In quantum mechanics, the **Wentzel-Kramers-Brillouin** or simply the **WKB approximation** is a method 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.

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