Question

The human eye is most sensitive to green light of wavelength 505 nm. Experiments have found that when people are kept in a dark room until their eyes adapt to the darkness, a $single$ photon of green light will trigger receptor cells in the rods of the retina. (a) What is the frequency of this photon? (b) How much energy (in joules and electron volts) does it deliver to the receptor cells? (c) To appreciate what a small amount of energy this is, calculate how fast a typical bacterium of mass 9.5 $\times$ 10${}^{-12}$ g would move if it had that much energy.

Short answer

(a) 5.94 × 10¹⁴ Hz; (b) 3.94 × 10⁻¹⁹ J, 2.46 eV; (c) 2.87 × 10⁻² m/s.

Step-by-step solution

Convert Wavelength to Frequency

The frequency of a photon can be calculated using the speed of light equation: $c=\lambda u$, where $c$ is the speed of light $(3\times {10}^8\text{ m/s})$, $\lambda$ is the wavelength $(505\text{ nm})$, and $u$ is the frequency. First, convert $\lambda$ from nm to meters: $505\text{ nm}=505\times {10}^{-9}\text{ m}$. Then rearrange the equation for frequency: $u=\frac{c}{\lambda }$. Substitute the given values: $u=\frac{3\times {10}^8}{505\times {10}^{-9}}=5.94\times {10}^{14}\text{ Hz}$.

Calculate Energy in Joules

The energy of a photon is given by the equation $E=hu$, where $h$ is Planck's constant $(6.626\times {10}^{-34}\text{ J s})$ and $u$ is the frequency. Substitute the frequency calculated in Step 1: $E=6.626\times {10}^{-34}\times 5.94\times {10}^{14}=3.94\times {10}^{-19}\text{ J}$.

Convert Energy to Electron Volts

To convert energy from joules to electron volts (eV), use the conversion factor $1\text{ eV}=1.602\times {10}^{-19}\text{ J}$. Thus, $E=\frac{3.94\times {10}^{-19}}{1.602\times {10}^{-19}}=2.46\text{ eV}$.

Calculate Bacterium's Speed

The kinetic energy of the bacterium can be equated to the photon energy: $\frac{1}{2}m{v}^2=E$. Convert the mass of the bacterium from grams to kilograms: $9.5\times {10}^{-12}\text{ g}=9.5\times {10}^{-15}\text{ kg}$. Rearrange the equation for velocity $v$: $v=\sqrt{\frac{2E}{m}}$. Substitute the known values: $v=\sqrt{\frac{2\times 3.94\times {10}^{-19}}{9.5\times {10}^{-15}}}=2.87\times {10}^{-2}\text{ m/s}$.

Key concepts

Photon Energy

Photon energy is a crucial concept in quantum mechanics, particularly when studying light and electromagnetic radiation. A photon is a particle of light, often described as a "packet" of energy. The energy of a single photon is directly related to its frequency, which is part of what makes wavelengths and colors of light unique.

The formula used to calculate the energy of a photon is:

• $E=h\cdot u$
• $E$ represents the energy of the photon
• $h$ is Planck's constant, approximately $6.626\times {10}^{-34}\text{ J s}$
• $u$ is the frequency of the photon

This equation highlights that even a minuscule amount of energy can be significant at a microscopic level, such as triggering receptor cells in the eye.

Moreover, understanding photon energy is vital in many fields including photonics and quantum computing, as photons are often used for transmitting information and performing computations.

Frequency Calculation

Frequency calculation is an essential step in determining the characteristics of light, which is important in both physics and engineering.

To find the frequency of light, the speed of light $c$ is considered constant (at about $3\times {10}^8\text{ m/s}$), and is related to a photon's wavelength $\lambda$ by the equation:

• $c=\lambda \cdot u$
• $u=\frac{c}{\lambda }$

Here, $u$ represents the frequency. For this calculation, make sure the wavelength is in meters, converting from nanometers by using $1\text{ nm}={10}^{-9}\text{ m}$.

Once the wavelength is converted, the frequency can be determined, which allows for further calculations such as finding the photon's energy.

Planck's Constant

Planck's constant is a fundamental figure in quantum physics. It serves as a bridge between the macroscopic and quantum worlds, helping to measure action in the quantum domain. Represented by the letter $h$, Planck's constant has a value of approximately $6.626\times {10}^{-34}\text{ J s}$.

This constant plays a critical role in calculating photon energy, where it signifies the proportionality factor between the frequency $u$ of a photon and its energy $E$, as seen in the equation $E=h\cdot u$.

Planck's constant not only aids in understanding the quantization of light but also underpins the concept of quantized energy levels in atoms, providing the basis for much of modern quantum mechanics.

In essence, Planck's constant allows us to understand phenomena at a microscopic scale, which would otherwise be invisible or inexplicable with classical physics alone.