Question

When we refer to a "bound" particle. we usually mean one for which there is no probability of finding it outside some finite confines. Could a bound particle be perfectly dead stationary, meaning a well-defined velocity of zero? Why or why nor?

Short answer

No. A bound particle cannot be perfectly stationary because it would violate the Heisenberg uncertainty principle, which states that the exact position and momentum (or velocity) of a particle cannot both be perfectly defined simultaneously. If a bound particle was perfectly static (velocity equal to zero), its position would become completely undefined, contradicting the definition of a bound particle.

Step-by-step solution

Understanding the Bounded Particle

In quantum mechanics, a bound particle is a particle that is confined to a certain region of space which could be described by a potential well. As it is confined, it's not free to move to any arbitrary position in space.

The Uncertainty Principle

According to the Heisenberg uncertainty principle in quantum physics, it's not possible to simultaneously measure the exact position and momentum (or equivalently, velocity) of a particle. This means that the more precisely the position is known, the less precisely the momentum can be known, and vice versa.

Applying the Uncertainty Principle to the Bounded Particle

If a bound particle were perfectly motionless, it would have a well-defined velocity of zero. But according to the uncertainty principle, if a particle's velocity is perfectly defined, its position becomes completely undefined. This contradicts with the definition of a bound particle, which is confined to a certain region of space. Therefore, it is not possible for a bound particle to be perfectly stationary.

Key concepts

Heisenberg Uncertainty Principle

In quantum mechanics, the Heisenberg Uncertainty Principle is a foundational concept. This principle tells us that there is a fundamental limit to the precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously. If you try to measure a particle's position very accurately, the measurement of its momentum becomes less precise, and vice versa. This isn't a limitation of our measurement tools, but rather a fundamental property of nature itself. Imagine trying to measure the position of a flying bird in the sky. As you focus on pinpointing its exact spot, the difficulty in knowing how fast and in what direction it’s moving increases. The same holds true for particles at the quantum level, which is why a bound particle can't have a precisely zero velocity.

Bound Particles

Bound particles are confined within a certain region of space. This confinement means that they can't be found just anywhere; their presence is limited to a specific area delineated by a potential barrier. To picture this, think of a ball being trapped within a bowl. The ball is free to move within the bowl but cannot escape without additional energy. In quantum mechanics, a bound particle similarly cannot escape its potential well without additional energy. The confinement of a bound particle implies a probability distribution describing where it may be found, but due to the uncertainty principle, this probability also includes some uncertainty in its momentum, meaning it can never be perfectly at rest.

Potential Well

A potential well is a concept used to describe the energy landscape in which a particle is found. Think of it as a valley in a hilly region where a marble might roll back and forth, unable to climb out without added energy. In quantum mechanics, a particle within a potential well is akin to having energy levels between defined limits. These wells can vary in shape and size, from the simple vertical walls of an idealized box to more complex shapes. The depth and shape of the well define the characteristics of any bound states or trapped particles. Also, within these potential wells, quantum effects mean that particles are "smeared out" within the confines of the well, never being at a single point at any given time due to their wave-like nature.

Particle Confinement

Particle confinement refers to the quantum state of a particle being restricted within a certain space. This can happen due to various factors like potential wells or physical boundaries. In quantum scenarios, unlike classical physics, confinement of a particle in a reduced volume increases its "quantum fuzziness," making its behavior less predictable. By confining a particle, its quantum wavefunction spreads over the available space, meaning it cannot have a precise position. This behavior is integral to understanding bound particles as they exhibit both wave-like properties and constraints imposed by their environment. If a particle is confined, we might know its position quite well, but its momentum remains elusive, making it impossible for such a particle to be completely motionless.