Question

Exercises \(78-88\) refer to a particle of mass \(m\) described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} 2 \sqrt{a^{3}} x e^{-a x} & x>0 \\ 0 & x<0 \end{array}\right. $$ Verify that the normalization constant \(2 \sqrt{a^{3}}\) is correct.

Short answer

To verify the normalization constant, perform the integral of the absolute square of the wave function over all space (or over the range from 0 to infinity, since the function is zero for \(x<0\)). The integral should evaluate to 1 if the normalization constant is correct.

Step-by-step solution

Determine the function to be integrated

First, observe that the normalization constant is to be verified for the wave function \(\psi(x)\), which is given as \(2 \sqrt{a^{3}} x e^{-a x}\) for \(x > 0\) and 0 for \(x < 0\). Thus, the function that needs to be integrated is the absolute square of \(\psi(x)\), denoted as \(|\psi(x)|^2\) or \(\psi(x)*\psi(x)\), over its range of non-zero values (from 0 to infinity). Here, the absolute square simply involves squaring the given function, as it is already real.

Perform the integration

The integral to be solved is \(\int_0^\infty |\psi(x)|^2 dx = \int_0^\infty (2 \sqrt{a^{3}} x e^{-a x})^2 dx\). This equals \(\int_0^\infty 4a^3 x^2 e^{-2a x} dx\). You can integrate this by parts or by using tabular integration, or if you recall the form of the gamma function, you could recognize this as a scaled gamma function. Whichever method you choose, the result should be 1, as this would confirm that the normalization constant is correct.

Interpret the result

If the result of the integral in Step 2 is 1, the normalization constant is correct. If it is not 1, then the proposed normalization constant is incorrect.

Key concepts

Quantum Mechanics

The realm of quantum mechanics is strikingly different from the classical world we perceive daily, delving into the behavior of particles at the subatomic level. A fundamental concept in this realm is the wave function, represented by \( \psi(x) \), which provides a probabilistic forecast of a particle's position and momentum. In essence, quantum mechanics suggests that particles exhibit both wave-like and particle-like characteristics, a dual nature encapsulated by the wave function.

For students grappling with understanding wave functions, it's essential to recognize that they are mathematical expressions describing the quantum state of a single particle or system of particles. They are the key to unlocking the probabilities of finding particles in particular states. The wave function's square modulus, \( |\psi(x)|^2 \), is defined as the probability density, translating the abstract wave function into meaningful predictions about where a particle is likely to be found upon measurement.

Particle Physics

Particle physics zooms in on the universe's smallest known components and their interactions. Examining the wave function in the context of particle physics allows scientists to delve into the ephemeral existence of subatomic particles such as electrons, protons, and quarks. It’s crucial to note that, within this domain, every particle is described by its own unique wave function, which adheres to the normalization condition.

This is because, in the quantum world, the wave function's interpretation depends on its ability to convey probabilities that sum to one—an absolute requirement to maintain the conservation of probability. Therefore, normalization is not merely a mathematical quirk but a reflection of the fundamental principles governing particle physics. Correct normalization ensures that the total probability of finding the particle somewhere in the universe is exactly one, a foundational constraint in both quantum mechanics and particle physics.

Probability Density

The concept of probability density is central to quantum mechanics. It's a measure that details how the probability of finding a particle varies over space, providing insight into the likelihood of a particle's presence at any given position.

When exploring the wave function, \( \psi(x) \), the quantity \( |\psi(x)|^2 \)—which is the wave function's absolute square—serves as the probability density. Hence, the need for normalization: to ensure that the total probability across all space equals one. When lent context by an actual particle described by a wave function like \( \psi(x)=2 \sqrt{a^{3}} x e^{-a x} \) for \( x > 0 \) and zero elsewhere, the physical interpretation becomes vivid. The particle has zero probability of being found at \( x < 0 \) and a non-uniform probability for \( x > 0 \) that depends on the value of \( x \) and the parameter \( a \). This variability is typical of quantum systems, where probability densities manifest diverse and often complex distributions.

Mathematical Integration

Mathematical integration serves as the backbone for quantifying areas and volumes but, within the context of quantum mechanics, it's pivotal for normalizing wave functions.

The process of normalization involves integrating the wave function's probability density over the entire space where the particle could exist. As illustrated in the given exercise, to verify the wave function's normalization constant, one must integrate \( |\psi(x)|^2 \) from zero to infinity. This integral quantifies the total probability and should equate to one if the normalization constant is correct.

The integration step is not just a mechanical process but a manifestation of an underlying physical truth: the cumulative probability of finding a particle in all possible places must be unity. To master these concepts, especially in complex cases, students should cultivate comfort with techniques such as integration by parts or identifying special functions like the gamma function for efficient computation.