Derive a formula from the uncertainty principle

[bobby.pdx]

Homework Statement

Derive from the uncertainty principle a formula for the relative spread of the spectral line that corresponds to the longest wavelength of the Lyman series.

Homework Equations

uncertainty principle:
σ_xσ_p≥\hbar/2

planck constant
\hbar=h/2pi
h=λp

Lyman series:
1/λ=R_H(1-1/n²)

λ=hc/E_i-E_f

The Attempt at a Solution

I'm not quite sure how to go about the problem. I have gathered some formulas I believe will help me out. If I substitute some of these formulas into the uncertainty principle I get

σ_xσ_p≥(hc/E_i-E_f)p/(4pi)

I'm not sure where to go from here. Any help would be greatly appreciated.

 

[hilbert2]

I don't think it's possible to solve the problem using only those equations. You are probably expected to approximate the "lifetime broadening" of the spectral line. You'd have to know the lifetime/transition rate of the excited state corresponding to the spectral line, and use the time-energy uncertainty relation $\Delta E \Delta t \geq \hbar / 2$.

 

[bobby.pdx]

Let's say the lifetime of the excited state is 10^-7 seconds. How would I go about deriving the formula from there?

 

[hilbert2]

Well, if you know the lifetime $\Delta t$, then the spectral linewidth is just $\Delta E \approx \frac{\hbar}{2\Delta t}$.

 

[bobby.pdx]

This seems right. The only thing is the problem then asks to use this formula to calculate this kind of spread of spectral lines for both hydrogen and tritium for this spectral line with and without the reduced mass correction to the Bohr model of both hydrogen and tritium. If this formula is correct then the answer would be the same for both hydrogen and tritium right?