Question

Under what circumstance does the integral \(\int_{x_{0}}^{\infty} x^{b} d x\) diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than \(|x|^{-1} / 2\) does as \(x\) gets large.

Short answer

The integral \(\int_{x_{0}}^{\infty} x^{b} d x\) diverges when \(b \leq -1\). Hence the wave function must fall to 0 faster than \(|x|^{-1} / 2\) as \(x\) gets large or it will be unphysical and not normalizable.

Step-by-step solution

Compute the Integral

To find whether the integral \(\int_{x_{0}}^{\infty} x^{b} d x\) diverges, we first compute the integral. Using the power rule for integration, the antiderivative of \(x^{b}\) is \(x^{b+1} / (b+1)\). So the integral is \((x^{b+1} / (b+1))|_{x_{0}}^\infty\).

Evaluate the Integral and Determine the Condition for Divergence

The result of the integral becomes \(\infty / (b+1)\) minus the finite term \(x_{0}^{b+1} / (b+1)\) as x gets infinitely large. This integral diverges (is infinite) if \(b+1 \leq 0\). Therefore, the condition for divergence is for \(b \leq -1\)

Relate the Result to Wave Function Behavior

The result implies that for the integral to converge, ie for the wave function to be normalizable, it must fall off faster than \(|x|^{-1}\) as \(x\) gets large. Which is indeed faster than \(|x|^{-1} / 2\). Therefore, a physically acceptable wave function must fall to 0 faster than \(|x|^{-1} / 2\) to be normalizable.

Key concepts

Quantum Mechanics and Wave Functions

Quantum mechanics delves into the behavior of particles at the smallest scales, where the classical concepts of physics no longer apply. Here, the state of a quantum system is described by a wave function—a mathematical entity that encapsulates all the information about a system's properties. An essential quality of a valid wave function is that it must be normalizable. This means that the total probability of finding a particle within all space must equal one.

To ensure this, the wave function's square modulus, when integrated over all space, should converge to one. If the integral diverges, the function cannot be normalized, and thus it doesn't represent a physically possible state. The behavior of wave functions at infinity directly impacts their normalizability, necessitating a fall off faster than certain rates to maintain physical acceptability.

Power Rule for Integration

A fundamental tool in calculus, the power rule for integration, is used to find the antiderivative of polynomial functions. It states that the integral of a power function, specifically in the form of \(x^n\), where \(n\) is a real number, is given by \(\frac{x^{n+1}}{n+1}\), provided that \(n\) differs from -1.

Applying this rule allows us to solve the integral of \(x^b\) by transforming it into \(\frac{x^{b+1}}{b+1}\), as long as \(b+1\) is not zero. The simplicity of this technique makes it an invaluable part of understanding the behavior of functions, particularly when evaluating whether certain integrals will converge or diverge.

Integral Divergence

The concept of integral divergence is instrumental in determining whether an integral results in a finite value or not. An integral \(\int_a^\infty f(x) dx\) is said to diverge if the limit as \(x\) approaches infinity does not yield a finite result.

For the given function \(x^b\), if \(b+1 \leq 0\), the values of the function do not decrease quickly enough as \(x\) approaches infinity to result in a finite area under the curve, thus the integral diverges. Recognizing conditions where divergence occurs is essential to fields such as quantum mechanics, where such divergences imply non-physical behaviors.

Normalizable Wave Function

A normalizable wave function is a keystone in quantum mechanics, ensuring that probability interpretations make sense. In the context of our initial problem, a wave function is considered normalizable if the integral of its square modulus over all space converges to one.

Using the condition that emerged in the previous analysis, a wave function must decrease faster than \(|x|^{-1/2}\) to be normalizable. This ensures that when squared, the function falls off faster than \(\frac{1}{|x|}\), allowing the integral over all space to be a finite, well-defined value and maintaining the physicality of the quantum state it represents.