Einstein–Brillouin–Keller method

The Einstein–Brillouin–Keller (EBK) method is a semiclassical technique (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr–Sommerfeld quantization which did not consider the caustic phase jumps at classical turning points.^[1] This procedure is able to reproduce exactly the spectrum of the 3D harmonic oscillator, particle in a box, and even the relativistic fine structure of the hydrogen atom.^[2]

History

The first paper by Albert Einstein in 1917 generalized Bohr–Sommerfeld quantization as part of the old quantum theory.^[3][1] Einstein also discussed that the method was not generalizable for chaotic systems.^[3][1] The paper was influential for Louis de Broglie thesis in 1924 and Erwin Schrödinger reformulation of quantum mechanics in terms of the Schrödinger equation in 1926.^[1]

Another correction to Bohr–Sommerfeld quantization was introduced by Léon Brillouin in 1926,^[4] who also introduced the same year the WKB approximation.^[5] Mathematician Rudolf Ernest Langer also introduced his Langer correction in 1937.^[5]

Einstein's formulation however was mostly ignored, until its independent discovery by Joseph Keller in 1958.^[1][6] Keller derived his equation from the WKB approximation and introduced the correction of caustic phase jumps at classical turning points.^[1] In 1972, mathematician Viktor Maslov formalized the concept in terms of Maslov indexes.^[5]

Ian C. Percival reintroduced Einstein's paper to the physics community in 1973,^[7] and coined the term Einstein–Brillouin–Keller method.^[1]

In 1976–1977, Michael Berry and Michael Tabor derived an extension to Gutzwiller trace formula for the density of states of an integrable system starting from EBK quantization.^[8][9][10]

Procedure

Given a separable classical system defined by conjugate coordinates ${\displaystyle (q_{i},p_{i});i\in \{1,2,\cdots ,d\}}$, in which every pair ${\displaystyle (q_{i},p_{i})}$ describes a closed function or a periodic function in ${\displaystyle q_{i}}$, the EBK procedure involves quantizing the line integrals of ${\displaystyle p_{i}}$ over the closed orbit of ${\displaystyle q_{i}}$:

${\displaystyle I_{i}={\frac {1}{2\pi }}\oint p_{i}dq_{i}=\hbar \left(n_{i}+{\frac {\mu _{i}}{4}}+{\frac {b_{i}}{2}}\right)}$

where ${\displaystyle I_{i}}$ is the action-angle coordinate, ${\displaystyle n_{i}}$ is a positive integer, and ${\displaystyle \mu _{i}}$ and ${\displaystyle b_{i}}$ are Maslov indexes. ${\displaystyle \mu _{i}}$ corresponds to the number of classical turning points in the trajectory of ${\displaystyle q_{i}}$ (Dirichlet boundary condition), and ${\displaystyle b_{i}}$ corresponds to the number of reflections with a hard wall (Neumann boundary condition).^[10]

Examples

1D Harmonic oscillator

The Hamiltonian of a simple harmonic oscillator is given by

${\displaystyle H={\frac {p^{2}}{2m}}+{\frac {m\omega ^{2}x^{2}}{2}}}$

where ${\displaystyle p}$ is the linear momentum and ${\displaystyle x}$ the position coordinate. The action variable is given by

${\displaystyle I={\frac {2}{\pi }}\int _{0}^{x_{0}}{\sqrt {2mE-m^{2}\omega ^{2}x^{2}}}\mathrm {d} x}$

where we have used that ${\displaystyle H=E}$ is the energy and that the closed trajectory is 4 times the trajectory from 0 to the turning point ${\displaystyle x_{0}={\sqrt {2E/m\omega ^{2}}}}$.

The integral turns out to be

${\displaystyle E=I\omega }$,

which under EBK quantization there are two soft turning points in each orbit ${\displaystyle \mu _{x}=2}$ and ${\displaystyle b_{x}=0}$. Finally, that yields

${\displaystyle E=\hbar \omega (n+1/2)}$,

which is the exact result for quantization of the quantum harmonic oscillator.

2D hydrogen atom

The Hamiltonian for a non-relativistic electron (electric charge ${\displaystyle e}$) in a hydrogen atom is:

${\displaystyle H={\frac {p_{r}^{2}}{2m}}+{\frac {p_{\varphi }^{2}}{2mr^{2}}}-{\frac {e^{2}}{4\pi \epsilon _{0}r}}}$

where ${\displaystyle p_{r}}$ is the canonical momentum to the radial distance ${\displaystyle r}$, and ${\displaystyle p_{\varphi }}$ is the canonical momentum of the azimuthal angle ${\displaystyle \varphi }$. Take the action-angle coordinates:

${\displaystyle I_{\varphi }={\text{constant}}=|L|}$

For the radial coordinate ${\displaystyle r}$:

${\displaystyle p_{r}={\sqrt {2mE-{\frac {L^{2}}{r^{2}}}+{\frac {e^{2}}{4\pi \epsilon _{0}r}}}}}$

${\displaystyle I_{r}={\frac {1}{\pi }}\int _{r_{1}}^{r_{2}}p_{r}dr={\frac {me^{2}}{4\pi \epsilon _{0}{\sqrt {-2mE}}}}-|L|}$

where we are integrating between the two classical turning points ${\displaystyle r_{1},r_{2}}$ (${\displaystyle \mu _{r}=2}$)

${\displaystyle E=-{\frac {me^{4}}{32\pi ^{2}\epsilon _{0}^{2}(I_{r}+I_{\varphi })^{2}}}}$

Using EBK quantization ${\displaystyle b_{r}=\mu _{\varphi }=b_{\varphi }=0,n_{\varphi }=m}$ :

${\displaystyle I_{\varphi }=\hbar m\quad$ ;\quad m=0,1,2,\cdots }

${\displaystyle I_{r}=\hbar (n_{r}+1/2)\quad$ ;\quad n_{r}=0,1,2,\cdots }

${\displaystyle E=-{\frac {me^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}(n_{r}+m+1/2)^{2}}}}$

and by making ${\displaystyle n=n_{r}+m+1}$ the spectrum of the 2D hydrogen atom ^[11] is recovered :

${\displaystyle E_{n}=-{\frac {me^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}(n-1/2)^{2}}}\quad$ ;\quad n=1,2,3,\cdots }

Note that for this case ${\displaystyle I_{\varphi }=|L|}$ almost coincides with the usual quantization of the angular momentum operator on the plane ${\displaystyle L_{z}}$. For the 3D case, the EBK method for the total angular momentum is equivalent to the Langer correction.

See also

• Physics portal

• Hamilton–Jacobi equation
• WKB approximation
• Quantum chaos