WKB Approximation in 3D

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1 1 WKB Approximation in 3D We see solutions ψr of the stationary Schrodinger equations for a spinless particle of energy E: 2 2m 2 ψ + V rψ = Eψ At rst, we just rewrite the Schrodinger equation in the following way: 2 ψ + 2 rψ = where r given by: 2 r 2m 2 E V r is the de Broglie wavenumber that the particle would classically have at position r. In other words, is the classical momentum of the particle at position r, given that its inetic energy there is E V r. Next, we represent an exact wavefunction using an auxiliary function W r: ψr = e i W r Note that W r has the same units as, and need not be real. It is straight-forward to show that the Schrodinger equation leads to the following exact equation for W r: Semiclassical limit: the lowest order W 2 = 2 2 r + i 2 W If we assume that the second term on the right-hand-side of the above equation can be neglected next to the rst term, 2 W W 2, then we approximately obtain: 2 W W 2 W r W r r where r is a wavevector whose magnitude is r. At this point, only the magnitude of = ˆ is determined, its orientation specied by a unit-vector ˆ which may have positional dependence seems arbitrary. Given the above form of W, we nd that: i 2 W i 2 We thus obtain a correction to the approximate equation W r due to the spatial variations of the wavevector. A closer scrutiny of the approximation condition reveals the physical meaning of the WKB approximation: 2 W W 2 2 V 4π E V r λr where: λr = 2π r is the local wavelength of the particle's de Broglie wave. The WKB approximation is valid if the signicant spatial variations of the potential energy V occur only at length-scales much larger than the particle's wavelength. The measure of potential's spatial variations also involves the energy scale of the potential, which is to be compared to the local classical inetic E V r energy of the particle. Clearly, the WKB approximation breas down wherever the total energy E of the particle becomes too close to the local potential energy. At this order of the WKB approximation, we obtain the wavefunction: from: ψr = e i W r ψ exp i W r = W + r W r = W + r

2 Semiclassical limit: the next order Now, start from the lowest order approximation W and include the ensuing estimate i 2 W i 2 as a correction in the equation for W : An improved solution can be written as: W 2 = 2 2 r + i 2 W i 2 W ˆ 2 + i 1 + i Note that this form does not exhaust all possible solutions one example is discussed later. If we wanted to, we could proceed iteratively: use the corrected W to recompute 2 W = W, substitute it bac into the equation for W, calculate an even more correct W, repeat indenitely: i 2 W i i W i i This would generate an expansion for W in powers of: 2 x 2 λ 1 2 where λ = 2π/ is the de Broglie wavelength. Since the inetic energy is K = E V = h 2 /2mλ 2, we may view the wavelength λ of aparticle with large K as being proportional to the Planc's constant h = 2π, so that the WKB expansion in the powers of x is eectively an expansion in powers of. The spirit of a semiclassical approximation is to consider small and retain only a few lowest order terms in the expansion over powers of. Keeping only: W ˆ 2 + i 1 + i from the iterative expansion, we nd: W r = W i + O 2 = W + + i + O 2 Keeping in mind that ψr = exp i W, we see that, in the classically allowed regions 2 >, the third purely imaginary term of W contributes only to the spatial variations of the wavefunction's amplitude the second term is real and gives rise to the wavefunction's phase variations as before. The integral over r taes place on any path from the origin to the position r. If we tae this integral over any closed path, with the end-point r coming bac to the origin, then the uniqueness of the wavefunction at every point in space implies: = 2πn, = These requirements are quite strong, given that they apply to every possible closed path. They restrict the allowed choices of ˆ and lead to energy quantization in bound-states. Specically, the second requirement implies that / 2 = φ should be a gradient of a scalar function φr. Then: W r = W + = W + + i + i 2 [ ] φr φ + O 2 + O 2 = W + + i 2 φ + O 2

3 3 and hence: ψr = e i W r Ae 1 2 φr exp i r Considering the following vector calculus identities: 2 = ˆ [ ] ˆ ˆ ˆ = [ˆ + ˆ] = ˆ + ˆ ˆ = ˆ ˆ + ˆˆ + ˆ ˆ = ˆ [ ˆ log ] + ˆ ˆ + log we could try to ensure / 2 = φ by imposing: [ ] ˆ ˆ log + ˆ ˆ =, φ = log Note that is an arbitrary constant with the same units as, but it's convenient to pic =. In this case, we would get the wavefunction familiar from the one-dimensional WKB approximation: ψr = e i W r Ae 1 2 φr exp i r = A exp i r r However, there is a problem. We restricted the orientation ˆ of the wavevector by the condition: [ ] ˆ ˆ log + ˆ ˆ = which contains two orthogonal vectors on its left-hand side, ˆ and ˆ ˆ. Both of these orthogonal vectors must individually vanish if the above condition were to hold: [ ] ˆ ˆ log =, ˆ ˆ = ˆ, ˆ = The gradient is uniquely determined by the potential V r, and aligning the unit-vectors ˆ everywhere with it will generally produce a non-vanishing divergence ˆ. An example is a sphericallysymmetric potential, where ˆ = is parallel to and = 2/r. We must conclude that actually: φ log Both φ and ˆ need to be determined some other way to satisfy: 2 = φ Note that this problem does not arise in one dimension, since there curls have no meaning and ˆ ˆx = const automatically yields ˆ =. Here we wor out some examples for spherically-symmetric potentials V r = V r: 2mE V r r = r = 2

4 4 = m dv Consult the form of gradient and divergence in spherical coordinates... Consider radially-propagating de Broglie waves, ˆ =, i.e. = : 2 2 = r + 1 d = 1 r 2 dr 2 = 2 r + d = φr = dφ φr = const+ Then, consider waves that rotate about the ẑ axis, ˆ = ˆϕ or r = r, θ, ϕˆϕ: = 1 d r sin θ dϕ ˆ 2 r + 1 d = const+logr 2 2 = = φr φr = const This is troubling, because it describes a wavefunction whose amplitude lacs radial dependence even though the potential varies with r. We must redo the analysis of W r from the Schrodinger equation and assume a more general solution than we considered before, or perhaps go to higher orders of the semiclassical expansion. Since we are looing for rotating solutions, we should consider a function: W r = mϕ + i log wr, θ which automatically maes the wavefunction single-valued as long as m is an integer. directly from the form of W r: W = i w w r + 1 w r θ ˆθ + m r sin θ ˆϕ [ w 2 W 2 = 2 w ] 2 w r r m 2 θ r 2 sin 2 θ We obtain If we neglect 2 W in the spirit of the lowest order WKB approximation, then the Schrodinger equation: W 2 = i 2 W becomes approximately: 2 w 1 r r 2 2 w = 2 r m2 θ r 2 sin 2 w 2 θ Separating variables wr, θ = frgθ yields: 1 2 df f 2 1 g 2 r 2 2 dg = 2 r m2 dθ r 2 sin 2 θ which clearly breas down into two independent single-variable dierential equations: 2 2 df dg + 2 rf 2 =, sin 2 θ = m 2 g 2 dθ Taing square roots of these equations reveals: ˆ df = ±irf f = f exp ±i r

5 5 sin θ dg [ ] m θ dθ = ±mg g = g tan 2 This is just an example of a WKB calculation. We are missing an enumeration of spherical harmonics in terms of the orbital quantum number, and also the function fr describes a radially propagating wave not a bound state. More sophisticated analysis along these lines, which includes some higherorder corrections from 2 W is necessary in order to properly approximate all states, including bound states. Probability current density The wavefunction written as: has the following exact current density: ψr = e i W r j = i ψ ψ ψ ψ = e i W W W + W 2m 2m The continuity equation in the classically allowed regions 2 > implies: j + ρ t = ρ t = j = W + W + i W + W W W = 2 Re [ 2 W + i W 2] = Re [ i 2 2] = This is automatically satised, since the Schrodinger equation itself ensures the continuity equation.