanding the speed of light as a constant allows physicists to make precise measurements and develop technologies based on the properties of electromagnetic radiation.