monochromatic light of wavelength 720 nm passes through a fast shutter, which stays open for a nanosecond. What will be the wavelength spread of the beam after the light emerges through the shutter?

To understand the wavelength spread of a beam of monochromatic light after passing through a fast shutter, we need to consider the principles of wave mechanics and the uncertainty principle in quantum physics. When light passes through a shutter that is open for a very short time, such as one nanosecond, it can lead to a spread in the wavelengths of the light due to the inherent uncertainties in energy and time.

Understanding the Concept of Wavelength Spread

Monochromatic light, like the one with a wavelength of 720 nm, is typically characterized by a single wavelength. However, when we introduce a time constraint, such as the shutter being open for just one nanosecond, we can expect some dispersion in the wavelengths. This phenomenon can be explained using the Heisenberg uncertainty principle, which states that the more precisely we know the position (or time) of a particle, the less precisely we can know its momentum (or energy), and vice versa.

Applying the Uncertainty Principle

The uncertainty principle can be mathematically expressed as:

• ΔE * Δt ≥ ħ/2

Where:

• ΔE is the uncertainty in energy.
• Δt is the uncertainty in time (in this case, 1 nanosecond or 1 x 10^-9 seconds).
• ħ (h-bar) is the reduced Planck's constant, approximately 1.055 x 10^-34 Js.

Given that Δt is 1 x 10^-9 seconds, we can calculate the uncertainty in energy (ΔE).

Calculating the Energy Uncertainty

Rearranging the uncertainty principle gives us:

• ΔE ≥ ħ/(2 * Δt)

Substituting the values:

• ΔE ≥ (1.055 x 10^-34 Js) / (2 * 1 x 10^-9 s)
• ΔE ≥ 5.275 x 10^-26 J

Relating Energy Uncertainty to Wavelength Spread

Next, we need to relate this energy uncertainty to a wavelength spread. The energy of a photon is given by:

• E = hc/λ

Where:

• h is Planck's constant (approximately 6.626 x 10^-34 Js).
• c is the speed of light (approximately 3 x 10⁸ m/s).
• λ is the wavelength (720 nm or 720 x 10^-9 m).

To find the wavelength spread (Δλ), we can differentiate the energy equation:

• ΔE = -hc/λ² * Δλ

Rearranging gives us:

• Δλ = -ΔE * λ² / (hc)

Substituting the known values:

• Δλ = - (5.275 x 10^-26 J) * (720 x 10^-9 m)² / (6.626 x 10^-34 Js * 3 x 10⁸ m/s)

Calculating this gives:

• Δλ ≈ 1.2 x 10^-12 m or 1.2 pm (picometers).

Final Thoughts

The wavelength spread of the beam after passing through the shutter for one nanosecond is approximately 1.2 picometers. This small spread illustrates how even a brief exposure can introduce uncertainties in the properties of light, emphasizing the fascinating interplay between time and energy in quantum mechanics.