# Derive a formula from the uncertainty principle

• bobby.pdx
In summary, to derive a formula for the relative spread of the spectral line corresponding to the longest wavelength of the Lyman series from the uncertainty principle, we can use the time-energy uncertainty relation and the lifetime/transition rate of the excited state. This formula is given by ΔE ≈ (hbar/2Δt), where Δt is the lifetime of the excited state. This calculation would result in the same answer for both hydrogen and tritium, regardless of the reduced mass correction to the Bohr model.
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:
σxσp≥$\hbar$/2

planck constant
$\hbar$=h/2pi
h=λp

Lyman series:
1/λ=RH(1-1/n2)

λ=hc/Ei-Ef

## 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

σxσp≥(hc/Ei-Ef)p/(4pi)

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

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##.

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

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

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?

## 1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, states that it is impossible to know the exact position and momentum of a particle at the same time. This means that the more precisely we know one of these properties, the less precisely we know the other.

## 2. How does the uncertainty principle relate to the formula?

The uncertainty principle is mathematically represented by the formula Δx * Δp ≥ h/4π, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant. This formula quantifies the limitation on our ability to measure both position and momentum of a particle simultaneously.

## 3. Can you provide an example of the uncertainty principle in action?

Yes, an example of the uncertainty principle in action is the double-slit experiment. In this experiment, a beam of particles is passed through two slits and observed on a screen. The pattern created by the particles on the screen is determined by their position and momentum, but due to the uncertainty principle, we cannot know both of these properties at the same time. Therefore, the pattern on the screen appears as a wave rather than individual particles.

## 4. How is the uncertainty principle important in quantum mechanics?

The uncertainty principle is a fundamental concept in quantum mechanics and plays a crucial role in understanding the behavior of particles at the subatomic level. It sets limits on the precision of our measurements and highlights the probabilistic nature of the quantum world.

## 5. Can the uncertainty principle be violated?

No, the uncertainty principle is a fundamental principle of quantum mechanics and has been proven experimentally countless times. It is a fundamental part of how the universe operates and cannot be violated.

Replies
3
Views
2K
• Quantum Physics
Replies
3
Views
1K
Replies
5
Views
3K
• Introductory Physics Homework Help
Replies
16
Views
1K
Replies
8
Views
2K
Replies
5
Views
2K
Replies
1
Views
3K
Replies
2
Views
2K