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 second-order PDE in two variables $x$ and $t$ can be written as $$A \frac{\partial^2 u}{\partial x^2} + 2B \frac{\partial^2 u}{\partial x \partial t} + C \frac{\partial^2 u}{\partial t^2} + \text{lower order terms} = 0$$ where $A$, $B$, and $C$ are coefficients. 3. **Classification rule:** - The PDE is **parabolic** if $B^2 - AC = 0$. - It is **elliptic** if $B^2 - AC < 0$. - It is **hyperbolic** if $B^2 - AC > 0$. 4. **Identify coefficients in Schrödinger equation:** Rewrite the equation as $$-\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2} + i \hbar \frac{\partial \phi}{\partial t} = 0$$ Note that there is no second derivative with respect to $t$, so $$A = -\frac{\hbar^2}{2m}, \quad B = 0, \quad C = 0$$ 5. **Calculate discriminant:** $$B^2 - AC = 0^2 - \left(-\frac{\hbar^2}{2m}\right)(0) = 0$$ 6. **Conclusion:** Since $B^2 - AC = 0$, the Schrödinger equation is a **parabolic** PDE. --- 7. **Problem Statement:** Solve the time-dependent Schrödinger equation $$-\frac{\hbar^2}{2m} \frac{\partial^2 \phi(x,t)}{\partial x^2} = -i \hbar \frac{\partial \phi(x,t)}{\partial t}$$ 8. **Rewrite equation:** $$i \hbar \frac{\partial \phi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2}$$ 9. **Method: Separation of variables** Assume solution of the form $$\phi(x,t) = X(x) T(t)$$ 10. **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). 11. **Time equation:** $$i \hbar \frac{dT}{dt} = E T \implies \frac{dT}{dt} = -\frac{iE}{\hbar} T$$ Solution: $$T(t) = T_0 e^{-i E t / \hbar}$$ 12. **Spatial equation:** $$-\frac{\hbar^2}{2m} \frac{d^2 X}{dx^2} = E X$$ Rearranged: $$\frac{d^2 X}{dx^2} + \frac{2mE}{\hbar^2} X = 0$$ 13. **General solution for $X(x)$:** Let $$k^2 = \frac{2mE}{\hbar^2}$$ Then $$X(x) = A e^{i k x} + B e^{-i k x}$$ 14. **Complete solution:** $$\phi(x,t) = \left(A e^{i k x} + B e^{-i k x}\right) e^{-i E t / \hbar}$$ 15. **Interpretation:** This represents a free particle wave function with energy $E$ and wave number $k$. **Final answer:** The Schrödinger equation is parabolic, and its general solution is $$\phi(x,t) = \left(A e^{i k x} + B e^{-i k x}\right) e^{-i E t / \hbar}$$ where $k = \sqrt{\frac{2mE}{\hbar^2}}$ and $A,B$ are constants determined by initial/boundary conditions.