1. **Problem Statement:** Determine if each given function is normalized over its specified domain and volume element. If not normalized, find the normalization constant $N$ such that $$\int |N \psi|^2 dV = 1.$$ 2. **Normalization Rule:** For a function $\psi$, normalization means $$\int |\psi|^2 dV = 1,$$ where $dV$ is the volume element of the space. If not normalized, find $N$ such that $$N^2 \int |\psi|^2 dV = 1 \Rightarrow N = \frac{1}{\sqrt{\int |\psi|^2 dV}}.$$ --- ### (a) $\psi(z) = \sin\left(\frac{2\pi z}{b}\right)$, $0 \le z \le b$, $dV = dz$ 3. Compute $$I = \int_0^b \sin^2\left(\frac{2\pi z}{b}\right) dz.$$ Use identity $$\sin^2 x = \frac{1 - \cos 2x}{2}.$$ 4. Substitute and integrate: $$I = \int_0^b \frac{1 - \cos\left(\frac{4\pi z}{b}\right)}{2} dz = \frac{1}{2} \int_0^b 1 dz - \frac{1}{2} \int_0^b \cos\left(\frac{4\pi z}{b}\right) dz.$$ 5. Evaluate integrals: $$\frac{1}{2} [z]_0^b - \frac{1}{2} \left[ \frac{b}{4\pi} \sin\left(\frac{4\pi z}{b}\right) \right]_0^b = \frac{b}{2} - 0 = \frac{b}{2}.$$ 6. Since $$I = \frac{b}{2} \neq 1,$$ function is not normalized. Normalization constant: $$N = \frac{1}{\sqrt{I}} = \sqrt{\frac{2}{b}}.$$ --- ### (b) $\psi(z) = \sin\left(\frac{2\pi z}{b}\right) + 2 \sin\left(\frac{3\pi z}{b}\right)$, $0 \le z \le b$ 7. Compute $$I = \int_0^b \left| \sin\left(\frac{2\pi z}{b}\right) + 2 \sin\left(\frac{3\pi z}{b}\right) \right|^2 dz.$$ 8. Expand square: $$I = \int_0^b \left[ \sin^2\left(\frac{2\pi z}{b}\right) + 4 \sin^2\left(\frac{3\pi z}{b}\right) + 4 \sin\left(\frac{2\pi z}{b}\right) \sin\left(\frac{3\pi z}{b}\right) \right] dz.$$ 9. Orthogonality of sine functions with different frequencies implies cross term integral is zero: $$\int_0^b \sin\left(\frac{2\pi z}{b}\right) \sin\left(\frac{3\pi z}{b}\right) dz = 0.$$ 10. Use previous result for each sine squared term: $$\int_0^b \sin^2\left(\frac{2\pi z}{b}\right) dz = \frac{b}{2},$$ $$\int_0^b \sin^2\left(\frac{3\pi z}{b}\right) dz = \frac{b}{2}.$$ 11. Calculate total integral: $$I = \frac{b}{2} + 4 \times \frac{b}{2} = \frac{b}{2} + 2b = \frac{5b}{2}.$$ 12. Normalization constant: $$N = \frac{1}{\sqrt{I}} = \sqrt{\frac{2}{5b}}.$$ --- ### (c) $\psi(z) = e^{-a z^2}$, $a > 0$, $z \in (-\infty, +\infty)$ 13. Compute $$I = \int_{-\infty}^{\infty} e^{-2 a z^2} dz.$$ 14. Use Gaussian integral formula: $$\int_{-\infty}^{\infty} e^{-\alpha z^2} dz = \sqrt{\frac{\pi}{\alpha}},$$ for $\alpha > 0$. Here, $\alpha = 2a$. 15. So, $$I = \sqrt{\frac{\pi}{2a}}.$$ 16. Normalization constant: $$N = \frac{1}{\sqrt{I}} = \left( \frac{2a}{\pi} \right)^{1/4}.$$ --- ### (d) $\psi(\phi) = e^{i a \phi} + e^{-i a \phi}$, $a > 0$, $\phi \in [0, 2\pi]$ 17. Compute $$I = \int_0^{2\pi} |e^{i a \phi} + e^{-i a \phi}|^2 d\phi.$$ 18. Simplify inside integral: $$|e^{i a \phi} + e^{-i a \phi}|^2 = (e^{i a \phi} + e^{-i a \phi})(e^{-i a \phi} + e^{i a \phi}) = 2 + 2 \cos(2 a \phi).$$ 19. Integral becomes: $$I = \int_0^{2\pi} 2 + 2 \cos(2 a \phi) d\phi = 2 \int_0^{2\pi} d\phi + 2 \int_0^{2\pi} \cos(2 a \phi) d\phi.$$ 20. Evaluate integrals: $$2 \times 2\pi + 2 \times 0 = 4\pi,$$ since integral of cosine over full period is zero. 21. Normalization constant: $$N = \frac{1}{\sqrt{I}} = \frac{1}{2 \sqrt{\pi}}.$$ --- ### (e) $\psi(r, \theta, \phi) = \cos \theta e^{-a r}$, $a > 0$, 3D spherical coordinates with $$dV = r^2 dr \sin \theta d\theta d\phi,$$ $$r \in [0, \infty), \theta \in [0, \pi], \phi \in [0, 2\pi].$$ 22. Compute $$I = \int_0^{2\pi} d\phi \int_0^{\pi} \sin \theta d\theta \int_0^{\infty} r^2 dr |\cos \theta e^{-a r}|^2.$$ 23. Separate integrals: $$I = \left( \int_0^{2\pi} d\phi \right) \left( \int_0^{\pi} \cos^2 \theta \sin \theta d\theta \right) \left( \int_0^{\infty} r^2 e^{-2 a r} dr \right).$$ 24. Evaluate angular integral over $\phi$: $$\int_0^{2\pi} d\phi = 2\pi.$$ 25. Evaluate $\theta$ integral using substitution $u = \cos \theta$, $du = -\sin \theta d\theta$: $$\int_0^{\pi} \cos^2 \theta \sin \theta d\theta = \int_{u=1}^{-1} u^2 (-du) = \int_{-1}^1 u^2 du = \frac{2}{3}.$$ 26. Evaluate radial integral: $$\int_0^{\infty} r^2 e^{-2 a r} dr = \frac{2}{(2a)^3} = \frac{1}{4 a^3}$$ using Gamma function or integration by parts. 27. Combine results: $$I = 2\pi \times \frac{2}{3} \times \frac{1}{4 a^3} = \frac{\pi}{3 a^3}.$$ 28. Normalization constant: $$N = \frac{1}{\sqrt{I}} = \sqrt{\frac{3 a^3}{\pi}}.$$ --- **Final answers:** - (a) $N = \sqrt{\frac{2}{b}}$ - (b) $N = \sqrt{\frac{2}{5b}}$ - (c) $N = \left( \frac{2a}{\pi} \right)^{1/4}$ - (d) $N = \frac{1}{2 \sqrt{\pi}}$ - (e) $N = \sqrt{\frac{3 a^3}{\pi}}$