20250402_185412.heic

# Quantum Mechanics ## 2.1 The Schrödinger Equation ### Time-Dependent Schrödinger Equation $-\frac{\hbar^2}{2m} \nabla^2 \Psi(\mathbf{r},t) + V(\mathbf{r},t)\Psi(\mathbf{r},t) = i\hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t}$ ### Time-Independent Schrödinger Equation $-\frac{\hbar^2}{2m...

# Quantum Mechanics ## 2.1 The Schrödinger Equation ### Time-Dependent Schrödinger Equation $-\frac{\hbar^2}{2m} \nabla^2 \Psi(\mathbf{r},t) + V(\mathbf{r},t)\Psi(\mathbf{r},t) = i\hbar \frac{\partial \Psi(\mathbf{r},t)}{\partial t}$ ### Time-Independent Schrödinger Equation $-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r})$ **Where:** * $\hbar$ is the reduced Planck constant ($\hbar = \frac{h}{2\pi}$) * $m$ is the mass of the particle * $\nabla^2$ is the Laplacian operator * $V(\mathbf{r},t)$ is the time-dependent potential energy * $V(\mathbf{r})$ is the time-independent potential energy * $\Psi(\mathbf{r},t)$ is the time-dependent wave function * $\psi(\mathbf{r})$ is the time-independent wave function * $i$ is the imaginary unit ($i^2 = -1$) * $E$ is the energy of the particle ## 2.2 Interpretation of the Wave Function * Born Interpretation: $|\Psi(\mathbf{r},t)|^2$ is the probability density of finding the particle at position $\mathbf{r}$ at time $t$. * $\int |\Psi(\mathbf{r},t)|^2 d^3r = 1$ (Normalization condition) ## 2.3 Operators and Observables | Observable | Operator | | :---------------- | :---------------------------------------------------------------------------- | | Position $\mathbf{r}$ | $\hat{\mathbf{r}} = \mathbf{r}$ | | Momentum $\mathbf{p}$ | $\hat{\mathbf{p}} = -i\hbar \nabla$ | | Kinetic Energy $T$ | $\hat{T} = -\frac{\hbar^2}{2m} \nabla^2$ | | Potential Energy $V$| $\hat{V} = V(\mathbf{r})$ | | Total Energy $E$ | $\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r})$ (Hamiltonian) | ## 2.4 Expectation Values * The expectation value of an observable $A$ is given by: $\langle A \rangle = \int \Psi^* (\mathbf{r},t) \hat{A} \Psi(\mathbf{r},t) d^3r$ ## 2.5 Uncertainty Principle * Heisenberg Uncertainty Principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$ * Where $\Delta x$ and $\Delta p$ are the standard deviations of position and momentum, respectively. ## 2.6 Example: Particle in a Box * Consider a particle confined to a one-dimensional box of length $L$ with potential $V(x) = 0$ inside the box ($0 \le x \le L$) and $V(x) = \infty$ outside the box. * The solutions to the time-independent Schrödinger equation are: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$, where $n = 1, 2, 3,...$ * The corresponding energy levels are: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, where $n = 1, 2, 3,...$