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# Lecture 22: The WKB Approximation The WKB approximation is a method for finding approximate solutions to linear differential equations of the form: $\frac{d^2\psi}{dx^2} + Q(x)\psi(x) = 0$, where $Q(x)$ is a slowly varying function. This method is particularly useful in quantum mechanics for so...

# Lecture 22: The WKB Approximation The WKB approximation is a method for finding approximate solutions to linear differential equations of the form: $\frac{d^2\psi}{dx^2} + Q(x)\psi(x) = 0$, where $Q(x)$ is a slowly varying function. This method is particularly useful in quantum mechanics for solving the time-independent Schrödinger equation in situations where the potential energy varies slowly compared to the de Broglie wavelength of the particle. ## 1. The Basic Idea The WKB approximation assumes that the solution to the differential equation can be written in the form: $\psi(x) = e^{\frac{i}{\hbar}S(x)}$, where S(x) is a complex function that can be expanded as a power series in $\hbar$: $S(x) = S_0(x) + \frac{\hbar}{i}S_1(x) + (\frac{\hbar}{i})^2S_2(x) +...$ The idea is that when $\hbar$ is small, the higher-order terms in the series become negligible, and the approximation becomes more accurate. ## 2. Derivation of the WKB Approximation To derive the WKB approximation, we substitute the assumed solution into the differential equation and solve for the terms in the power series for S(x). Substituting $\psi(x) = e^{\frac{i}{\hbar}S(x)}$ into the differential equation, we get: $\frac{d^2}{dx^2}e^{\frac{i}{\hbar}S(x)} + Q(x)e^{\frac{i}{\hbar}S(x)} = 0$ Using the chain rule to evaluate the first term: $\frac{d}{dx}(\frac{i}{\hbar}\frac{dS}{dx}e^{\frac{i}{\hbar}S(x)}) + Q(x)e^{\frac{i}{\hbar}S(x)} = 0$ $\frac{i}{\hbar}\frac{d^2S}{dx^2}e^{\frac{i}{\hbar}S(x)} + (\frac{i}{\hbar})^2(\frac{dS}{dx})^2e^{\frac{i}{\hbar}S(x)} + Q(x)e^{\frac{i}{\hbar}S(x)} = 0$ Dividing by $e^{\frac{i}{\hbar}S(x)}$ and multiplying by $\hbar^2$, we obtain: $i\hbar\frac{d^2S}{dx^2} - (\frac{dS}{dx})^2 + \hbar^2Q(x) = 0$ Now, substituting the power series expansion for S(x) into this equation: $i\hbar\frac{d^2}{dx^2}(S_0(x) + \frac{\hbar}{i}S_1(x) +...) - (\frac{d}{dx}(S_0(x) + \frac{\hbar}{i}S_1(x) +...))^2 + \hbar^2Q(x) =0$ Expanding and collecting terms with the same powers of $\hbar$, we get: **Zeroth Order Term ($\hbar^0$)**: $-(\frac{dS_0}{dx})^2 + \hbar^2Q(x) = 0 $ $\frac{dS_0}{dx} = \pm \hbar\sqrt{Q(x)}$ $S_0(x) = \pm \int{\hbar\sqrt{Q(x)}dx}$ **First Order Term ($\hbar^1$)**: $i\frac{d^2S_0}{dx^2} - 2\frac{dS_0}{dx}\frac{dS_1}{dx} = 0$ $i\frac{d^2S_0}{dx^2} = 2\frac{dS_0}{dx}\frac{dS_1}{dx}$ $\frac{dS_1}{dx} = \frac{i}{2}\frac{\frac{d^2S_0}{dx^2}}{\frac{dS_0}{dx}}$ Integrating both sides with respect to x: $S_1(x) = \frac{i}{2}ln(\frac{dS_0}{dx}) + C$ where C is a constant of integration. ## 3. The WKB Solutions Using the expressions for $S_0(x)$ and $S_1(x)$, we can write the approximate solutions to the differential equation: $\psi(x) = e^{\frac{i}{\hbar}S(x)} = e^{\frac{i}{\hbar}(S_0(x) + \frac{\hbar}{i}S_1(x) +...)}$ Keeping only the first two terms in the expansion. $\psi(x) \approx e^{\frac{i}{\hbar}S_0(x)}e^{S_1(x)}$ Substituting the expressions for $S_0(x)$ and $S_1(x)$: $\psi(x) \approx e^{\frac{i}{\hbar}(\pm \int{\hbar\sqrt{Q(x)}dx)}}e^{\frac{i}{2}ln(\frac{dS_0}{dx}) + C}$ $\psi(x) \approx e^{\pm i\int{\sqrt{Q(x)}dx}}e^{ln(\sqrt{\frac{dS_0}{dx}})}e^C$ $\psi(x) \approx e^{\pm i\int{\sqrt{Q(x)}dx}}\sqrt{\frac{dS_0}{dx}}e^C$ $\psi(x) \approx e^{\pm i\int{\sqrt{Q(x)}dx}}\sqrt{\sqrt{Q(x)}}e^C$ $\psi(x) \approx \frac{1}{\sqrt{Q(x)}}(C_1e^{i\int{\sqrt{Q(x)}dx}} + C_2e^{-i\int{\sqrt{Q(x)}dx}})$ where $C_1$ and $C_2$ are arbitrary constants. ## 4. Conditions for the Validity of the The WKB Approximation The WKB approximation is valid when the function $Q(x)$ varies slowly compared to the de Broglie wavelength of the particle. This condition can be expressed mathematically as: $|\frac{dQ}{dx}|