Question

Exercises \(78-88\) refer to a particle of mass \(m\) described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} 2 \sqrt{a^{3}} x e^{-a x} & x>0 \\ 0 & x<0 \end{array}\right. $$ Sketch the wave function. Is it smooth?

Short answer

The wave function for \(x<0\) is a straight line along the x-axis while for \(x \geq 0\) it's an exponentially decreasing function. With a common derivative at \(x = 0\), the wave function is smooth.

Step-by-step solution

Understanding the Wave Function

Analyze the given wave function \(\Psi(x)\), which is piece-wise defined. For \(x \geq 0\) the wave function is expressed by a term of the form \[2 \sqrt{a^{3}} x e^{-a x}\], and for \(x<0\) it is simply \(0\).

Graphing the Wave Function

Sketch the given wave function \(\psi(x)\). Due to the piece-wise nature of the function, the plot for \(x < 0\) would be a line along the x-axis. For \(x \geq 0\), the plot shape depends on the value of \(a\), but will decrease exponentially as we move away from 0.

Evaluating Smoothness

Evaluate if the function is smooth. Generally, a function is smooth if it has derivatives of all orders everywhere in its domain. In this case, since the function \(\psi(x)\) is differentiable for \(x < 0\) and \(x \geq 0\) separately, the crucial part to check is the point \(x = 0\). Verify that the derivative from the positive side equals zero which is the value of the derivative from the negative side. Therefore, this wave function is indeed smooth.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics, including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Unlike classical mechanics, where physical properties such as position and momentum are definite, in quantum mechanics, these properties are probabilistic and are described by wave functions.

The wave function, symbolized as \( \psi \) in exercises like the one presented, is a mathematical function providing information about the probability amplitude of a particle's position and momentum. Important principles such as the superposition of states and entanglement are part of the quantum mechanics realm. Through the analysis of \( \psi(x) \) for a particle, one can predict probabilities of finding the particle at a certain location, rather than exact outcomes.

Particle Mass

In the given exercise, the mass of the particle, denoted as \( m \), is a fundamental property that quantifies the amount of matter within the particle. In quantum mechanics, the mass of a particle contributes to determining its wave function, which describes its quantum state. The mass directly influences the particle's behavior under the influence of a force due to Newton's second law of motion \( F=ma \).

Moreover, the mass also affects how the wave function evolves over time. This is governed by the Schrödinger equation, a key equation in quantum mechanics that uses the mass of the particle in its formulation. Understanding the role of mass is pivotal in predicting how a particle moves and interacts with other particles and forces within the quantum realm.

Exponential Decay

Exponential decay is a process by which a quantity decreases at a rate proportional to its current value. In the context of the wave function given in the exercise, \( 2 \sqrt{a^{3}} x e^{-ax} \) includes the term \( e^{-ax} \) which signifies that the particle's probability amplitude decays exponentially as you move away from the origin \( x=0 \).

This type of decay is ubiquitous in physics and can describe a wide variety of systems, including radioactive decay in nuclear physics and attenuation of a signal in communications engineering. Understanding the nature of exponential decay is essential for interpreting the behavior of wave functions in quantum systems, as it highlights the likelihood of finding a particle at increasing distances from the point of origin.

Smoothness of Functions

The concept of smoothness of a function is deeply rooted in calculus and is essential for understanding the behavior of mathematical functions and their graphs. In simplistic terms, a function is considered smooth if it is continuously differentiable at all points in its domain, meaning it has no sharp corners or edges. For a wave function like the one sketched in the exercise, smoothness is important because it ensures the physical plausibility of a particle's quantum state.

In the exercise, to determine smoothness, one must check the derivatives of the function at all points, especially at any points where the function could change abruptly, such as at \( x=0 \) where the piece-wise parts of the function converge. The wave function provided is smooth because it maintains continuous derivatives, particularly at the boundary between the exponential section and the zero section. This ensures the suitability of the wave function for describing quantum states without any discontinuities or physically unrealistic behaviors.