WKB approximation

This method is named after physicists Gregor Wentzel, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926. In 1923, Earlier appearances of essentially equivalent methods are: Francesco Carlini in 1817, The important contribution of Jeffreys, Wentzel, Kramers, and Brillouin to the method was the inclusion of the treatment of turning points, connecting the evanescent and oscillatory solutions at either side of the turning point. For example, this may occur in the Schrödinger equation, due to a potential energy hill. ==Formulation==

The WKB method approximates the solution of a differential equation whose highest derivative is multiplied by a small parameter \varepsilon. For a differential equation \varepsilon \frac{d^ny}{dx^n} + a(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + k(x)\frac{dy}{dx} + m(x)y= 0,assume a solution of the form of an asymptotic series expansion y(x) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty} \delta^n S_n(x)\right]in the limit \delta \rightarrow 0. The asymptotic scaling of \delta in terms of \varepsilon will be determined by the equation. See the example below. Substituting the above ansatz into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms S_n (x)in the expansion. WKB theory is a special case of multiple scale analysis. ==An example==

This example comes from the text of Carl M. Bender and Steven Orszag.n_\max \approx \frac{2}{\varepsilon} \left| \int_{x_0}^{x_\ast} \sqrt{-Q(z)}\,dz \right| , \delta^{n_\max}S_{n_\max}(x_0) \approx \sqrt{\frac{2\pi}{n_\max}} e^{-n_\max}, where x_0 is the point at which y(x_0) needs to be evaluated and x_{\ast} is the (complex) turning point where Q(x_{\ast}) = 0, closest to x = x_0. The number n_\max can be interpreted as the number of oscillations between x_0 and the closest turning point. If \varepsilon^{-1}Q(x) is a slowly changing function, \varepsilon\left| \frac{dQ}{dx} \right| \ll Q^2 , ^{\text{[might be }Q^{3/2}\text{?]}} the number n_\max will be large, and the minimum error of the asymptotic series will be exponentially small. == Application in non-relativistic quantum mechanics ==

The above example may be applied specifically to the one-dimensional, time-independent Schrödinger equation,-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x),which can be rewritten as\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} \left( V(x) - E \right) \Psi(x).Approximation away from the turning points The wave function can be rewritten as the exponential of another function (closely related to the action), which could be complex, \Psi(\mathbf x) = e^{i S(\mathbf{x}) / \hbar}, so that its substitution in Schrödinger's equation gives:i\hbar \nabla^2 S(\mathbf x) - \left(\nabla S(\mathbf x)\right)^2 = 2m \left( V(\mathbf x) - E \right),Next, the semi-classical approximation is used. This means that each function is expanded as a power series in \hbar:S = S_0 + \hbar S_1 + \hbar^2 S_2 + \cdots Substituting in the equation, and only retaining terms up to first order in \hbar, one gets\left(\nabla S_0+\hbar \nabla S_1\right)^2 - i\hbar\left(\nabla^2 S_0\right) = 2m\left(E-V(\mathbf x)\right), which gives the following two relations:\begin{align} \left(\nabla S_0\right)^2 = 2m \left(E - V(\mathbf x)\right) &= \left(p(\mathbf x)\right)^2 \\[1ex] 2\nabla S_0 \cdot \nabla S_1 - i \nabla^2 S_0 &= 0. \end{align}These can be solved for one-dimension systems. The first equation can be solved byS_0(x) = \pm \int \sqrt{ 2m \left( E - V(x)\right) } \,dx=\pm\int p(x) \,dx, and the second equation computed for the possible values of the above, is generally expressed as:\Psi(x) \approx C_+ \frac{ e^{+ \frac i \hbar \int p(x)\,dx} }{\sqrt } In the classically allowed region, namely the region where V(x) the integrand in the exponent is imaginary and the approximate wave function is oscillatory. In the classically forbidden region V(x) > E, the solutions are growing or decaying. It is evident in the denominator that both of these approximate solutions become singular near the classical turning points, where E = V(x) , and cannot be valid. (The turning points are the points where the classical particle changes direction.) Hence, when E > V(x), the wave function can be chosen to be expressed as:\Psi(x') \approx \frac{1}{\sqrt} \left[ C \cos\left(\frac 1 \hbar \int \left|p(x)\right| dx + \alpha\right) + D \sin\left(- \frac 1 \hbar \int \left|p(x)\right| dx +\alpha\right)\right] and for V(x) > E,\Psi(x') \approx \frac{ C_{+} e^{- \frac{1}{\hbar} \int |p(x)|\,dx}}{\sqrt} + \frac{ C_{-} e^{+ \frac{1}{\hbar} \int |p(x)|\,dx} }{ \sqrt } . The integration in this solution is computed between the classical turning point and the arbitrary position x' . Validity of WKB solutions From the condition:\left(S_0'(x)\right)^2 - \left(p(x)\right)^2 + \hbar \left(2 S_0'(x)S_1'(x)-iS_0''(x)\right) = 0 It follows that: \hbar\left| 2 S_0'(x)S_1'(x)\right| + \hbar\left| i S_0''(x)\right| \ll \left|(S_0'(x))^2\right| + \left| (p(x))^2\right| For which the following two inequalities are equivalent since the terms in either side are equivalent, as used in the WKB approximation: \begin{align} \hbar \left| S_0''(x)\right| &\ll \left|(S_0'(x))^2\right| \\ 2\hbar \left| S_0'S_1' \right| &\ll \left|(p'(x))^2\right| \end{align} The first inequality can be used to show the following: \begin{align} \hbar \left| S_0''(x)\right| &\ll \left|p(x)\right|^2 \\[6pt] \frac{1}{2} \frac{\hbar} \left|\frac{dp^2}{dx}\right| &\ll \left|p(x)\right|^2 \\[6pt] \lambda \left|\frac{dV}{dx}\right| &\ll \frac{\left|p\right|^2}{m}\\ \end{align} where |S_0'(x)|= |p(x)| is used and \lambda(x) is the local de Broglie wavelength of the wave function. The inequality implies that the variation of potential is assumed to be slowly varying. This condition can also be restated as the fractional change of E-V(x) or that of the momentum p(x) , over the wavelength \lambda , being much smaller than 1 . Similarly it can be shown that \lambda(x) also has restrictions based on underlying assumptions for the WKB approximation that:\left|\frac{d\lambda}{dx}\right| \ll 1 which implies that the de Broglie wavelength of the particle is slowly varying.\begin{align} \Psi(x) &= C_A \operatorname{Ai}\left( \sqrt[3]{U_1} \cdot (x - x_1) \right) + C_B \operatorname{Bi}\left( \sqrt[3]{U_1} \cdot (x - x_1) \right) \\[6pt] &= C_A \operatorname{Ai}\left( u \right) + C_B \operatorname{Bi}\left( u \right). \end{align}Although for any fixed value of \hbar, the wave function is bounded near the turning points, the wave function will be peaked there, as can be seen in the images above. As \hbar gets smaller, the height of the wave function at the turning points grows. It also follows from this approximation that:\begin{align} \frac{1}{\hbar}\int p(x) \, dx &= \sqrt{U_1} \int \sqrt{x-a}\, dx \\ &= \frac 2 3 \left[\sqrt[3]{U_1} \left(x-a\right)\right]^{\frac 3 2} = \frac 2 3 u^{\frac 3 2}. \end{align} Connection conditions It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of , this matching procedure will not work: The function obtained by connecting the solution near +\infty to the classically allowed region will not agree with the function obtained by connecting the solution near -\infty to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy , which will give an approximation to the exact quantum energy levels.The wavefunction's coefficients can be calculated for a simple problem shown in the figure. Let the first turning point, where the potential is decreasing over x, occur at x=x_1 and the second turning point, where potential is increasing over x, occur at x=x_2 . Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions. \begin{align} \Psi_{V>E}(x) &\approx u^{-\frac{1}{4}} \left[ A \exp\left(\tfrac 2 3 u^\frac{3}{2}\right) + B \exp\left(-\tfrac 2 3 u^\frac{3}{2}\right) \right] \\[6pt] \Psi_{E>V}(x) &\approx u^{-\frac{1}{4}} \left[C \cos\left(\tfrac 2 3 u^\frac{3}{2} - \alpha \right) + D \sin\left(\tfrac 2 3 u^\frac{3}{2} - \alpha\right) \right]\\ \end{align} First classical turning point For U_1 ie. decreasing potential condition or x=x_1 in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero. Considering Bairy functions to be the required connection formula, we get: \begin{align} \operatorname{Bi}(u) &\to -\frac{1}{\sqrt \pi}\frac{1}{\sqrt[4]{u}} \sin\left(\frac 2 3 |u|^{\frac 3 2} - \frac \pi 4\right) & \text{where} \quad u \to -\infty\\[1ex] \operatorname{Bi}(u) &\to \frac{1}{\sqrt \pi}\frac{1}{\sqrt[4]{u}} \exp\left(\frac 2 3 u^{\frac 3 2}\right) & \textrm{where} \quad u \to +\infty \end{align} We cannot use Airy function since it gives growing exponential behaviour for negative x. When compared to WKB solutions and matching their behaviours at \pm \infty , we conclude: A=-D=N , B=C=0 and \alpha = \frac \pi 4 . Thus, letting some normalization constant be N , the wavefunction is given for increasing potential (with x) as: It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual eigenfunction. In particular, the Maslov-corrected Bohr–Sommerfeld energies are good approximations to the actual eigenvalues of the Schrödinger operator. Specifically, the error in the energies is small compared to the typical spacing of the quantum energy levels. Thus, although the "old quantum theory" of Bohr and Sommerfeld was ultimately replaced by the Schrödinger equation, some vestige of that theory remains, as an approximation to the eigenvalues of the appropriate Schrödinger operator. General connection conditions Thus, from the two cases the connection formula is obtained at a classical turning point, x = a: Since the classical particle's velocity goes to zero at the turning points, it spends more time near the turning points than in other classically allowed regions. This observation accounts for the peak in the wave function (and its probability density) near the turning points. Applications of the WKB method to Schrödinger equations with a large variety of potentials and comparison with perturbation methods and path integrals are treated in Müller-Kirsten. == Examples in quantum mechanics ==

Although WKB potential only applies to smoothly varying potentials, Bound states within 2 rigid wall The potential of such systems can be given in the form: V(x) = \begin{cases} \infty & \text{if } x > x_2 \\ V(x) & \text{if } x_2 \geq x \geq x_1\\ \infty & \text{if } x where x_1 . For E \geq V(x) between x_1 and x_2 which are thus the classical turning points, by considering approximations far from x_1 and x_2 respectively we have two solutions: \begin{align} \Psi_{\text{WKB}}(x) &= \frac{A}{\sqrt} \sin\left(\frac 1 \hbar \int_x^{x_1} |p(x)| dx \right) \\ \Psi_{\text{WKB}}(x) &= \frac{B}{\sqrt} \sin\left(\frac 1 \hbar \int_x^{x_2} |p(x)| dx \right) \end{align} Since wavefunctions must vanish at x_1 and x_2 . Here, the phase difference only needs to account for n \pi which allows B = (-1)^n A . Hence the condition becomes: \int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = n\pi \hbar where n = 1,2,3,\cdots but not equal to zero since it makes the wavefunction zero everywhere. E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. This result is also consistent with the use of equation from bound state of one rigid wall without needing to consider an alternative potential. Quantum tunneling The potential of such systems can be given in the form: V(x) = \begin{cases} 0 & \text{if } x x_2 \\ \end{cases} where x_1 . Its solutions for an incident wave is given as \psi(x) = \begin{cases} A e^{i k_0 x} + B e^{-i k_0 x} & \text{if } x x_2 \end{cases} with k_0 = p_0/\hbar, where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential. This is a fair assumption for wide potential barriers through which the wavefunction is not expected to grow to high magnitudes. By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:\frac {|D|^2} {|A|^2} = \frac{4}{(1+{a_1^2}/{p_0^2} )} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right) where a_1 = |p(x_1)| and a_2 = |p(x_2)| . Using \mathbf J(\mathbf x,t) = \frac{i\hbar}{2m} \left(\psi^* \nabla\psi - \psi\nabla\psi^*\right) we express the values without signs as: \begin{align} J_{\text{inc.}} &= \tfrac{\hbar}{2m} \left(\tfrac{2p_0}{\hbar}|A|^2\right) \\ J_{\text{ref.}} &= \tfrac{\hbar}{2m} \left(\tfrac{2p_0}{\hbar}|B|^2\right) \\ J_{\text{trans.}} &= \tfrac{\hbar}{2m} \left(\tfrac{2p_0}{\hbar}|D|^2\right) \end{align} Thus, the transmission coefficient is found to be: T = \frac {|D|^2} {|A|^2} = \frac{4}{\left(1 + {a_1^2}/{p_0^2} \right)} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right) where {{nowrap|p(x) = \sqrt {2m \left( E - V(x)\right)} ,}} a_1 = |p(x_1)| and a_2 = |p(x_2)| . The result can be stated as T \sim ~ e^{-2\gamma} where \gamma = \int_{x_1}^{x_2} |p(x')| dx' . ==See also==