A molecule in a laser trap is confined to a region of 1um( micrometer). An incoming photon of known energy is scattered off it and bounces back in the direction it came, where its energy is measured. How broad is the measured energy peak ? Give your answer in eV, with one significant figure
To determine the broadness of the measured energy peak, we need to consider the uncertainty principle of quantum mechanics. According to the principle, there is an inherent limit to the precision with which both the position and momentum of a particle can be known simultaneously.
In this case, the molecule is confined to a region of 1 μm, which represents the uncertainty in its position. The uncertainty in momentum is related to the wavelength of the photon used.
The uncertainty principle can be expressed as follows:
Δx * Δp >= h/2π,
where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is the Planck's constant (6.62607015 × 10^(-34) J·s).
Since the momentum of a photon is given by p = h/λ, where λ is the wavelength of the photon, we can rewrite the uncertainty principle as:
Δx * (h/λ) >= h/2π.
Simplifying the equation, we find:
Δx >= λ/2π.
The energy of a photon can be determined using the equation E = hc/λ, where h is Planck's constant (6.62607015 × 10^(-34) J·s), c is the speed of light (299,792,458 m/s), and λ is the wavelength of the photon.
To find the broadness of the measured energy peak, we need to calculate the wavelength uncertainty, Δλ. This can be done by substituting Δx = 1 μm into the equation above and solving for Δλ:
1 μm >= Δλ/2π.
Rearranging the equation, we find:
Δλ <= 2π μm.
Finally, we can calculate the width of the measured energy peak by substituting the calculated Δλ into the energy equation:
ΔE = hc/Δλ.
Plugging in the constants (h = 6.62607015 × 10^(-34) J·s, c = 299,792,458 m/s) and converting μm to meters, we get:
ΔE = (6.62607015 × 10^(-34) J·s * 299,792,458 m/s) / (2π * 10^(-6) m).
Calculating this value, we find that the measured energy peak is approximately 0.066 eV (with one significant figure).