Energy of a Photon

The energy of a single photon is directly proportional to its frequency. This relationship is defined by the Planck Equation: $E = hf$ where $E$ is energy in Joules (J), $f$ is frequency in Hertz (Hz), and $h$ is Planck’s constant ($6.63 \times 10^{-34}$ J s).

By substituting the wave equation ($c = f\lambda$), energy can also be expressed in terms of wavelength: $E = \frac{hc}{\lambda}$ where $c$ is the speed of light ($3.00 \times 10^8$ m/s) and $\lambda$ is the wavelength in meters (m).

These equations demonstrate that energy is inversely proportional to wavelength; as the wavelength of radiation increases, the energy carried by each individual photon decreases.

Step 1: Identify Knowns: Determine if the problem provides frequency ($f$) or wavelength ($\lambda$). Ensure all values are in standard SI units (e.g., convert nanometers to meters).

Step 2: Select Formula: Use $E = hf$ if frequency is known, or $E = \frac{hc}{\lambda}$ if wavelength is known. For calculations involving multiple photons, the total energy is $E_{total} = n \cdot E_{photon}$, where $n$ is the number of photons.

Step 3: Constant Application: Use the standard values for $h$ ($6.63 \times 10^{-34}$ J s) and $c$ ($3.00 \times 10^8$ m/s). These are universal constants and do not change regardless of the medium or source.

Step 4: Power and Intensity: To find the number of photons emitted per second, divide the total power (Watts or J/s) by the energy of a single photon ($n = \frac{P}{E_{photon}}$).

Unit Conversions: Wavelengths are frequently given in nanometers ($nm$). Always convert to meters by multiplying by $10^{-9}$ before plugging into the formula $E = \frac{hc}{\lambda}$.

Order of Magnitude Check: Photon energies for visible light are typically in the range of $10^{-19}$ Joules. If your answer is significantly different (e.g., $10^{14}$ or $10^{-34}$), re-check your powers of ten.

Rounding and Precision: Use the full value of constants provided in your data sheet until the final step. Rounding $h$ or $c$ prematurely can lead to significant errors in the final result.

Interpreting 'Monochromatic': If a question mentions monochromatic light, it means all photons in that beam have the exact same frequency and therefore the exact same energy.

Confusing Frequency and Wavelength: Students often mistakenly assume that a 'larger' wavelength means more energy. Remember the inverse relationship: larger $\lambda$ means smaller $f$ and lower $E$.

Mixing Units: Using frequency in MHz without converting to Hz, or using $c$ in $km/s$ instead of $m/s$, will result in incorrect energy values.

Energy vs. Power: Power is the total energy per second. Do not confuse the power of a laser with the energy of one of its photons; a high-power laser might just be emitting a massive number of low-energy photons.