# On the Application of the Einstein-Brillouin-Keller (EBK) Method

Consider the Born-Sommerfeld quantization condition (modified) [see Einstein–Brillouin–Keller (EBK)]

$$I_{i} = \frac{1}{2\pi}S_{i} = \frac{1}{2\pi}\oint p_{i} dq_{i} = \hbar \left(n_{i} + \frac{\mu_{i}}{4} + \frac{b_{i}}{2} \right),\tag{1}$$

when applied to the semi-harmonic oscillator with potential,

$$V(x) = \frac{1}{2}m\omega^{2}x^{2}\text{ for }x>0\text{ and }V(x) = \infty\text{ otherwise.} \tag{2}$$

We here have a turning point at $$x=x_{1}$$, say $$x_{1} = a$$ and obtain the expression for this as

$$a = \frac{1}{\omega}\sqrt{\frac{2E}{m}}.\tag{3}$$

We can find that upon integration, $$\int_{0}^{a}k(x) dx = \frac{\pi E}{2\hbar \omega}$$. Now, in the quantization condition, the Maslov indices take the values $$\mu = 1$$ and $$b = 1$$ for the reasons that there is one turning point and since there is one reflection at the hard wall (and also since $$\Psi(0) = 0$$ holds) respectively. Upon doing this substitution, we get

$$I_{x} = \frac{1}{2\pi}S_{x} = \frac{1}{2\pi}\int p(x) dx = \left(n + \frac{3}{4} \right)\hbar,\tag{4}$$

and this, when equated to what was found previously doesn't give the correct expression for $$E_{n}$$. This seems to work only when we take $$S_{x} = 2\int p(x) dx$$, since this correctly gives $$E_{n} = \left(2n+\frac{3}{2}\right)\hbar\omega$$. How do we account for this factor of "$$2$$"? Is this to do with phase changes due to the reflections or that we take into consideration both possibilities of solutions with $$E and $$E>V(x)$$ at the turning point?

Links to other questions about EKB: for an outline of general features: 1, for derivation and additional references: 2, about the Maslov index in Bohr-Sommerfeld quantization condition: 3, for references: 4.

• Hi Spoilt Milk, I removed your second question as Phys.SE works best with 1 question per post. Commented Jun 25, 2020 at 12:29

Hint: Eq. (1) is for a closed orbit, i.e. the particle traverses the interval $$[0,a]$$ twice, i.e. forth & back. This explains the factor 2.