Applying uncertainty principle to decaying states

[cosmicopium]

Homework Statement

I quote from my text, "The decay of excited states in atoms and nuclei often leave the system in another, albeit lower-energy, excited state. (a) One example is the decay between two excited states of the nucleus of ^48Ti. The upper state has a lifetime of 1.4 ps, the lower state 3.0 ps. What is the fractional uncertainty deltaE/E in the energy of 1.3117-MeV gamma rays connecting the two states? (b) Aother example is the H_alpha line of the hydrogen Balmer series. In this case the lifetime of both states is about the same, 10-8 s. What is the uncertainty in the energy of the H_alpha photon?"

Homework Equations

delta_E*delta_t is greater than or equal to h-bar/2

The Attempt at a Solution

Those lifetimes they give are of each state, not of photon itself. I thought about trying to find the energy of each state, by using the above version of the uncertainty principle, and then subtracting to get the energy difference, which should be the energy of the photon. However, I keep getting the wrong answer. Any ideas? The book contains no examples of this type of problem.
My book, which has been wrong in the past, says the answers are: (a) 5.310-10 eV (b) 1.3210-7 eV

 

[mark726]

Thank you for your question. It seems like you are on the right track in trying to use the uncertainty principle to solve this problem. However, I believe there may be a small mistake in your approach. Let me explain.

The uncertainty principle states that the product of the uncertainty in energy (delta_E) and the uncertainty in time (delta_t) must be greater than or equal to h-bar/2. In this case, we are given the lifetime of each state, not the uncertainty in time. Therefore, we need to use the inverse of the lifetime (1/t) as the uncertainty in time.

Now, for part (a), we can use the formula delta_E*delta_t is greater than or equal to h-bar/2 to find the uncertainty in energy. We know that the energy difference between the two states is 1.3117 MeV. Using this information, we can set up the following equation:

delta_E * (1/1.4 ps) = h-bar/2

Solving for delta_E gives us a value of approximately 4.64x10^-13 MeV. This is the uncertainty in energy for the transition between the two states. To find the fractional uncertainty, we need to divide this value by the energy difference (1.3117 MeV) and multiply by 100%. This gives us a fractional uncertainty of approximately 3.54x10^-11%.

For part (b), we can use the same approach. The only difference is that now both states have the same lifetime of 10^-8 s. Therefore, the uncertainty in time for each state is also 10^-8 s. Using the same formula, we get:

delta_E * (1/10^-8 s) = h-bar/2

Solving for delta_E gives us a value of approximately 6.58x10^-8 eV. Again, to find the fractional uncertainty, we divide this value by the energy difference (1.3123 eV) and multiply by 100%. This gives us a fractional uncertainty of approximately 5.00x10^-6%.

I hope this helps clarify the approach for solving this problem. Please let me know if you have any further questions. Good luck with your studies!