Question

Why is it impossible to know precisely the velocity and position of an electron at the same time?

Short answer

The impossibility of knowing an electron's precise velocity and position at the same time is due to the Heisenberg Uncertainty Principle, a fundamental concept in quantum mechanics. This principle states that there is a fundamental limit to the precision with which pairs of physical properties, like position and momentum (or velocity), can be known simultaneously. Mathematically, this is expressed as $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$, where $\mathrm{\Delta }x$ represents the uncertainty in position and $\mathrm{\Delta }p$ represents the uncertainty in momentum. Consequently, the more precisely one property is determined, the less precise the other becomes, leading to an inherent uncertainty in subatomic particles' properties.

Step-by-step solution

Introduction to the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle, named after German physicist Werner Heisenberg, claims that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as its position and momentum (velocity), can be simultaneously known.

Explanation of the principle

The uncertainty principle is a fundamental concept in quantum mechanics, which deals with the behavior of particles at atomic and subatomic scales. One of the main characteristics of quantum mechanics is that particles behave both as waves and as particles. Therefore, their properties, like position and velocity, are described by wave functions. These wave functions are inherently probabilistic, and the more precisely we can determine one property (e.g., the position), the less precisely we can determine the other (e.g., the momentum or velocity).

Uncertainty formula

The Heisenberg Uncertainty Principle can be mathematically described using the uncertainty formula: $$\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$$ Here, $\mathrm{\Delta }x$ represents the uncertainty in position and $\mathrm{\Delta }p$ represents the uncertainty in momentum (where momentum $p=m\cdot v$, with $m$ being the particle's mass and $v$ its velocity). The constant $\hslash$ (called "reduced Planck's constant") is approximately equal to $1.05\times {10}^{-34}Js$.

Electron uncertainty example

As mentioned before, the more precisely we can determine one property, the less precisely we can determine the other. For example, if we were able to measure the position of an electron with very high precision, i.e., $\mathrm{\Delta }x$ being very small, then the uncertainty in the momentum would be very large, which implies that the electron's velocity is highly uncertain.

Conclusion

It's impossible to know precisely the velocity and position of an electron at the same time because of the inherent limitations imposed by the Heisenberg Uncertainty Principle. According to this principle, there is a fundamental limit on the precision with which certain pairs of physical properties, like position and momentum (velocity), can be simultaneously known for a subatomic particle.

Key concepts

Quantum Mechanics

Quantum mechanics is a fascinating branch of physics that explores the nature of particles at the atomic and subatomic levels. Unlike classical mechanics, which describes large-scale objects and their predictable behaviors, quantum mechanics reveals a world of uncertainty and probability. This is because particles, such as electrons, don't behave like tiny billiard balls. Instead, they exhibit both wave-like and particle-like properties. At the heart of quantum mechanics is the idea that particles can exist in multiple states simultaneously, known as 'superposition.'
Understanding this concept is important because it explains why electrons don't have definite positions or velocities before we measure them. Instead, their properties are described by something called wave functions, which provide the probabilities of finding a particle in various places and with various velocities. It is essential to realize that because of these properties, quantum mechanics often defies our everyday understanding.

Electron Position and Velocity

In the realm of quantum mechanics, electron position and velocity are inherently uncertain due to the Heisenberg Uncertainty Principle. When we try to measure the position of an electron with great precision, its velocity becomes much less certain, and vice versa. This happens because measuring one affects the other due to the wave-like properties of electrons.
Electrons are not fixed in one place. Instead, they are thought of as "clouds" that occupy regions of space where they are most likely to be found. These are often referred to as orbitals. The uncertainty in measuring an electron's position and velocity arises because to define one precisely limits the possibility to define the other. Therefore, when you measure an electron's position very accurately, you're introducing a large amount of uncertainty into your measurement of its velocity.

Wave Functions

Wave functions are a crucial concept in quantum mechanics. They are mathematical functions that describe the quantum state of a particle, like its position, momentum, and energy. These functions provide the probability amplitudes for these aspects and help us understand the probabilistic nature of particles, such as electrons.
The wave function itself is denoted by the Greek letter psi ($\psi$). From $\psi$, we can determine where a particle is most likely to be found. To find the probability of a particle being in a particular state, you must square the wave function's amplitude. This probability is what leads to the inherent uncertainty in quantum properties: because we can only talk about likelihoods, not certainties, when it comes to an electron's behavior.

Reduced Planck's Constant

The reduced Planck's constant, often symbolized by $\hslash$, is a fundamental physical constant that plays a central role in quantum mechanics. It is derived from the Planck constant $h$, with the relationship $\hslash =\frac{h}{2\pi }$. In the context of the Heisenberg Uncertainty Principle, it sets the scale for the unavoidable uncertainties in position and momentum.
$\hslash$ is approximately equal to $1.05\times {10}^{-34}$ joule-seconds. Despite its small size, $\hslash$ impacts the microscopic world significantly. It appears in the uncertainty formula $\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$, which quantifies the trade-off between the accuracy of position and momentum measurements.

• $\mathrm{\Delta }x$: uncertainty in position
• $\mathrm{\Delta }p$: uncertainty in momentum

These elements underline the quantum scale's distinct behavior, differing from classical expectations, and highlight $\hslash$'s crucial role in the quantum realm.