Frequency, wavelength, and photon energy are fundamental concepts in quantum physics and electromagnetic theory. Frequency (f) is the number of wave cycles that pass a fixed point per unit of time. Wavelength (λ) is the distance between two consecutive crests or troughs of a wave. Photon energy (E) is the amount of energy carried by a single photon, which is the elementary particle of light and all other forms of electromagnetic radiation.

The relationships between frequency, wavelength, and photon energy are expressed by the following equations:

\[ c = f \lambda \]

\[ E = hf = \frac{hc}{\lambda} \]

Where:

- c is the speed of light in vacuum (approximately 3 × 10⁸ m/s)
- f is the frequency (measured in Hz)
- λ (lambda) is the wavelength (measured in meters)
- E is the photon energy (measured in joules, J)
- h is Planck's constant (approximately 6.626 × 10⁻³⁴ J·s)

Let's calculate the photon energy of a light wave with a frequency of 5 × 10¹⁴ Hz:

- Identify the known values:
- f = 5 × 10¹⁴ Hz
- h ≈ 6.626 × 10⁻³⁴ J·s

- Apply the photon energy formula: \[ E = hf \]
- Substitute the known values: \[ E = (6.626 \times 10^{-34} \text{ J·s})(5 \times 10^{14} \text{ Hz}) \]
- Perform the calculation: \[ E = 3.313 \times 10^{-19} \text{ J} \]

Let's visualize the relationship between frequency, wavelength, and photon energy:

This visual representation shows:

- The wave nature of light, represented by the sinusoidal curve (blue)
- The wavelength (λ), which is the distance between two consecutive peaks
- The frequency (f), represented by the number of wave cycles per unit time (green)
- The photon energy (E), which is proportional to the frequency (red)

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