# $\hbar^2$ Correction to the Bohr-Sommerfeld Quantization Condition

We can get the Bohr-Sommerfeld quantization from the WKB method as answered. Since we use approximation, there should be an error in the system, I know this is not right all the time; in some conditions approximation can yield the real result. So, is there a way to find the $$\hbar^2$$ correction?

• Linked, as well as this. Actually, there are striking failures of the approximation even at a level below $\hbar^2$. The best link is through deformation quantization. Nov 5, 2023 at 23:58

OK, from the point of WKB, if we could find $$S_2$$, coefficient of the term $$(\hbar/i)^2$$, and plug it into the Schrödinger equation, we would have find the correction. From the WKB, we already know that $$S_0 = \int p(x)dx$$ and $$S_1 = - \frac{1}{2} \ln p + C$$. We also have the condition, $$S_1''+(S_1')^2+2S_0'S_2' = 0$$, so plugging $$S_1$$and $$S_0$$ written in terms of $$p$$, we got $$S_2$$ = $$\frac{1 \cdot p'}{4 \cdot p^2} + \frac{1}{8} \int \frac{p'^2}{p^3}dx$$. WKB says that, $$\psi = e^{\frac{i}{\hbar}s}$$ and $$s = s_0 + \frac{\hbar}{i}s_1 + \frac{\hbar}{i}^2 s_2$$. If we plug $$s$$ containing the $$s_2$$ term to the Schrödinger equation, we can find the correction.