1. **Problem Statement:** We need to determine which of the three proposed wavefunctions is physically acceptable for a one-dimensional quantum system defined on the interval $x \geq a$. 2. **Key Criteria for Acceptable Wavefunctions:** - The wavefunction $\Psi(x)$ must be normalizable, meaning $\int_a^{\infty} |\Psi(x)|^2 dx$ must be finite. - The wavefunction must be finite and well-behaved (no singularities) in the domain. - The wavefunction should satisfy boundary conditions relevant to the physical system. 3. **Analyze Aldrin's wavefunction:** $$\Psi(x) = N \tan(ax), \quad x \geq a$$ - The tangent function has singularities at $x = \frac{\pi}{2a}, \frac{3\pi}{2a}, \ldots$ which lie in $[a, \infty)$ for some $a$. - This means $\Psi(x)$ is not finite everywhere in the domain. - Also, $\tan(ax)$ does not decay at infinity, so the integral of $|\Psi(x)|^2$ diverges. - **Conclusion:** Aldrin's wavefunction is not acceptable. 4. **Analyze Kim's wavefunction:** $$\Psi(x) = N x^{1/2} e^{-ax}, \quad x \geq a$$ - The exponential decay $e^{-ax}$ ensures the wavefunction goes to zero as $x \to \infty$. - The factor $x^{1/2}$ grows slowly but is dominated by the exponential decay. - The integral $\int_a^{\infty} x e^{-2ax} dx$ converges, so the wavefunction is normalizable. - The wavefunction is finite and well-behaved for $x \geq a$. - **Conclusion:** Kim's wavefunction is acceptable. 5. **Analyze Xyrose's wavefunction:** $$\Psi(x) = N \sin(ax), \quad x \geq a$$ - The sine function oscillates indefinitely and does not decay as $x \to \infty$. - The integral $\int_a^{\infty} \sin^2(ax) dx$ diverges. - The wavefunction is not normalizable. - **Conclusion:** Xyrose's wavefunction is not acceptable. 6. **Final Conclusion:** Only Kim's wavefunction satisfies the physical requirements of being normalizable and finite over the domain $x \geq a$. **Answer:** Kim is most likely correct.