1. **Problem Statement:** Show that the Schrödinger equation $$-\frac{\hbar^2}{2m} \frac{\partial^2 \phi(x,t)}{\partial x^2} = -i \hbar \frac{\partial \phi(x,t)}{\partial t}$$ is a parabolic partial differential equation. 2. **Recall the general form of second-order PDEs:** $$A \frac{\partial^2 u}{\partial x^2} + 2B \frac{\partial^2 u}{\partial x \partial t} + C \frac{\partial^2 u}{\partial t^2} + \text{lower order terms} = 0$$ The classification depends on the discriminant $D = B^2 - AC$: - Elliptic if $D < 0$ - Parabolic if $D = 0$ - Hyperbolic if $D > 0$ 3. **Rewrite Schrödinger equation:** $$-\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2} + i \hbar \frac{\partial \phi}{\partial t} = 0$$ 4. **Identify coefficients:** - $A = -\frac{\hbar^2}{2m}$ - $B = 0$ (no mixed derivative term) - $C = 0$ (no second derivative in time) 5. **Calculate discriminant:** $$D = B^2 - AC = 0^2 - \left(-\frac{\hbar^2}{2m}\right)(0) = 0$$ 6. **Conclusion:** Since $D=0$, the Schrödinger equation is a parabolic PDE. --- 7. **Problem Statement:** Solve the time-dependent Schrödinger equation for a free particle: $$-\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2} = -i \hbar \frac{\partial \phi}{\partial t}$$ 8. **Rewrite:** $$i \hbar \frac{\partial \phi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2}$$ 9. **Use separation of variables:** Assume $$\phi(x,t) = X(x) T(t)$$ Substitute into equation: $$i \hbar X(x) \frac{dT}{dt} = -\frac{\hbar^2}{2m} T(t) \frac{d^2 X}{dx^2}$$ Divide both sides by $X(x) T(t)$: $$i \hbar \frac{1}{T} \frac{dT}{dt} = -\frac{\hbar^2}{2m} \frac{1}{X} \frac{d^2 X}{dx^2} = E$$ where $E$ is a separation constant (energy eigenvalue). 10. **Solve spatial part:** $$\frac{d^2 X}{dx^2} + \frac{2mE}{\hbar^2} X = 0$$ General solution: $$X(x) = A e^{i k x} + B e^{-i k x}$$ where $$k = \frac{\sqrt{2mE}}{\hbar}$$ 11. **Solve temporal part:** $$i \hbar \frac{dT}{dt} = E T$$ $$\Rightarrow \frac{dT}{dt} = -\frac{i E}{\hbar} T$$ Solution: $$T(t) = C e^{-i E t / \hbar}$$ 12. **Combine solutions:** $$\phi(x,t) = \left(A e^{i k x} + B e^{-i k x}\right) e^{-i E t / \hbar}$$ 13. **Interpretation:** This represents a free particle wave function as a superposition of plane waves with energy $E$ and wave number $k$. **Final answer:** The Schrödinger equation is parabolic, and its general free particle solution is $$\phi(x,t) = \left(A e^{i k x} + B e^{-i k x}\right) e^{-i E t / \hbar}$$ where $k = \frac{\sqrt{2mE}}{\hbar}$.