To determine if the given functions $\psi(x)$ are suitable wave functions, we need to check if they satisfy certain criteria, which include:

1. Normalization: A wave function $\psi(x)$ must be normalizable, meaning the integral of the square of the absolute value of the function over the domain must converge to a finite value: $\int |\psi(x)|^2 dx < \infty$ This ensures the probability interpretation holds.

2. Continuity: The wave function should be continuous and finite over the domain of interest.

Let's analyze each case.

(a) $\psi(x) = A e^{-\alpha x^2}, \alpha > 0$

• Domain: $-\infty < x < \infty$.
• Behavior: As $x \to \pm \infty$, $\psi(x) \to 0$ exponentially.
• Normalization: Since the function is Gaussian, it is normalizable over $-\infty < x < \infty$.
• Conclusion: This is a suitable wave function.

(b) $\psi(x) = A e^{-\alpha x}, \alpha > 0$

• Domain: $0 < x < \infty$.
• Behavior: As $x \to \infty$, $\psi(x) \to 0$ exponentially. However, as $x \to 0$, the function diverges.
• Normalization: This function diverges at $x = 0$, so it cannot be normalized over $0 < x < \infty$.
• Conclusion: This is not a suitable wave function.

(c) $\psi(x) = \frac{A}{x}$

• Domain: $0 < x < \infty$.
• Behavior: As $x \to \infty$, $\psi(x) \to 0$, but as $x \to 0$, $\psi(x) \to \infty$.
• Normalization: This function diverges at $x = 0$, and hence cannot be normalized.
• Conclusion: This is not a suitable wave function.

(d) $\psi(x) = \frac{A}{\sqrt{|x|}}$

• Domain: $-\infty < x < \infty$.
• Behavior: As $x \to \infty$, $\psi(x) \to 0$, but as $x \to 0$, $\psi(x)$ diverges.
• Normalization: This function diverges at $x = 0$, so it cannot be normalized over the whole real line.
• Conclusion: This is not a suitable wave function.

(e) $\psi(x) = \frac{1}{x^{1/4}} e^{-\sqrt{x}}$

• Domain: $0 < x < \infty$.
• Behavior: As $x \to \infty$, $\psi(x) \to 0$ exponentially. As $x \to 0$, $\psi(x)$ diverges due to the $x^{-1/4}$ term.
• Normalization: Since it diverges as $x \to 0$, it cannot be normalized over the domain $0 < x < \infty$.
• Conclusion: This is not a suitable wave function.

Summary:

• (a) is a suitable wave function.
• (b), (c), (d), and (e) are not suitable wave functions.

Would you like further details or have any questions about these results?

Here are five related questions to deepen understanding:

1. What is the significance of normalization in quantum mechanics?
2. How does the behavior of a wave function at infinity affect its normalizability?
3. Why is continuity an important criterion for wave functions?
4. Can a wave function with singularities at certain points be used in quantum mechanics?
5. What are the physical interpretations of different types of wave function behavior at the boundaries?

Tip: When analyzing wave functions, always check their behavior at both the limits and any points where they could potentially diverge (e.g., near zero).