WKB approximation

The semiclassical WKB approximation from quantum mechanics (named after Gregor Wentzel , Hendrik Anthony Kramers and Léon Brillouin ) provides an approximation of the solution of the one-dimensional, stationary Schrödinger equation . The approximation is based on the assumption that the potential only 'slowly' changes with position, i.e. H. via the expansion of a wavelength , and therefore a solution can be found from the constant potential . ${\ displaystyle V (x)}$ ${\ displaystyle V (x) = V_ {0}}$

Under this assumption, the approximate solution of the Schrödinger equation is:

{\ displaystyle \ psi (x) = \ left ({\ frac {\ text {const}} {2m [EV (x)]}} \ right) ^ {1/4} \ exp \ left (\ pm {\ frac {i} {\ hbar}} \ int \ mathrm {d} x '{\ sqrt {2m (EV (x'))}} \ right)}

The two signs stand for two independent solutions.

history

The approximation was published almost simultaneously and independently of one another in 1926 by the physicists Gregor Wentzel , Hendrik Anthony Kramers and Léon Brillouin in the context of quantum mechanics, whose initials gave it its name. But it can also be found earlier in the work of various mathematicians and physicists such as Francesco Carlini (1817, in celestial mechanics), George Green (1837), Joseph Liouville (1837), John William Strutt, 3rd Baron Rayleigh (1912), Richard Goose (1915), Harold Jeffreys (1923). It is therefore sometimes also called the WKBJ (additionally after Jeffreys) or the Liouville-Green method. Even Werner Heisenberg used the method in 1924 in his thesis on hydrodynamics .

Derivation

From the one-dimensional stationary Schrödinger equation

{\ displaystyle - {\ frac {\ hbar ^ {2}} {2m}} {\ frac {d ^ {2}} {dx ^ {2}}} \ psi (x) + V (x) \ psi ( x) = E \ psi (x)}

If the potential is constant, the solution is the plane wave ${\ displaystyle V (x) = V_ {0}}$

{\ displaystyle \ psi (x) = A \ exp \ left (\ pm {\ frac {i} {\ hbar}} p_ {0} x \ right)}

with . In the case of a slow change in the potential, i.e. a potential that can be regarded as constant in the order of magnitude of the deBroglie wavelength, one can assume and choose a solution approach analogous to the problem with constant potential as follows. ${\ displaystyle p_ {0} = {\ sqrt {2m (E-V_ {0})}}}$ ${\ displaystyle p (x) = {\ sqrt {2m (EV (x))}}}$

{\ displaystyle \ psi (x) = A \ exp \ left ({\ frac {i} {\ hbar}} S (x) \ right)}

Inserted into the Schrödinger equation one obtains

{\ displaystyle - {\ frac {i \ hbar} {2m}} {\ frac {d ^ {2} S (x)} {dx ^ {2}}} + {\ frac {1} {2m}} \ left [{\ frac {dS (x)} {dx}} \ right] ^ {2} + V (x) -E = 0}

No approximation has been made so far. We can now in powers of the following develop ${\ displaystyle S (x)}$ ${\ displaystyle \ hbar}$

{\ displaystyle S (x) = S_ {0} (x) + \ hbar S_ {1} (x) + {\ frac {\ hbar ^ {2}} {2}} S_ {2} (x) + \ dots}

This is put into the Schrödinger equation:

{\ displaystyle {- {\ frac {i \ hbar} {2m}} {\ frac {d ^ {2}} {dx ^ {2}}} \ left (S_ {0} (x) + \ hbar S_ { 1} (x) + \ dots \ right) + {\ frac {1} {2m}} \ left [{\ frac {d} {dx}} \ left (S_ {0} (x) + \ hbar S_ { 1} (x) + \ dots \ right) \ right] ^ {2} + V (x) -E = 0}}

You can now calculate these terms up to the desired order and collect them according to the power of . ${\ displaystyle \ hbar}$

Each term belonging to a power of must then vanish individually. ${\ displaystyle \ hbar}$

For the second order the Schrödinger equation reads:

{\ displaystyle - {\ frac {i \ hbar} {2m}} \ left [{\ frac {d ^ {2}} {dx ^ {2}}} S_ {0} (x) + \ hbar {\ frac {d ^ {2}} {dx ^ {2}}} S_ {1} (x) \ right] + {\ frac {1} {2m}} \ left [{\ frac {d} {dx}} S_ {0} (x) + \ hbar {\ frac {d} {dx}} S_ {1} (x) \ right] ^ {2} + V (x) -E = 0}

{\ displaystyle {\ Leftrightarrow \ hbar ^ {0} \ left [{\ frac {1} {2m}} \ left ({\ frac {dS_ {0} (x)} {dx}} \ right) ^ {2 } + V (x) -E \ right] + \ hbar ^ {1} \ left [- {\ frac {i} {2m}} {\ frac {d ^ {2} S_ {0} (x)} { dx ^ {2}}} + {\ frac {1} {m}} {\ frac {dS_ {0} (x)} {dx}} {\ frac {dS_ {1} (x)} {dx}} \ right] + \ hbar ^ {2} \ left [{\ frac {1} {2m}} \ left ({\ frac {dS_ {1} (x)} {dx}} \ right) ^ {2} - {\ frac {i} {2m}} {\ frac {d ^ {2} S_ {1} (x)} {dx ^ {2}}} \ right] = 0}}

For the differential equation in the term zero order in ${\ displaystyle \ hbar}$

{\ displaystyle {\ frac {1} {2m}} \ left [{\ frac {dS_ {0} (x)} {dx}} \ right] ^ {2} + V (x) -E = 0}

one finds a solution through

{\ displaystyle S_ {0} (x) = \ pm \ int {\ sqrt {2m (EV (x '))}} dx'}

and it follows

{\ displaystyle \ psi (x) = A \ exp \ left (\ pm {\ frac {i} {\ hbar}} \ int {\ sqrt {2m (EV (x '))}} dx' \ right). }

This result describes solutions of a one-dimensional Schrödinger equation in the limit case , which has an equivalent counterpart in point mechanics with the Hamilton-Jacobi equation . ${\ displaystyle \ hbar \ rightarrow 0}$

The Taylor series expansion of the exponential function, however, shows the mathematical inconsistency of this simplest approximation in the context of the classic convergence criteria, since every summand that occurs diverges due to division by and thus the sum is not well-defined without suitable regularization. In addition, the description of tunneling is problematic in this approximation , because on the one hand an approximate solution for a Schrödinger equation is to be constructed, for the solution of which the validity of Born's probability postulate is assumed, on the other hand the Limes assumes a classic Hamiltonian operating principle, which is incompatible with intrinsically quantum mechanical tunneling processes is. ${\ displaystyle \ hbar = 0}$ ${\ displaystyle (V (x)> E)}$ ${\ displaystyle \ hbar \ rightarrow 0}$

An additional first-order consideration of the action function in fixes the constant A and completes the semiclassical WKB approximation. A precise calculation shows that this approximation is only good if the momentum of the wave function is significantly larger than the local variation of the potential (see above). Despite the effect discretization, resolved to the first order, this shows the proximity of the approach to the classical ray-optical approximation, which is used in geometric optics through the eikonal and in the Hamilton-Jacobi theory. ${\ displaystyle \ hbar}$ ${\ displaystyle p_ {0} = \ hbar / \ lambda}$

Consequences for the transmission through a barrier

The WKB approximation is used to approximate non-rectangular potential barriers. To do this, the barrier is broken down into many thin, rectangular partial barriers.

The tunnel probabilities of the individual segments are multiplied for the tunnel probability through the barrier. This results in ${\ displaystyle \ left | T \ right | ^ {2}}$

{\ displaystyle \ ln \ left | T \ right | ^ {2} \ approx -2 \! \ int \ limits _ {\ text {Barrier}} \! \ kappa (x) \, \ mathrm {d} x, }

in which

{\ displaystyle \ kappa (x) = {\ sqrt {\ frac {2m (V (x) -E)} {\ hbar ^ {2}}}}.}

As explained in the previous section, the barrier must be piece-wise constant compared to the wavelength of the matter wave to justify the approximation. ${\ displaystyle \ lambda}$

For sufficiently large wavelengths, e.g. B. near the turning points of classical particle movements , this can no longer be the case. In these regions, there must be constant connection through exact solutions, the Airy functions . ${\ displaystyle p \ propto 1 / \ lambda \ rightarrow 0}$

literature

Brillouin, Léon : La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives . In: Comptes Rendus de l'Academie des Sciences . 183, 1926, pp. 24-26.
Hendrik Anthony Kramers : Wave mechanics and half-integer quantization . In: Journal of Physics . Issue 39, No. 10 . Springer, 1926, ISSN 0939-7922 , p. 828-840 , doi : 10.1007 / BF01451751 .
Wentzel, Gregor : A generalization of the quantum conditions for the purposes of wave mechanics . In: Journal of Physics . 38, No. 6-7, 1926, pp. 518-529. bibcode : 1926ZPhy ... 38..518W . doi : 10.1007 / BF01397171 .
The WKB approximation is covered in most quantum mechanics textbooks, e.g. B. Nolting Basic Course Theoretical Physics 5/2 - Quantum Mechanics - Methods and Applications , Springer Verlag 2001, Chapter 7.4 (Quasi-classical approximation), pp. 190ff

Individual evidence

^ Jeffreys, On certain approximate solutions of linear differential equations of the second order , Proc. London Math. Soc. 23, 1923, p. 428
↑ J. Calvert History of the WKB approximation
↑ N. Fröman, PO Fröman On the history of the so called WKB method from 1817 to 1926 , in J. Bang, J. De Boer Semiclassical descriptions of atomic and nuclear collisions , Amsterdam 1985
^ Walter Blum, Helmut Rechenberg, Hans-Peter Dürr (editor): Heisenberg, Gesammelte Werke, A / 1, Springer Verlag 1985, p. 19, commentary by Subrahmanyan Chandrasekhar Hydrodynamic stability and turbulence (1922-1948) , abstract

[1] Jeffreys, On certain approximate solutions of linear differential equations of the second order , Proc. London Math. Soc. 23, 1923, p. 428

[2] J. Calvert History of the WKB approximation

[3] N. Fröman, PO Fröman On the history of the so called WKB method from 1817 to 1926 , in J. Bang, J. De Boer Semiclassical descriptions of atomic and nuclear collisions , Amsterdam 1985

[4] Walter Blum, Helmut Rechenberg, Hans-Peter Dürr (editor): Heisenberg, Gesammelte Werke, A / 1, Springer Verlag 1985, p. 19, commentary by Subrahmanyan Chandrasekhar Hydrodynamic stability and turbulence (1922-1948) , abstract