Problem 30 - College Physics

Question

If an excited state of an atom has a lifetime of \(2.0 \times 10^{-7} \mathrm{~s},\) what is the minimum error associated with the measurement of the energy of this state?

Step-by-Step Solution

Answer

The minimum error in energy measurement is \( 2.6364295 \times 10^{-28} \) J.

Step 1: Identify Known Quantities

We are given that the lifetime of the excited state, denoted as \( \Delta t \), is \( 2.0 \times 10^{-7} \) seconds.

Step 2: Introduce Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle relates the uncertainty in energy \( \Delta E \) and the uncertainty in time \( \Delta t \) as follows: \[ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \] where \( \hbar \) is the reduced Planck's constant (\( \hbar = 1.0545718 \times 10^{-34} \, \text{Js} \)).

Step 3: Rearrange Formula for Energy Uncertainty

To find the minimum error in energy, rearrange the inequality to solve for \( \Delta E \): \[ \Delta E \geq \frac{\hbar}{2 \cdot \Delta t} \]

Step 4: Calculate Minimum Energy Uncertainty

Substitute the given values of \( \Delta t \) and \( \hbar \) into the rearranged formula: \[ \Delta E \geq \frac{1.0545718 \times 10^{-34}}{2 \times 2.0 \times 10^{-7}} \] Calculate the result to find the minimum energy uncertainty.

Step 5: Perform Final Calculation

Compute the division: \( \Delta E \geq \frac{1.0545718 \times 10^{-34}}{4 \times 10^{-7}} \) \( \Delta E \geq 2.6364295 \times 10^{-28} \text{ J} \).

Step 6: Minimum Error in Energy

The minimum error associated with the measurement of the energy of this state is \( 2.6364295 \times 10^{-28} \text{ J}. \)

Key Concepts

Energy Uncertainty

The idea of energy uncertainty is central to understanding the behavior of quantum systems. Quantum mechanics teaches us that it's impossible to precisely measure two conjugate properties, like energy and time, simultaneously. The Heisenberg Uncertainty Principle captures this idea with the equation:
\[ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \]
Here, \( \Delta E \) represents the uncertainty in energy, while \( \Delta t \) indicates the uncertainty in time. This formula essentially tells us that if the lifetime of a state is very short (\( \Delta t \) is small), then the uncertainty in energy (\( \Delta E \)) must be large. Conversely, a longer lifetime results in a lower energy uncertainty.

• Uncertainty in measurement is a crucial insight of quantum mechanics.
• It reflects a fundamental limit rather than a flaw in technological precision.

When solving problems, the energy uncertainty can be calculated if the time uncertainty is known. This principle helps physicists understand the stability and reactions of particles and atoms.

Lifetime of Excited State

The lifetime of an excited state, denoted by \( \Delta t \), refers to the time span an atom remains in an excited energy state before returning to a lower energy state. This period is vital for determining the uncertainty in energy as per the Heisenberg Uncertainty Principle.

• Short lifetimes lead to greater uncertainties in energy.
• This results in spectral lines being broader, an effect seen in many optical experiments.

For instance, in the exercise provided, the excited state lasts for \(2.0 \times 10^{-7}\) seconds. This brevity showcases a relatively high uncertainty in energy. Studying excited state lifetimes is important in fields such as spectroscopy and photophysics, providing insights into material properties and behaviors.

Reduced Planck's Constant

The reduced Planck's constant, denoted as \( \hbar \), is a fundamental constant in quantum mechanics. It is defined as \( \hbar = \frac{h}{2\pi} \), where \( h \) is the Planck's constant. The value of \( \hbar \) is approximately \( 1.0545718 \times 10^{-34} \) Js.
This constant is pivotal in the realm of quantum mechanics because it appears in various equations, including the Heisenberg Uncertainty Principle.

• It quantifies the scale at which quantum mechanical phenomena become relevant.
• \( \hbar \) is used in contexts involving wavefunction descriptions and angular momentum calculations.

By using \( \hbar \) rather than the full Planck's constant, equations become simplified, making them easier to use in practical calculations. It serves as a reminder of nature's intrinsic limits on certain measurements.