The eyes of certain reptiles pass a single visual signal to the brain when the visual receptors are struck by photons of a wavelength of \(850 \mathrm{nm}\). If a total energy of \(3.15 \times 10^{-14} \mathrm{J}\) is required to trip the signal, what is the minimum number of photons that must strike the receptor?

Planck's constant, denoted as \( h \), is a fundamental constant in quantum mechanics. It represents the proportionality constant between the minimal incremental energy that can be gained or lost by an atom or molecule, and the frequency of the electromagnetic radiation absorbed or emitted. With a value of approximately \( 6.626 \times 10^{-34} \) joule-seconds (J s), it sets the scale for the quantum of action in the quantum realm. This infuses quantization into the energy levels of particles in a system and forms the foundation for the energy calculation of a photon. Understanding this constant is crucial because it shows up in various formulas related to quantum mechanics and is used to determine the energy of a photon given its frequency or wavelength.

The speed of light, commonly denoted as \( c \), is a universal physical constant central to much of physics, particularly in electromagnetism, relativity, and quantum mechanics. It is the speed at which all massless particles and associated fields travel in a vacuum, and it has a precise value of \( 3.00 \times 10^8 \) meters per second (m/s). This speed is incredibly fast and is used when calculating the energy of a photon. Since the energy of a photon is dependent on its frequency, and frequency is inversely related to the wavelength for a given speed of light, knowledge of \( c \) allows us to connect the energy of photons to their wavelengths, enabling us to understand phenomena such as the photoelectric effect and the colors emitted from heated objects.

The energy of a photon is determined by its frequency or inversely by its wavelength. In physics, the energy \( E \) of a single photon can be calculated with the well-known equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the photon's wavelength. The equation underscores that photons with higher frequencies (and therefore shorter wavelengths) carry more energy, which can be seen in practice when comparing the effects of different types of electromagnetic radiation, such as visible light versus X-rays. This photon energy is particularly significant because it explains how light can cause chemical reactions, as in photosynthesis, or how it can trigger the release of electrons from a metal surface, leading to the photoelectric effect.

Wavelength conversion is the process of changing the measurement of light's wavelength into different units, which is often necessary when performing calculations such as determining a photon's energy. Light's wavelength is typically measured in meters but can be presented in nanometers (nm) or other units depending on the context. To use the photon energy equation effectively, the wavelength needs to be in meters because the constants \( h \) and \( c \) are in joule-seconds and meters per second, respectively. For instance, converting from nanometers to meters involves dividing by the factor of \( 1 \times 10^9 \), since there are that many nanometers in a meter. This step is essential as it ensures that the units align correctly in the energy calculation and helps to avoid errors in the final result, thereby providing accurate insight into the behavior of photons at different wavelengths.