Question

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

Short answer

Answer: Using the Heisenberg Uncertainty Principle formula and solving for the uncertainties in time (Δt), we can calculate the maximum existence times for both electron-positron pairs and proton-antiproton pairs. The specific values of Δt will depend on the calculated uncertainties in energy (ΔE) for each pair, which were determined using the rest mass energies of the particles.

Step-by-step solution

Determine the rest mass energies of the particles

We'll begin by calculating the rest mass energies (E) of the electron and proton using the equation: E = mc^2 where m is the mass of the particle, and c is the speed of light. For an electron, m = 9.11 x 10^{-31} kg, and For a proton, m = 1.67 x 10^{-27} kg Keep in mind that the antiparticles have the same mass, so their rest mass energies will be equal.

Calculate the uncertainties in energy

To find the uncertainties in energy (ΔE), we must consider both the particle and the antiparticle: For the electron-positron pair, ΔE = 2 * E_electron For the proton-antiproton pair, ΔE = 2 * E_proton

Apply the Heisenberg Uncertainty Principle

Now that we have the uncertainties in energy, we can use the Heisenberg Uncertainty Principle formula to find the uncertainties in time (Δt): ΔE * Δt ≥ ħ/2 Solving for Δt, we get: Δt ≥ ħ/(2 * ΔE) We will now plug in our calculated values for ΔE and the reduced Planck constant (ħ = 1.05 x 10^{-34} Js).

Calculate the maximum existence time for electron-positron pairs

For the electron-positron pair, substitute the appropriate values into the Δt formula: Δt_electron-positron ≥ (1.05 x 10^{-34} Js)/(2 * ΔE_electron-positron) Calculate the result to find the maximum existence time.

Calculate the maximum existence time for proton-antiproton pairs

For the proton-antiproton pair, substitute the appropriate values into the Δt formula: Δt_proton-antiproton ≥ (1.05 x 10^{-34} Js)/(2 * ΔE_proton-antiproton) Calculate the result to find the maximum existence time. Now you have found the maximum existence times for both the electron-positron pairs and the proton-antiproton pairs based on the energy-time uncertainty principle.

Key concepts

Particle-Antiparticle Pair Creation

The fascinating phenomenon of particle-antiparticle pair creation is rooted in quantum mechanics. It involves a fleeting moment where energy momentarily transforms into a particle and its opposite, known as an antiparticle. This can occur even in a vacuum, seemingly out of 'empty space.' Think of it much like a loan from the universe – energy is 'borrowed' to create these pairs, but they must annihilate each other and repay the energy quickly, often in tiny fractions of a second.

For students grappling with this concept, an analogy could be helpful: imagine if you could briefly create a pair of magnetic toys by taking a loan from an energy 'bank,' and then they immediately snap back together and disappear, repaying the energy 'loan.' This is akin to how particles and antiparticles spring in and out of existence.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a cornerstone of quantum physics that places a fundamental limit on our ability to precisely know certain pairs of physical properties at the same time. One of the most famous pairs is position and momentum. However, it also applies to energy and time. According to this principle, the more accurately we know the energy of a system, the less precisely we can know the time over which the system exists, and vice versa. This principle can be mathematically expressed as \[\begin{equation} ΔEΔt ≥ \frac{ħ}{2} \text{,}\bbox[yellow] \where ΔE \text{is the uncertainty in energy,} Δt \text{is the uncertainty in time, and} ħ \text{(h-bar) is the reduced Planck constant.}\end{equation}\]

Understandably, it can be challenging for students to wrap their heads around this; thus, a useful way to think of it is that there's a trade-off or a 'blurriness' in our understanding of energy and time for very small scales. Imagine trying to take a quick, clear snapshot of a rapidly moving object – the faster it moves (akin to shorter time), the blurrier the image (akin to uncertain energy).

Rest Mass Energy Calculation

The Rest Mass Energy of a particle is a concept deeply embedded in Einstein's theory of relativity. It is perhaps best understood through the iconic equation \[\begin{equation} E = mc^2 \bbox[yellow] \text{, where } E \text{ denotes energy, } m \text{ mass, and } c \text{ the speed of light in a vacuum.}\end{equation}\]

When we refer to the rest mass energy, we're talking about the energy that an object contains simply by virtue of having mass and existing - it's almost like the 'inherent energy' of stuff. For students, think of it like the potential of a still water reservoir; even when calm and still, it holds the potential to generate energy. To calculate this for a given particle, you plug the mass of the particle (the same for its antiparticle) and the speed of light into Einstein's equation. The product is the energy associated with the particle at rest, not moving relative to an observer.

Planck Constant

The Planck constant is a quantum of action that plays a vital role in the quantum world. It is defined as the proportionality constant between the minimum increment of energy, also known as a quantum, of a particle and the frequency of its associated electromagnetic wave. Symbolized as h, its value is approximately 6.62607015 × 10^-34 J·s, with the reduced Planck constant (denoted as ħ or h-bar) being h divided by 2π.

Students might find it abstract, so let's simplify this: think of the Planck constant as the 'DNA' of the quantum realm – a fundamental component that helps dictate the behavior and properties of particles. It's as crucial to understanding quantum physics as knowing that water is H2O is to understanding chemistry.