Problem 35 - College Physics

Question

(a) If the lifetime of excited state \(\mathrm{A}\) is longer than that of another excited state \(\mathrm{B}\), then the width of a spectral line due to natural broadening for a transition from state A to the ground state will be (1) smaller than, (2) the same as, (3) greater than that for a transition from state B to the ground state. Why? (b) Calculate the ratio of the width of a spectral line due to natural broadening for a transition from an excited state with a lifetime of \(10^{-12} \mathrm{~s}\) to the ground state to that from a state with a lifetime of \(10^{-8}\) s to the ground state.

Step-by-Step Solution

Answer

(a) Smaller than. (b) The ratio is 10000.

Step 1: Understanding Natural Broadening

The natural broadening of a spectral line is inversely related to the lifetime of the excited state from which the transition occurs. This relationship is described by the uncertainty principle. The formula relating the lifetime (\(\tau\)) and the energy width (\(\Gamma\)) is \[\Gamma = \frac{\hbar}{\tau}\], where \(\hbar\) is the reduced Planck's constant.

Step 2: Comparing Lifetimes and Widths

For part (a), if the lifetime of the excited state \(A\) is longer than that of state \(B\), then according to the formula \(\Gamma = \frac{\hbar}{\tau}\), the width \(\Gamma_A\)will be smaller than \(\Gamma_B\) because \(\tau_A > \tau_B\). Thus, the correct choice is (1) smaller than.

Step 3: Calculating the Width Ratio

For part (b), use the formula \(\Gamma = \frac{\hbar}{\tau}\) for both states and calculate the width ratio. Let \(\tau_1 = 10^{-12} \, \text{s}\) and \(\tau_2 = 10^{-8} \, \text{s}\). The widths are \(\Gamma_1 = \frac{\hbar}{10^{-12}}\) and \(\Gamma_2 = \frac{\hbar}{10^{-8}}\), so the ratio is \[\frac{\Gamma_1}{\Gamma_2} = \frac{\frac{\hbar}{10^{-12}}}{\frac{\hbar}{10^{-8}}} = \frac{10^{-8}}{10^{-12}} = 10^4.\]

Key Concepts

Spectral Lines

Spectral lines are unique fingerprints of atoms and molecules, revealing the emission or absorption of light at specific wavelengths. These lines come from electrons transitioning between different energy levels in an atom. When electrons move from a higher energy level to a lower one, they emit a photon with a specific energy corresponding to a particular wavelength. This emission gives rise to the characteristic lines observed in a spectrum.

In addition to their spectral identity, spectral lines can have different widths. The width can be influenced by various factors, including natural broadening, Doppler broadening, and collisional broadening. Natural broadening is intrinsic to the atom or molecule and arises due to the limited lifetime of excited states.

Understanding the properties of spectral lines is essential in fields like astrophysics and chemistry, where they are used to infer the composition and conditions of stars and gases. These unique spectral signatures provide a wealth of information about the environment in which the molecules exist.

Uncertainty Principle

The uncertainty principle is a fundamental concept in quantum mechanics, emphasizing the intrinsic limitations in observing certain pairs of complementary properties of particles. It primarily states that the more precisely one property is measured, the less precisely the complementary property can be known.

The most famous example involves position and momentum, but for spectral lines, the principle relates the energy width to the lifetime of an excited state. Mathematically, it can be represented as \[ riangle E riangle t \approx rac{ar{h}}{2},\]where \( riangle E \) is the uncertainty in energy (energy width), and \( riangle t \) is the uncertainty in time (related to the lifetime of the state).

Thus, a shorter lifetime results in a larger energy uncertainty or broader spectral line. This relationship explains why spectral lines from states with shorter lifetimes appear broader due to natural broadening. In essence, the uncertainty principle highlights a fundamental characteristic of the quantum world.

Energy Width

Energy width refers to the range of energy within which an atomic transition occurs, directly impacting the appearance of spectral lines. For atomic transitions, the energy width is affected by the lifetime of the excited state, with shorter lifetimes leading to broader spectral lines and vice versa.

The relationship between the energy width \( riangle E \) and the lifetime \( riangle t \) can be described by the simple formula:\[ riangle E = rac{ar{h}}{ riangle t},\]where \( ar{h}\) is the reduced Planck's constant. By observing the energy width of a spectral line, researchers can infer details about the state lifetime. For example, the exercise highlights how a state with a lifetime of \( 10^{-12} \) seconds has a broader spectral line than a state with a lifetime of \( 10^{-8} \) seconds, embodying this inverse relationship.

These energy widths and associated line profiles are crucial for spectroscopy, as they offer insights into the physical conditions of the emitting or absorbing material. They become important tools in the study of both terrestrial and astronomical phenomena.