# Uncertainty Principle and the lifetime of an excited state

• CRD_98
In summary, to estimate the lifetime of an excited state of an atom with a natural emission line width of 3.00 × 10−4 eV, you can use the equations ΔE Δt = (1/2) (h/2π) or ΔE Δt = (h/2π), with the latter being the more commonly accepted answer. Both equations give an approximate estimate and it is up to the instructor to decide which one to use.
CRD_98

## Homework Statement

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Estimate the lifetime of the excited state of an atom whose natural emission line width is 3.00 × 10−4 eV.

## Homework Equations

$$\Delta E \Delta t = \frac{1}{2} \times \frac{h}{2\pi}$$
$$\Delta E \Delta t = \frac{h}{2\pi}$$

## The Attempt at a Solution

I think all I have to do is rearrange for Δt then sub in the energy given as ΔE, but I'm not sure when I use which equation above, the correct answer is using the second, but I thought the first equation was a more accurate improved version of the second?

Wikipedia favors the first of your two equations where delta is the standard deviation of the quantity. Using either formula, you only get an approximate estimate. I think most instructors would accept either answer.

mccoy1

## What is the Uncertainty Principle?

The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, is a fundamental principle in quantum mechanics that states that it is impossible to know both the exact position and momentum of a particle at the same time. This means that there will always be some level of uncertainty in our measurements of these properties.

## How does the Uncertainty Principle relate to the lifetime of an excited state?

The Uncertainty Principle applies to the lifetime of an excited state because the more precisely we measure the energy of an excited state, the less precisely we can measure the time it takes for that state to decay. This is because the energy and time measurements are related through the Uncertainty Principle.

## Why is the Uncertainty Principle important in quantum mechanics?

The Uncertainty Principle is important because it sets a fundamental limit on our ability to measure and understand the behavior of quantum systems. It highlights the inherently probabilistic nature of quantum mechanics and has implications for how we view the world at the smallest scales.

## Can the Uncertainty Principle be violated?

No, the Uncertainty Principle is a fundamental principle of quantum mechanics and has been supported by numerous experiments. It is a fundamental limitation of our ability to measure certain properties of particles and cannot be violated.

## How is the Uncertainty Principle related to the concept of wave-particle duality?

The Uncertainty Principle is closely related to the concept of wave-particle duality, which states that particles can exhibit both wave-like and particle-like behavior. The Uncertainty Principle arises because in order to accurately describe a particle's position, we need to use a wave-like description, and vice versa for momentum. This duality is a fundamental aspect of quantum mechanics and is essential to understanding the Uncertainty Principle.

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