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The relationship between momentum and energy for a photon is pivotal in understanding how atomic and subatomic processes work. Photons are unique because, unlike particles with mass, they have energy directly related to their momentum. This relationship is governed by the equation \( E = pc \), where \( E \) is the energy, \( p \) is the momentum, and \( c \) is the speed of light. Since a photon is massless, its energy depends entirely on its momentum and the speed of light, rather than mass and velocity as in classical mechanics. This is crucial for calculating certain phenomena in quantum mechanics.

When calculating energy from momentum, we use the constant speed of light (\( 3 \times 10^8 \, \text{m/s} \)), making computations straightforward and reflective of the photon's fundamental properties. This relationship is powerful because it allows the translation of quantum behavior into energetic terms, which are more intuitive to us.

Determining the wavelength of a photon is an essential part of understanding its properties and behavior. The wavelength \( \lambda \) can be calculated using the formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the photon's momentum.

Since photons do not possess a rest mass, their wavelength is a function of Planck’s fundamental constant \( 6.626 \times 10^{-34} \, \text{Js} \). This formula derives from the quantum nature of photons, linking a photon's momentum directly to its corresponding wavelength. Knowing a photon's wavelength can provide insights into its energy type, behavior, and the part of the electromagnetic spectrum it belongs to.

This calculation is significant because it translates a photon’s dynamic state, described by its momentum, into a spatial characteristic, its wavelength. This helps bridge concepts between particle and wave nature, a core aspect of wave-particle duality.

The electromagnetic spectrum is a comprehensive range of all types of electromagnetic radiation. Each type corresponds to a different wavelength and energy level, placing each type in a unique position on the spectrum.

From gamma rays with the shortest wavelengths to radio waves with the longest, understanding where a photon's wavelength places it provides valuable insights into its properties and potential applications. The visible light spectrum is only a small part of this vast range but is crucial for numerous biological and technological processes.

In our case, a wavelength of 804 nm places the photon in the visible range, near the infrared region. This means the photon is just on the edge of what is perceivable to human vision, bridging visible light and heat radiation. By knowing a photon's place on the spectrum, scientists can predict its interactions with matter and practical uses, such as communications, imaging, and medical therapies.

Planck's constant is a fundamental constant of nature that plays a crucial role in quantum mechanics, specifically impacting the quantization of energy. It connects the energy of photons to their frequencies, following the relation \( E = hf \), where \( E \) is energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \), and \( f \) is frequency.

In our wavelength calculation context, Planck’s constant helps express the photon's energy in terms of its wavelength instead of frequency. This is vital in many fields such as optics, quantum physics, and even electrical engineering because it allows us to calculate and relate energies in different forms.

Without Planck’s constant, the link between the minute scales of photons and wider observable energy outputs would be much harder to define. This constant is an integral part of understanding modern physics, as it forms the bridge between the particle and wave descriptions of light.