WKB Approximation
Also called the Semiclassical Approximation for Quantum Mechanics.
\(S(\vec{x},t)=\pm\int^x dx' \sqrt{2m(E-V(x'))}-Et = W(x)-Et\).
Continuity of One-Dimension
\(\frac{\partial}{\partial t}\mathcal{P} + \frac{\partial}{\partial x}j = 0\).
\(j=\frac{\mathcal{P}}{m}\frac{\partial S}{\partial x}\).
Stationary Case
\(\frac{\partial \mathcal{P}}{\partial t} = 0\).
So, \(\frac{\partial j}{\partial x} = 0\). Hence, \(\frac{\partial}{\partial x}\left[\mathcal{P}\frac{\partial S}{\partial x}\right] = 0\). So, \(\rho\frac{d W}{dx}\) is a constant, \(\mathcal{C}\). Then, \(\sqrt{\mathcal{P}} = \frac{\mathcal{C}}{\sqrt[4]{E-V(x)}}\), so \(\mathcal{P}=\frac{\mathcal{C}^2}{\sqrt{E-V(x)}}\).
Thus, we are able to get a wavefunction without any solving: \(\Psi = \frac{\mathcal{C}}{\sqrt[4]{E-V(x)}}\exp(\pm\frac{i}{\hbar}\int^xdx' \sqrt{2m(E-V(x'))}-\frac{i}{\hbar}Et)\). Wentzel-Kramers-Brillouin solution. Thus, \(\mathcal{P}\sim\frac{1}{v_{classical}}\).
WKB Criteria
\(\hbar\left|\frac{d^2}{dx^2}W\right|\ll\left|\frac{dW}{dx}\right|^2\). So, \(\hbar\frac{2m\left|\frac{dV}{dx}\right|}{2\sqrt{2m(E-V(x))}}\ll|2m(E-V(x))|\). Hence, \(\frac{\hbar}{\sqrt{2m(E-V(x))}}=\frac{1}{k}=\frac{\lambda_B}{2\pi}\ll \frac{2(E-V(x))}{\left|\frac{dV}{dx}\right|}\), where \(k\) is the wavenumber, related to the momentum by \(\hbar\). This gives us a deBrolie wavelength to compare.
So, WKB is valid in the low wavelength cases.
Problem Areas
Turning points pose problem areas for WKB, but they can be solved as linear wavefunction pieces to splice the region solutions together.
Small Example
Finite Well but with rounded edges, energy such that it is a bound state.
For regions I, III: \(\Psi=\frac{\mathcal{C}}{\sqrt[4]{2m(V(x)-E)}}\exp(\pm \frac{1}{\hbar}\int^x \sqrt{2m(V(x')-E)})dx'\)
For regions II: \(\Psi=\frac{\mathcal{C}}{\sqrt[4]{2m(E-V(x))}}\exp(\pm \frac{i}{\hbar}\int^x \sqrt{2m(E-V(x'))})dx'\)
For turning point solutions to splice together the regions: \(\Psi = Ax+B\) \(-\frac{\hbar^2}{2m}\Psi''=(E-V(x))\Psi(x) \Rightarrow\) Linear solutions.
Alternatively, \(-\frac{\hbar^2}{2m}\Psi''+\frac{dV}{dx}|_{x_{1,2}}(x-x_{1,2})\Psi=0\). Gives a Bessel equation - Airy function.
Energies
Rigid \(\Rightarrow\) \(V\to\infty\).
\(\int_{x_1}^{x_2}dx\sqrt{2m(E_n-V(x))} = \pi\hbar g_n\).
Case 1 - No Rigid Walls
\(g_n=n+1/2\)
Case 2 - 1 Rigid Wall
\(g_n=n+3/4\)
Case 3 - 2 Rigid Walls
\(g_n=n+1\)
Notes
If \(n\) is large, each \(g_n\) is approximately equal.
Also, as \(n\) increases, we get closer to classical typically.
Examples
Energies of QMHO
\(E=\frac{m\omega^2 x_{1}^2}{2}\). \(\int_{-x_1}^{x_1}dx\sqrt{2m(\frac{m\omega^2 x_1^2}{2}-\frac{m\omega^2x^2}{2})}=\frac{\pi E}{\omega}=\pi\hbar(n+1/2)\). \(x_1=\pm\sqrt{\frac{2E}{m\omega^2}}\). WKB: \(E=\hbar\omega(n+1/2)\)
Bouncing Neutrons
Hard floor. So, the potential is infinite in the negative region and a linear potential at \(x\gt 0\).
Tunneling
Use WKB to find the wavefunction for a more complicated potential barrier, use typical solutions (if possible) for outside propogators.
\(T\approx \exp(-2\gamma), \gamma=\frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx\)