Editing Quantum tunnelling (section)

=== WKB approximation ===
{{Main|WKB approximation}}

The wave function is expressed as the exponential of a function:
<math display="block">\Psi(x) = e^{\Phi(x)},</math> where <math display="block">\Phi''(x) + \Phi'(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).</math>
<math>\Phi'(x)</math> is then separated into real and imaginary parts:
<math display="block">\Phi'(x) = A(x) + i B(x),</math> where ''A''(''x'') and ''B''(''x'') are real-valued functions.

Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
<math display="block">A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).</math>[[File:Wigner function for tunnelling.ogv|upright=1.2|thumb|Quantum tunneling in the [[phase space formulation]] of quantum mechanics. [[Wigner quasiprobability distribution|Wigner function]] for tunneling through the potential barrier <math>U(x) = 8e^{-0.25 x^2}</math> in atomic units (a.u.). The solid lines represent the [[level set]] of the [[Hamiltonian mechanics|Hamiltonian]] <math>H(x,p) = p^2 / 2 + U(x) </math>.]]

To solve this equation using the semiclassical approximation, each function must be expanded as a [[power series]] in <math>\hbar</math>. From the equations, the power series must start with at least an order of <math>\hbar^{-1}</math> to satisfy the real part of the equation; for a good classical limit starting with the highest power of the [[Planck constant]] possible is preferable, which leads to
<math display="block">A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x)</math>
and
<math display="block">B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x),</math>
with the following constraints on the lowest order terms,
<math display="block">A_0(x)^2 - B_0(x)^2 = 2m \left( V(x) - E \right)</math>
and
<math display="block">A_0(x) B_0(x) = 0.</math>

At this point two extreme cases can be considered.

'''Case 1'''

If the amplitude varies slowly as compared to the phase <math>A_0(x) = 0</math> and
<math display="block">B_0(x) = \pm \sqrt{ 2m \left( E - V(x) \right) }</math>
which corresponds to classical motion. Resolving the next order of expansion yields
<math display="block">\Psi(x) \approx C \frac{ e^{i \int dx \sqrt{\frac{2m}{\hbar^2} \left( E - V(x) \right)} + \theta} }{\sqrt[4]{\frac{2m}{\hbar^2} \left( E - V(x) \right)}}</math>

'''Case 2'''

If the phase varies slowly as compared to the amplitude, <math>B_0(x) = 0</math> and
<math display="block">A_0(x) = \pm \sqrt{ 2m \left( V(x) - E \right) }</math>
which corresponds to tunneling. Resolving the next order of the expansion yields
<math display="block">\Psi(x) \approx \frac{ C_{+} e^{+\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} + C_{-} e^{-\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{\sqrt[4]{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}</math>

In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points <math>E = V(x)</math>. Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.

To start, a classical turning point, <math>x_1</math> is chosen and <math>\frac{2m}{\hbar^2}\left(V(x)-E\right)</math> is expanded in a power series about <math>x_1</math>:
<math display="block">\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \cdots</math>

Keeping only the first order term ensures linearity:
<math display="block">\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1).</math>

Using this approximation, the equation near <math>x_1</math> becomes a [[differential equation]]:
<math display="block">\frac{d^2}{dx^2} \Psi(x) = v_1 (x - x_1) \Psi(x).</math>

This can be solved using [[Airy function]]s as solutions.
<math display="block">\Psi(x) = C_A Ai\left( \sqrt[3]{v_1} (x - x_1) \right) + C_B Bi\left( \sqrt[3]{v_1} (x - x_1) \right)</math>

Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.

Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between <math>C,\theta</math> and <math>C_{+},C_{-}</math> are
<math display="block">C_{+} = \frac{1}{2} C \cos{\left(\theta - \frac{\pi}{4}\right)}</math>
and
: [[File:Quantum Tunnelling animation.gif|upright=1.3|thumb|Quantum tunnelling through a barrier. At the origin (''x'' = 0), there is a very high, but narrow potential barrier. A significant tunnelling effect can be seen.|alt=]]<math>C_{-} = - C \sin{\left(\theta - \frac{\pi}{4}\right)}</math>

With the coefficients found, the global solution can be found. Therefore, the [[transmission coefficient (physics)|transmission coefficient]] for a particle tunneling through a single potential barrier is
<math display="block">T(E) = e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left[ V(x) - E \right]}},</math>
where <math>x_1,x_2</math> are the two classical turning points for the potential barrier.

For a rectangular barrier, this expression simplifies to:
<math display="block">T(E) = e^{-2\sqrt{\frac{2m}{\hbar^2}(V_0-E)}(x_2-x_1)}.</math>