Question

P14.9 Which of the following functions are acceptable wave functions over the indicated interval? a. \(e^{-x^{2} / 2}-\infty < x < \infty\) \(\begin{array}{ll}\text { b. } e^{-i x} & 0 < x < 2 \pi\end{array}\) c. \(x^{2} e^{-2 \pi i x} \quad 0 < x < \infty\) d. \(x e^{-x} \quad 0 < x < \infty\) Explain your answers.

Short answer

All four functions a, b, c, and d are acceptable wave functions since they are continuous, have continuous first derivatives, and are normalizable over their respective intervals.

Step-by-step solution

Check continuity and first derivative

: For each function, we will check if it is continuous and if its first derivative is continuous over the given interval. a. \(e^{-x^{2} / 2}\) is continuous over the interval \(-\infty < x < \infty\) since it is an exponential function. The first derivative is also continuous. b. \(e^{-ix}\) is also continuous over the interval \(0 < x < 2\pi\) as it is an exponential function as well. Similarly, its first derivative is continuous. c. \(x^2 e^{-2\pi i x}\) is continuous over the interval \(0 < x < \infty\), as it is a product of a continuous polynomial function and continuous exponential function. Its first derivative is continuous too. d. \(x e^{-x}\) is continuous over the interval \(0 < x < \infty\), as it is a product of a continuous polynomial function and continuous exponential function. Its first derivative is continuous as well. All given functions satisfy the first condition.

Check normalizability

: For each function, we will check if it is normalizable by integrating the square of the wave function over the given interval. a. For function \(e^{-x^{2} / 2}\), calculate the integral over \(-\infty < x < \infty\): \[\int_{-\infty}^{\infty} e^{-x^2} dx\] This integral converges, so the function is normalizable. b. For function \(e^{-ix}\), calculate the integral over \(0 < x < 2\pi\): \[\int_{0}^{2\pi} e^{-2ix} dx\] This integral converges, so the function is normalizable. c. For function \(x^2 e^{-2\pi i x}\), calculate the integral over \(0 < x < \infty\): \[\int_{0}^{\infty} x^4 e^{-4\pi i x} dx\] This integral converges, so the function is normalizable. d. For function \(x e^{-x}\), calculate the integral over \(0 < x < \infty\): \[\int_{0}^{\infty} x^2 e^{-2x} dx\] This integral converges, so the function is normalizable. All given functions satisfy the second condition.

Conclusion

: Since all four functions a, b, c, and d are continuous and have continuous first derivatives, and are normalizable over their respective intervals, all of these functions are acceptable wave functions.

Key concepts

Continuous Functions as a Prerequisite for Wave Functions

Wave functions have an important role in quantum mechanics, and one primary requirement is their continuity over a given interval. A continuous function does not have any breaks, jumps, or sharp corners. In simpler terms, if you can draw the whole function without lifting your pencil from the paper, it is continuous.
For the functions given in the exercise, continuity is critical. Each function in the exercise is either an exponential or a product involving exponential and polynomial components. These types of functions are known for being inherently continuous.

• Exponential functions, like \(e^{-x^2/2}\) and \(e^{-ix}\), smoothly decrease or oscillate with no interruptions.
• Polynomial functions, such as \(x^2\) in \(x^2 e^{-2\pi i x}\) and \(x\) in \(x e^{-x}\), are continuous as well.
• When combined, as in \(x^2 e^{-2\pi i x}\) or \(x e^{-x}\), they remain continuous.

Each of these functions is continuous over its specified interval, making them eligible as wave functions based on this criterion.

Normalizability of Wave Functions

When we discuss wave functions, another important term surfaces: normalizability. This property ensures that the total probability of finding a particle described by the wave function is equal to one.
Mathematically, for a wave function \(\psi(x)\), normalizability requires that the integral of its square over the entire space is finite:\[\int_{a}^{b} |\psi(x)|^2 \, dx < \infty\]
For the wave functions considered:

• For \(e^{-x^2/2}\), \(e^{-ix}\), and their respective integrals, the outcomes are finite and hence they are normalizable across their domain.
• The function \(x^2 e^{-2\pi i x}\) and \(x e^{-x}\) also yield finite values when squared and integrated, confirming their normalizability.

Each integral converges, ensuring that the particle's existence is bound within the given interval and the function is deemed a good candidate for describing quantum phenomena.

First Derivative Continuity of Wave Functions

The continuity of the first derivative of a wave function is also a crucial factor in quantum mechanics. This characteristic ensures a smooth transition and no abrupt changes in the slope of the wave function. Essentially, if a function’s "rate of change" is smooth, it leads to predictable physical behaviors.
In this exercise, verifying the first derivative's continuity involves examining the components of each wave function:

• Exponential functions like \(e^{-x^2/2}\) and \(e^{-ix}\), including their derivatives, inherently possess continuous derivatives.
• In polynomial-exponential products like \(x^2 e^{-2\pi i x}\) and \(x e^{-x}\), each element individually and consequently their derivatives are continuous.

Continuity of the first derivative further fortifies the suitability of these functions as wave functions. They not only ensure smoothness in formulation but also reflect physical realizability adhering to quantum mechanical principles.