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----
-title: "Wentzel-Kramers-Brillouin approximation"
-firstLetter: "W"
-publishDate: 2021-02-22
-categories:
-- Quantum mechanics
-- Physics
-
-date: 2021-02-22T21:38:35+01:00
-draft: false
-markup: pandoc
----
-
-# Wentzel-Kramers-Brillouin approximation
-
-In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB
-approximation** is a technique to approximate the wave function $\psi(x)$ of
-the one-dimensional time-independent Schrödinger equation. It is an example
-of a **semiclassical approximation**, because it tries to find a
-balance between classical and quantum physics.
-
-In classical mechanics, a particle travelling in a potential $V(x)$
-along a path $x(t)$ has a total energy $E$ as follows, which we
-rearrange:
-
-$$\begin{aligned}
-    E = \frac{1}{2} m \dot{x}^2 + V(x)
-    \quad \implies \quad
-    m^2 (x')^2 = 2 m (E - V(x))
-\end{aligned}$$
-
-The left-hand side of the rearrangement is simply the momentum squared,
-so we define the magnitude of the momentum $p(x)$ accordingly:
-
-$$\begin{aligned}
-    p(x) = \sqrt{2 m (E - V(x))}
-\end{aligned}$$
-
-Note that this is under the assumption that $E > V$, which is always the
-case in classical mechanics, but not necessarily so in quantum
-mechanics, but we stick with it for now. We rewrite the Schrödinger
-equation:
-
-$$\begin{aligned}
-    0
-    = \dv[2]{\psi}{x} + \frac{2 m}{\hbar^2} (E - V) \psi
-    = \dv[2]{\psi}{x} + \frac{p^2}{\hbar^2} \psi
-\end{aligned}$$
-
-If $V(x)$ were constant, and by extension $p(x)$ too, then the solution
-is easy:
-
-$$\begin{aligned}
-    \psi(x)
-    = \psi(0) \exp(\pm i p x / \hbar)
-\end{aligned}$$
-
-This form is reminiscent of the generator of translations. In practice,
-$V(x)$ and $p(x)$ vary with $x$, but we can still salvage this solution
-by assuming that $V(x)$ varies slowly compared to the wavelength
-$\lambda(x) = 2 \pi / k(x)$, where $k(x) = p(x) / \hbar$ is the
-wavenumber. The solution then takes the following form:
-
-$$\begin{aligned}
-    \psi(x)
-    = \psi(0) \exp\!\Big(\!\pm\! \frac{i}{\hbar} \int_0^x \chi(\xi) \dd{\xi} \Big)
-\end{aligned}$$
-
-$\chi(\xi)$ is an unknown function, which intuitively should be related
-to $p(x)$. The purpose of the integral is to accumulate the change of
-$\chi$ from the initial point $0$ to the current position $x$.
-Let us write this as an indefinite integral for convenience:
-
-$$\begin{aligned}
-    \psi(x)
-    = \psi(0) \exp\!\bigg( \!\pm\! \frac{i}{\hbar} \Big( \int \chi(x) \dd{x} - C \Big) \bigg)
-\end{aligned}$$
-
-Where $C = \int \chi(x) \dd{x} |_{x = 0}$ is the initial point of the definite integral.
-For simplicity, we absorb the constant $C$ into $\psi(0)$.
-We can now clearly see that:
-
-$$\begin{aligned}
-    \psi'(x) = \pm \frac{i}{\hbar} \chi(x) \psi(x)
-    \quad \implies \quad
-    \chi(x) = \pm \frac{\hbar}{i} \frac{\psi'(x)}{\psi(x)}
-\end{aligned}$$
-
-Next, we insert this ansatz for $\psi(x)$ into the Schrödinger equation
-to get:
-
-$$\begin{aligned}
-    0
-    &= \pm \frac{i}{\hbar} \dv{(\chi \psi)}{x} + \frac{p^2}{\hbar^2} \psi
-    = \pm \frac{i}{\hbar} \chi' \psi \pm \frac{i}{\hbar} \chi \psi' + \frac{p^2}{\hbar^2} \psi
-    = \pm \frac{i}{\hbar} \chi' \psi - \frac{1}{\hbar^2} \chi^2 \psi + \frac{p^2}{\hbar^2} \psi
-\end{aligned}$$
-
-Dividing out $\psi$ and rearranging gives us the following, which is
-still exact:
-
-$$\begin{aligned}
-    \pm \frac{\hbar}{i} \chi'
-    = p^2 - \chi^2
-\end{aligned}$$
-
-Next, we expand this as a power series of $\hbar$. This is why it is
-called *semiclassical*: so far we have been using full quantum mechanics,
-but now we are treating $\hbar$ as a parameter which controls the
-strength of quantum effects:
-
-$$\begin{aligned}
-    \chi(x) = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + ...
-\end{aligned}$$
-
-The heart of the WKB approximation is its assumption that quantum effects are
-sufficiently weak (i.e. $\hbar$ is small enough) that we only need to
-consider the first two terms, or, more specifically, that we only go up to
-$\hbar$, not $\hbar^2$ or higher. Inserting the first two terms of this
-expansion into the equation:
-
-$$\begin{aligned}
-    \pm \frac{\hbar}{i} \chi_0'
-    &= p^2 - \chi_0^2 - 2 \frac{\hbar}{i} \chi_0 \chi_1
-\end{aligned}$$
-
-Where we have discarded all terms containing $\hbar^2$. At order
-$\hbar^0$, we then get the expected classical result for $\chi_0(x)$:
-
-$$\begin{aligned}
-    0 = p^2 - \chi_0^2
-    \quad \implies \quad
-    \chi_0(x) = p(x)
-\end{aligned}$$
-
-While at order $\hbar$, we get the following quantum-mechanical
-correction:
-
-$$\begin{aligned}
-    \pm \frac{\hbar}{i} \chi_0'
-    = - 2 \frac{\hbar}{i} \chi_0 \chi_1
-    \quad \implies \quad
-    \chi_1(x) = \mp \frac{1}{2} \frac{\chi_0'(x)}{\chi_0(x)}
-\end{aligned}$$
-
-Therefore, our approximated wave function $\psi(x)$ currently looks like
-this:
-
-$$\begin{aligned}
-    \psi(x)
-    &\approx \psi(0) \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int \chi_0(x) \dd{x} \Big) \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big)
-\end{aligned}$$
-
-We can reduce the latter exponential using integration by substitution:
-
-$$\begin{aligned}
-    \exp\!\Big( \!\pm\! \int \chi_1(x) \dd{x} \Big)
-    &= \exp\!\Big( \!-\! \frac{1}{2} \int \frac{\chi_0'(x)}{\chi_0(x)} \dd{x} \Big)
-    = \exp\!\Big( \!-\! \frac{1}{2} \int \frac{1}{\chi_0}\:d\chi_0 \Big)
-    \\
-    &= \exp\!\Big( \!-\! \frac{1}{2} \ln\!\big(\chi_0(x)\big) \Big)
-    = \frac{1}{\sqrt{\chi_0(x)}}
-    = \frac{1}{\sqrt{p(x)}}
-\end{aligned}$$
-
-In the WKB approximation for $E > V$, the solution $\psi(x)$ is thus
-given by:
-
-$$\begin{aligned}
-    \boxed{
-        \psi(x) \approx \frac{A}{\sqrt{p(x)}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big)
-    }
-\end{aligned}$$
-
-What if $E < V$? In classical mechanics, this is not allowed; a ball
-cannot simply go through a potential bump without the necessary energy.
-However, in quantum mechanics, particles can **tunnel** through barriers.
-
-Conveniently, all we need to change for the WKB approximation is to let
-the momentum take imaginary values:
-
-$$\begin{aligned}
-    p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)}
-\end{aligned}$$
-
-And then take the absolute value in the appropriate place in front of
-$\psi(x)$:
-
-$$\begin{aligned}
-    \boxed{
-        \psi(x) \approx \frac{A}{\sqrt{|p(x)|}} \exp\!\Big( \!\pm\! \frac{i}{\hbar} \int p(x) \dd{x} \Big)
-    }
-\end{aligned}$$
-
-In the classical region ($E > V$), the wave function oscillates, and
-in the quantum-mechanical region ($E < V$) it is exponential. Note that for
-$E \approx V$ the approximation breaks down, due to the appearance of
-$p(x)$ in the denominator.
-
-
-## References
-1.  D.J. Griffiths, D.F. Schroeter,
-    *Introduction to quantum mechanics*, 3rd edition,
-    Cambridge.
-2.  R. Shankar,
-    *Principles of quantum mechanics*, 2nd edition,
-    Springer.