A particular laser consumes 150.0 watts of electrical power and produces a stream of \(1.33 \times 10^{19} 1064-\mathrm{nm}\) photons per second. What is the percent efficiency of the laser in converting electrical power to light?

At the heart of quantum physics and understanding light phenomena is Planck's equation, which shows the relationship between the energy of a photon and its frequency. This principle is crucial for calculating the efficiency of devices like lasers. Planck's constant, denoted as 'h', is a fundamental constant with a value of approximately 6.626 x 10^-34 Joule seconds.

To get the energy of a photon, we use the formula:
\( E = hu \)
where \( E \) is the energy, \( h \) is Planck's constant, and \( u \) (the Greek letter nu) represents the frequency of the photon. In the context of a laser, knowing the energy output per photon helps determine the laser's overall efficiency in converting power into light.

Each photon, a particle of light, carries a distinct amount of energy, contributing to the behavior and function of lasers. The photon energy is determined by the aforementioned Planck's equation. To find the energy of a photon with a given wavelength, one must invert the wavelength to obtain frequency and then apply Planck's constant:

\( E = \frac{hc}{{\text l}} \)

Here, \( c \) is the speed of light in a vacuum, approximately 3.00 x 10⁸ meters per second, and \( \text l \) is the wavelength in meters. For a laser emitting a stream of photons, calculating the energy per photon is the first step in understanding the laser's power output.

The wavelength \( ( \text l ) \) of a photon and its frequency \( (u) \) are inversely related, connected through the speed of light \( (c) \):
\( u = \frac{c}{\text l} \)

This formula indicates that as the wavelength increases, the frequency decreases and vice versa, provided that the speed of light remains constant.

When it comes to lasers and their efficiencies, understanding this relationship is key. Longer wavelengths, for example, carry less energy per photon than shorter wavelengths, due to the lower frequency. For practical laser use, especially in calculations for energy output and efficiency, we must consider this relationship to determine the number of photons needed to achieve the desired power output and, subsequently, the laser's efficiency in producing light from electrical energy.