Question

According to the energy-lime uncertainty principle. the lifetime \(\Delta f\) of a state and the uncertainty \(\Delta E\) in its energy are inversely proportional. Hydrogen's \(656 \mathrm{~nm}\) red spectral line is the result of an electron making a transition "downward" from a quantum state whose lifetime is about \(10^{-8} s\) (a) What inherent uncertainty in the energy of the emitted photon docs this imply? (Note: Unfortunately. we might use the symbol \(\Delta E\) for the energy difference - i.e., the energy of the photon - but here if means the uncertain in that energy difference.) (b) To what range in wavelengths does this correspond? (As noted in Exercise \(2.57\). the uncertainty principle is one contributor to the broadening of spectral lines.) (c) Obtain a general formula relating \(\Delta \lambda\) to \(\Delta t\).

Short answer

(a) The inherent uncertainty in the energy of the emitted photon is given by the energy uncertainty principle. (b) The range in wavelengths is determined by the energy uncertainty. (c) The general formula relating \(\Delta \lambda\) to \(\Delta t\) is \( \Delta \lambda = \lambda^2 \hbar / (2hc \Delta t) \)

Step-by-step solution

Calculate the Uncertainty in Energy

Apply the energy-time Uncertainty Principle formula: \( \Delta E \Delta t \geq \hbar / 2 \), where \( \hbar \) is the reduced Planck constant. Solve for \(\Delta E\) to get the uncertainty in energy: \( \Delta E = \hbar / (2 \Delta t) \)

Transform Energy Uncertainty into Wavelength Range

The energy equation is \( E = hc / \lambda \), where \( h \) is the Planck constant, \( c \) is the speed of light, \( \lambda \) is the wavelength. This equation can be rearranged to show how changes in energy correspond to changes in wavelength. Differentiate the energy equation to get \( dE = - hc / \lambda^2 d\lambda \), and use the above relation \(\Delta E = hc / \lambda^2 \Delta \lambda\) to solve for \(\Delta \lambda\).

Derive the General Formula

Substitute \(\Delta E\) from step 1 into the expression for \(\Delta \lambda\) obtained in step 2, this provides a general formula relating \(\Delta \lambda\) and \(\Delta t\) as \( \Delta \lambda = \lambda^2 \hbar / (2hc \Delta t) \)

Key concepts

Energy-Wavelength Relationship

The relationship between energy and wavelength is a fundamental concept in physics, often illustrated through the equation \( E = \frac{hc}{\lambda} \). Here, \( E \) stands for energy, \( h \) is the Planck constant, \( c \) represents the speed of light, and \( \lambda \) is the wavelength. This equation tells us that energy is inversely proportional to wavelength. Therefore, as the wavelength decreases, the energy of the wave increases, and vice versa.
Understanding this relationship is critical in many fields of science and technology. For example, in electronics and optics, this principle helps to design devices that manipulate light at various wavelengths to perform specific tasks. Additionally, in quantum mechanics, it is essential to comprehend how energy and wavelength influence the behavior of particles, especially those obeying the rules of wave-particle duality.
This concept is also central to understanding phenomena such as spectral line broadening, where small changes in energy due to environmental conditions, like temperature or pressure, lead to changes in wavelength.

Spectral Line Broadening

Spectral line broadening is an important concept that deals with the increase in width of spectral lines. When atoms or molecules emit or absorb light, they produce lines at specific wavelengths. However, several factors can cause these lines to broaden.
One of the key contributors to broadening is uncertainty in the energy of quantum states, as described by the uncertainty principle. This principle indicates that if the lifetime of a quantum state is short, its energy becomes more uncertain. As this energy uncertainty translates to a range in wavelengths, the spectral lines spread out, appearing broader.
Other factors adding to spectral line broadening include:

• **Doppler Broadening**: Caused by the motion of atoms or molecules relative to the observer, such as in stars or other astronomical objects.
• **Pressure Broadening**: Results from collisions between particles, which alter energy levels and lead to a widening of the spectral lines.
• **Natural Broadening**: A fundamental broadening that happens even without external influences, largely due to the finite lifetime of excited states.

Identifying and quantifying these effects is crucial in fields like astronomy and spectroscopy, where precise measurements of wavelengths are needed to infer properties of distant stars and galaxies.

Quantum State Lifetime

The lifetime of a quantum state refers to how long a particle, such as an electron in an atom, remains in a particular energy level before transitioning to another level. This lifetime is directly linked to the uncertainty in energy through the uncertainty principle: \( \Delta E \Delta t \geq \frac{\hbar}{2} \).
In layman's terms, if a quantum state has a very short lifetime, there's more inherent uncertainty in the energy associated with that state. Conversely, a longer lifetime reduces the energy uncertainty, contributing to sharper spectral lines.
Quantum state lifetime is a vital consideration in technologies like lasers, where precise emission of energy at specific wavelengths is crucial. For instance, in telecommunications, lasers with long-lived quantum states provide stable and coherent light sources for efficient data transmission. Similarly, in medical imaging, accurately controlling quantum state lifetimes allows for clearer images and more accurate diagnostics.
From a theoretical perspective, understanding quantum state lifetimes helps physicists explain processes like atomic transitions and interactions that underpin much of modern quantum mechanics.

Reduced Planck Constant

The reduced Planck constant, often denoted as \( \hbar \) (pronounced "h-bar"), is a fundamental constant in quantum mechanics. It is defined as \( \hbar = \frac{h}{2\pi} \), where \( h \) is the Planck constant. This constant plays a pivotal role in many quantum equations, particularly those dealing with angular momentum and the uncertainty principle.
Reduced Planck constant simplifies the mathematical formulation of quantum mechanics, providing a consistent unit of measure for action, energy, and time that fits naturally with the circular symmetries seen in wavefunctions.
In the context of the uncertainty principle, \( \hbar \) helps quantify the trade-off between the precision of a particle's position and momentum, or energy and time. The appearance of \( \hbar \) in fundamental equations signifies the transition from classical to quantum physics, emphasizing the discrete nature of quantum phenomena.

• In quantum mechanics, \( \hbar \) encapsulates the idea that energy is quantized, appearing in discrete amounts.
• It manifests in calculations like angular momentum quantization: \( L = n\hbar \), where \( n \) is an integer, illustrating how angular momentum in quantum systems comes in set steps.

Thus, \( \hbar \) is not just a bridge between energy, time, and other physical quantities, but a fundamental property shaping our understanding of the quantum realm.