Question

An atom in a metastable state has a lifetime of 5.2 ms. What is the uncertainty in energy of the metastable state?

Short answer

The energy uncertainty is approximately $1.0148\times {10}^{-32}$ joules.

Step-by-step solution

Understanding Lifetime and Energy Uncertainty

The energy uncertainty of a quantum state, such as a metastable state of an atom, can be determined using the Heisenberg uncertainty principle. This principle in relation to energy ($\mathrm{\Delta }E$) and time ($\mathrm{\Delta }t$) is given as: $\mathrm{\Delta }E\cdot \mathrm{\Delta }t\ge \frac{\hslash }{2}$ where $\hslash$ is the reduced Planck’s constant ($\hslash =1.0545718\times {10}^{-34}J\cdot s$). Here, $\mathrm{\Delta }t$ is provided as the lifetime of the metastable state.

Set Up the Inequality for Uncertainty

Given the relation $\mathrm{\Delta }E\cdot \mathrm{\Delta }t=\frac{\hslash }{2}$, we can solve for the uncertainty in energy $\mathrm{\Delta }E$. $\mathrm{\Delta }t=5.2$ ms = $5.2\times {10}^{-3}$ s.

Solve for Energy Uncertainty

To find $\mathrm{\Delta }E$, rearrange the formula: $\mathrm{\Delta }E=\frac{\hslash }{2\cdot \mathrm{\Delta }t}$. Substituting the values, we get: $$\mathrm{\Delta }E=\frac{1.0545718\times {10}^{-34}}{2\times 5.2\times {10}^{-3}}$$.

Calculating the Result

Perform the calculation: $$\mathrm{\Delta }E\approx \frac{1.0545718\times {10}^{-34}}{2\times 5.2\times {10}^{-3}}\approx 1.0148\times {10}^{-32}J$$. Therefore, the uncertainty in energy of the metastable state is approximately $1.0148\times {10}^{-32}$ joules.

Key concepts

Metastable State

A metastable state refers to a temporary state of an atom where it is in an excited configuration. Unlike typical excited states, a metastable state has an unusually long lifetime before decaying to a more stable state. This prolonged existence makes it interesting to scientists as it provides a window to study various phenomena in quantum mechanics, such as energy levels and transitions.

The extended lifetime of a metastable state is due to restricted transition possibilities to a lower energy state. Such restrictions often occur when transitions to lower energy levels are "forbidden" or highly improbable under normal circumstances. This phenomenon is a key reason why some atomic states can last milliseconds, like the one mentioned with a 5.2 ms lifetime, in contrast to typical excited states that decay in nanoseconds.

Metastable states are incredibly significant in areas like lasers and phosphorescence, where sustained emission of light or energy over time is harnessed for practical applications.

Energy Uncertainty

Energy uncertainty is a principle that arises from the Heisenberg uncertainty principle, which posits that it is impossible to know both the energy and duration of a quantum system's state with absolute precision. In the context of a metastable state, this means that the exact energy level of the state cannot be perfectly defined if the lifetime of the state is known, and vice versa.

This uncertainty is expressed through the formula $$\mathrm{\Delta }E\cdot \mathrm{\Delta }t\ge \frac{\hslash }{2}$$ where $\mathrm{\Delta }E$ is the energy uncertainty, $\mathrm{\Delta }t$ is the time uncertainty or the lifetime of the state, and $\hslash$ is the reduced Planck's constant. The inequality highlights how knowledge about either time or energy impacts the precision with which the other can be determined.

Understanding energy uncertainty is crucial in quantum mechanics as it underscores the intrinsic limitations in measuring quantum states, influencing theories that tackle the behavior of atoms and particles at the smallest scales.

Reduced Planck's Constant

The reduced Planck's constant, often denoted by $\hslash$, is a fundamental physical constant crucial in quantum mechanics. It is defined as the Planck's constant $h$ divided by $2\pi$, or $\hslash =\frac{h}{2\pi }\approx 1.0545718\times {10}^{-34}J\cdot s$.

$\hslash$ is widely used as it naturally appears in quantum physics equations that describe the properties and behaviors of particles at quantum levels. In the context of the uncertainty principle, $\hslash$ bridges the connection between the measurable energy of a system and the time duration over which it can be accurately measured.

This constant is pivotal in understanding the scale at which quantum effects become significant. It plays a critical role in formulas that govern quantum entities like electrons in an atom or photons in light, signifying its ubiquitous presence in the equations that describe the microscopic universe.