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# Quantum Mechanics ## 2. The Schrodinger Equation ### Time-Dependent Schrodinger Equation $$ i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x,t) \right] \Psi(x,t) $$ where * $\Psi(x,t)$ is the wave function * $V(x,t)$ is the pote...

# Quantum Mechanics ## 2. The Schrodinger Equation ### Time-Dependent Schrodinger Equation $$ i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \left[ -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x,t) \right] \Psi(x,t) $$ where * $\Psi(x,t)$ is the wave function * $V(x,t)$ is the potential energy function ### Time-Independent Schrodinger Equation $$ E\psi(x) = \left[ -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right] \psi(x) $$ where * $E$ is the energy * $\psi(x)$ is the time-independent wave function ## 3. Solving the Time-Independent Schrodinger Equation ### Example: Particle in a Box $$ V(x) = \begin{cases} 0, & 0 < x < a \\ \infty, & \text{otherwise} \end{cases} $$ The general solution is $$ \psi(x) = A\sin(kx) + B\cos(kx) $$ where $k = \sqrt{2mE/\hbar^2}$. Boundary conditions: $\psi(0) = \psi(a) = 0$. This implies $B = 0$ and $ka = n\pi$, where $n$ is an integer. Thus, the allowed energies are $$ E_n = \frac{n^2\pi^2\hbar^2}{2ma^2} $$ and the normalized wave functions are $$ \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left( \frac{n\pi x}{a} \right) $$ ## 4. Expectation Values The expectation value of an operator $\hat{Q}$ is $$ \langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \hat{Q} \Psi(x,t) dx $$ For example, the expectation value of position is $$ \langle x \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) x \Psi(x,t) dx $$ and the expectation value of momentum is $$ \langle p \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \left( -i\hbar \frac{\partial}{\partial x} \right) \Psi(x,t) dx $$ ## 5. Uncertainty Principle $$ \sigma_x \sigma_p \geq \frac{\hbar}{2} $$ where * $\sigma_x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ is the standard deviation of position * $\sigma_p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$ is the standard deviation of momentum ## 6. Potential Step ### Potential $$ V(x) = \begin{cases} 0, & x < 0 \\ V_0, & x \geq 0 \end{cases} $$ ### Wave Functions For $E > V_0$: $$ \psi(x) = \begin{cases} A e^{ik_1 x} + B e^{-ik_1 x}, & x < 0 \\ C e^{ik_2 x} + D e^{-ik_2 x}, & x \geq 0 \end{cases} $$ where $k_1 = \sqrt{2mE/\hbar^2}$ and $k_2 = \sqrt{2m(E-V_0)/\hbar^2}$. ### Boundary Conditions At $x = 0$: 1. $\psi(x)$ is continuous: $A + B = C + D$ 2. $\psi'(x)$ is continuous: $ik_1(A - B) = ik_2(C - D)$ Diagram of a potential step with an incoming wave from the left, a reflected wave back to the left, and a transmitted wave to the right. The potential energy is 0 for $x < 0$ and $V_0$ for $x \ge 0$. The x-axis represents position and the y-axis represents potential energy.