Question

Particle-antiparticle pairs are occasionally created out of empty space. Looking at energy-time uncertainty, how long would such particles be expected to exist if they are: a) an electron/positron pair? b) a proton/antiproton pair?

Short answer

Answer: According to the energy-time uncertainty principle, the expected time uncertainties for a particle-antiparticle pair created out of empty space are: a) electron/positron pair: Δt_electron ≳ 6.42 × 10^-22 s b) proton/antiproton pair: Δt_proton ≳ 3.52 × 10^-25 s

Step-by-step solution

Determine the rest mass energy of electron/positron and proton/antiproton pairs

First, we need to find the rest mass energy for an electron/positron pair and a proton/antiproton pair. The masses of electron and proton are given as: m_electron = 9.10938356 × 10^-31 kg m_proton = 1.67262192 × 10^-27 kg Now, calculate the rest mass energy for each pair using the formula E = mc^2, where c = 3 × 10^8 m/s is the speed of light: E_electron = m_electron * c^2 ≈ 8.19 × 10^-14 J E_proton = m_proton * c^2 ≈ 1.50 × 10^-10 J

Use the energy-time uncertainty principle to calculate the time uncertainty

Now that we have the rest mass energy for each particle-antiparticle pair, we can use the energy-time uncertainty principle to find the time uncertainty for each pair: Δt ≥ ħ / (2 * ΔE) Here, ħ = h /(2 * π) is the reduced Planck constant, and h ≈ 6.62607015 × 10^-34 Js is the Planck constant. Firstly, calculate the value of ħ: ħ = h /(2 * π) ≈ 1.054571817 × 10^-34 Js Now, calculate the time uncertainty for an electron/positron pair: Δt_electron ≥ (1.054571817 × 10^-34 Js) / (2 * 8.19 × 10^-14 J) Δt_electron ≥ 6.42 × 10^-22 s And for a proton/antiproton pair: Δt_proton ≥ (1.054571817 × 10^-34 Js) / (2 * 1.50 × 10^-10 J) Δt_proton ≥ 3.52 × 10^-25 s In conclusion, the time uncertainties for a particle-antiparticle pair created out of empty space are: a) electron/positron pair: Δt_electron ≳ 6.42 × 10^-22 s b) proton/antiproton pair: Δt_proton ≳ 3.52 × 10^-25 s

Key concepts

Particle-Antiparticle Pair Creation

In the quantum world, empty space isn't as empty as one might think. Due to the fascinating principles of quantum mechanics, particle-antiparticle pairs can spontaneously materialize and then quickly disappear. These pairs are typically composed of a particle and its corresponding antiparticle. For instance, an electron can appear alongside a positron, its antimatter equivalent.
This phenomenon doesn't happen all the time but is part of the quantum 'background noise' that exists in any given region of space. The process is called pair creation, and it's a temporary event; the particles usually exist for only a fleeting moment before annihilating each other.
The brief existence of these pairs is connected to the energy-time uncertainty principle, which suggests that energy can fluctuate in short time durations, leading to such temporary creations.

Rest Mass Energy

To understand pair creation, the concept of rest mass energy is key. Simply put, rest mass energy is the energy held by a particle when it is not moving. According to Einstein's famous equation, \[ E = mc^2 \]this energy can be calculated by multiplying the mass of the particle by the speed of light squared.
For example, an electron has a very small mass, so its rest mass energy is also quite small, around 8.19 × 10^-14 Joules. In contrast, a proton, being much heavier, has a larger rest mass energy of about 1.50 × 10^-10 Joules.
These values illustrate a fundamental principle: heavier particles require more energy to exist, even temporarily, because they have larger rest mass energies.

Planck Constant

The Planck constant is a fundamental value deeply embedded in quantum mechanics. It provides the scale at which quantum mechanical effects become significant.
In the context of the energy-time uncertainty principle, we use a version called the reduced Planck constant, denoted by \( \hbar \), which is \[ \hbar = \frac{h}{2\pi} \approx 1.054571817 \times 10^{-34} \text{ Js} \]where \( h \approx 6.62607015 \times 10^{-34} \text{ Js} \). This value helps us determine the time interval in which energy fluctuations (like particle-antiparticle pair creations) are significant.
The smaller the value of \( \hbar \), the shorter the time a particle-antiparticle pair can exist based on its rest mass energy. It's essential in linking time, energy, and the scale at which quantum effects occur.

Quantum Fluctuations

Quantum fluctuations are temporary changes in the amount of energy in a point in space, as predicted by quantum mechanics. These fluctuations allow for the creation and annihilation of particle-antiparticle pairs.
They occur because of the inherent uncertainty in quantum mechanics, specifically through the energy-time uncertainty principle, which states: \[ \Delta t \geq \frac{\hbar}{2 \Delta E} \]Here, \( \Delta t \) is the time uncertainty and \( \Delta E \) is the energy uncertainty. This principle implies that the shorter the time interval, the higher the energy fluctuation can be.
Such fluctuations are a vivid reminder of the probabilistic nature of the quantum world, where energy can "borrow" from the universe to create particles briefly. As a result, quantum fluctuations are fundamental in understanding how the universe remains dynamic, even in a seemingly empty space.