What is the Uncertainty in Wavelength for an Excited Atomic State?

In summary, an excited atomic state with a lifetime of 1 ms has an uncertainty in its energy of 5.25x10^-32. The uncertainty and fractional uncertainty in the wavelength of the photon emitted during its decay is a few hundred meters and a fraction of 500 nm, respectively. To find the uncertainty in the wavelength, it is necessary to work out the energy associated with the transition and use the non-relativistic electron's dispersion relation to calculate the wavelength uncertainty.
  • #1
chill_factor
903
5

Homework Statement



An excited atomic state has a lifetime of 1 ms.

What is the uncertainty in its energy?

The photon emitted during its decay is 550 nm in wavelength. What is the uncertainty and fractional uncertainty in its wavelength?

Homework Equations



ΔEΔt≥hbar/2

The Attempt at a Solution



a. Straightforward plugging into the equation.

ΔE = hbar/(2Δt) = 5.25x10^-32

b. Use ΔE=hΔf to find the frequency.

Δf = 79.6 s^-1

if I were to plug this into ΔλΔf=c, it results in a very large Δλ which is unphysical.

Δλ = ?
 
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  • #2
that is the right method. the result is not a wavelength, but an uncertainty in the wavelength, and it should be a fraction of 500nm
 
  • #3
The resulting uncertainty is in the hundreds of meters though =(
 
  • #4
While it's true that ##\lambda f = c##, it doesn't follow that ##\Delta \lambda \Delta f=c##.
 
  • #5
work out the energy associated with the transition, then work out the ratio of the uncertainty of the energy and the energy of the transition. the energy is related to the wavelength of a particle through the dispersion relation. assume a non relativistic electron's dispersion relation, and work out the wavelength uncertainty from there.
 

FAQ: What is the Uncertainty in Wavelength for an Excited Atomic State?

1. What is wavelength uncertainty?

Wavelength uncertainty refers to the degree of variation or error in the measurement of a wavelength. It is a measure of the precision or accuracy of the wavelength measurement.

2. How is wavelength uncertainty calculated?

Wavelength uncertainty is typically calculated by taking the standard deviation of multiple measurements of the same wavelength. This provides a measure of the spread of values and therefore the level of uncertainty in the measurement.

3. What factors contribute to wavelength uncertainty?

There are several factors that can contribute to wavelength uncertainty, including instrumental limitations, human error in measurement, and environmental factors such as temperature and pressure.

4. How can wavelength uncertainty be reduced?

Wavelength uncertainty can be reduced by using more precise and accurate instruments, ensuring proper calibration of instruments, and taking multiple measurements to calculate an average value. Additionally, controlling environmental factors can also help reduce uncertainty.

5. Why is wavelength uncertainty important in scientific research?

Wavelength uncertainty is important in scientific research because it affects the reliability and validity of experimental results. A high level of uncertainty can lead to inaccurate conclusions and hinder the progress of scientific knowledge. It is therefore crucial to understand and minimize wavelength uncertainty in order to ensure the accuracy and precision of scientific data.

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