$$ \begin{array}{lccccc} \lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\ \hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\ & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} \end{array} $$ A ruby laser produces radiation of wavelength \(633 \mathrm{nm}\) in pulses whose duration is \(1.00 \times 10^{-9} \mathrm{~s}\). (a) If the laser produces \(0.376 \mathrm{~J}\) of energy per pulse, how many photons are produced in each pulse? (b) Calculate the power (in watts) delivered by the laser per pulse. \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)

When we talk about photon energy calculation, we are looking at the process of determining the amount of energy each individual photon carries. This energy can be expressed using the equation:
\( E = hu \),
where \( E \) represents the energy of the photon, \( h \) is the Planck constant, and \( u \) (nu) is the frequency of the electromagnetic wave associated with the photon.
The frequency \( u \) can be found from the speed of light \( c \) and the photon's wavelength \( \lambda \) using the relationship:
\( u = \frac{c}{\lambda} \).
Substituting the values for \( c \) and \( \lambda \) in this relationship gives us the frequency. Once we have the frequency, we can calculate the energy of a single photon. For instance, in the context of the textbook exercise, the energy per photon emitted by a ruby laser was found to be \( 3.14 \times 10^{-19} \) joules. Understanding this process is crucial for explaining phenomena such as the photoelectric effect and the operation of lasers.

The Planck constant, denoted as \( h \), is a fundamental physical constant that plays a pivotal role in quantum mechanics, particularly in the quantization of energy. Its value is approximately \( 6.63 \times 10^{-34} \) joule-seconds (J⋅s). The constant is named after Max Planck, the physicist who proposed that energy is quantized and can be calculated as a multiple of an elementary unit \( h \).

The Planck constant enables us to tie together the energy of photons with their frequency using the equation \( E = hu \), where \( E \) is the energy, \( u \) is the frequency, and \( h \) is the Planck constant. This relation is a cornerstone of quantum physics and explains why photons with higher frequencies (such as X-rays) have more energy than those with lower frequencies (like radio waves). Understanding the significance of the Planck constant helps us comprehend the discrete nature of energy at the quantum level.

The concept of laser pulse power is related to the measurement of the energy a laser emits in a given pulse duration. It can be thought of as the rate at which energy is deployed from a laser in a single pulse. The power \( P \) of a laser pulse is calculated using the formula:
\( P = \frac{E}{t} \),
where \( E \) is the energy delivered by the pulse and \( t \) is the duration of the pulse. Power is measured in watts (W), which is defined as joules per second (J/s).

In the context of our exercise, the power delivered by a ruby laser was calculated to be 376,000 watts (or 376 kW). This high power output, even for a very short duration, explains why lasers are so effective in applications that require precise and focused energy delivery, such as cutting materials or performing delicate surgical procedures.

The frequency of electromagnetic waves is a fundamental characteristic that describes the number of wavelengths that pass a certain point in a given period of time. It is measured in hertz (Hz), with one hertz equating to one wave cycle per second. Electromagnetic waves, which include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, vary widely in frequency.

Frequency can be directly related to the wavelength \( \lambda \) of the wave and the speed of light \( c \) through the equation:
\( u = \frac{c}{\lambda} \).
Ruby lasers, for example, emit light with a specific wavelength and thus a specific frequency. By knowing the wavelength of the emitted light, we can calculate its frequency and hence determine its position within the electromagnetic spectrum. This is crucial for identifying the type of light and understanding its behavior, energy, and potential interactions with matter.