Question

Quantum-mechanical stationary states are of the general form \(\Psi(x, t)=\psi(x) e^{-i \omega t},\) For the basic plane wave (Chapter 4 ), this is \(\Psi(x, t)=A e^{i k x} e^{-i \omega t}=\) \(A e^{i(k x-\omega t)},\) and for a particle in a box, it is \(\Psi(x, r)=\) A \(\sin (k x) e^{-i \omega t}\). Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?

Short answer

The unique features of waves moving in essentially two directions include interference and reflection leading to standing waves, which are only seen in the case of a particle in a box. This is supported by the evaluation of the probability densities where the uniformity in the plane wave supports pure momentum and variances in a particle in a box reveals movement in two distinct directions.

Step-by-step solution

Understanding the characteristics of the wave functions

Start by understanding the given quantum mechanical stationary states for both a basic plane wave and a particle in a box. A plane wave, \(\Psi(x, t)=A e^{i(k x-\omega t)}\), moves in a single direction and its wave function corresponds to a particle with well-defined momentum. For a particle in a box, the wave function is \(\Psi(x, r)= A \sin (k x) e^{-i \omega t}\), which is not just sinusoidal, but also has traits of movements in two directions.

Identifying Unique Characteristics

Quantum particles moving in one direction (like a plane wave) possess pure momentum, meaning they are uniform and unidirectional. Therefore, they lack features such as interference and/or reflection that are typical when waves move in essentially two directions (as in a box). These phenomena produce standing waves, a key characteristic of the particle in a box scenario.

Evaluating Probability Densities

The probability density for a quantum state is given by \(|\Psi(x, t)|^2\). Applying this to both cases, for a plane wave, the probability density is \(|A|^2\), meaning it's uniform (confirming the pure momentum property). For a particle in a box, the probability density is \(|A|^2 \sin^2(kx)\), which varies with position (due to sinusoidal nature), demonstrating the traits of moving in two distinct directions.

Key concepts

Plane Wave

Imagine you're tossing a stone into a calm pond and watching the ripples travel outward—this image closely mirrors the concept of a plane wave in quantum mechanics. A plane wave is the simplest form of a wave function, which represents the state of a quantum particle moving in one specific direction. It's mathematically expressed as \(\Psi(x, t)=A e^{i(kx-\omega t)}\), where \(A\) represents the amplitude, \((k\) is the wave number related to the momentum of the particle, and \((\textbackslash\omega\) stands for the angular frequency correlated with the particle's energy.

In this serene pond of our imagination, a plane wave is akin to a single, uniform wavefront, propelling across the pond without changing shape—symbolizing a particle with a well-defined, unvarying momentum. This wave nature stands in contrast to phenomena like reflection or interference, which you don't observe in this simple picture because there's only one direction of movement. That's what makes the plane wave a uniform probability distribution \(\|A\|^2\), perfect for visualizing a particle in a vast, open space without any boundaries to complicate its path.

Particle in a Box

Now, let's switch our setting and imagine a particle that's not free to roam an endless space but is instead confined within a microscopic 'box'—a fundamental scenario in quantum mechanics known as the particle in a box model. Here, the particle is restricted to a specific region and cannot escape, leading to its wave function taking the form \(\Psi(x, t)=A \sin(kx) e^{-i\omega t}\). Unlike the unobstructed plane wave, this wave function suggests the particle bounces back and forth between the walls of the box, resulting in a standing wave.

This model gives rise to unique attributes, such as nodes (points where the probability of finding the particle is zero) and anti-nodes (points of maximum probability). Standing waves are the standing ovation of waves—where certain points remain still while others vibrate with maximum energy. Think of a guitar string—plucked and fixed at both ends—where vibrations create standing wave patterns. The probability density for our quantum particle, \(\|A\|^2 \sin^2(kx)\), reflects this behavior with a sinusoidal fluctuation, capturing the essence of the particle's existence as trapped within the box's confines.

Probability Density

The term probability density might sound abstract but think of it as a map that shows where you're most likely to find something. In quantum mechanics, this 'something' is a particle, and the probability density tells us how likely it is to find that particle at a particular location. Mathematically, it's defined by \(\|\Psi(x, t)\|^2\), which represents the square of the wave function's absolute value.

For our free-traveling particle represented by a plane wave, this value doesn't change with location—the map of likelihood is uniform across all points. In contrast, the particle in a box has a dynamic map, one that peaks at certain points and dips to zero at others, resembling a vibrant city with areas of bustling activity and quiet neighborhoods. These variations in the probability density graphically portray the complex dance of a constrained particle—showing how probability ebbs and flows along the space of its minuscule stage, the box.