WKB approximation derivation for $E<V$

Question

I understand that we can write any complex wavefunction on polar form $A\exp(iθ)$ with both $A,θ$ real. Following the logic of Griffiths on WKB (here, page 291):

1. We write the energy wavefunction in the previous form.
2. For $E>V$, we insert the previous form in S.E and demand $A^"=0$. The reason we can do this approximation is because we are lead to equation (8.6), which can give indeed real $A$ for $A^"=0$.

But what happens when $E<V $? In this case if we write again the wavefunction in the form $A\exp(iθ)$ with $A,θ$ real, then we cannot apply $A^"=0$, because equation (8.6) would not be able to give real $A$ (since $p^2$ will now be negative).

So what do we do to overcome this problem for $E<V $?

Answer 1

OP has a point: The polar form of the complex wavefunction $\psi$ is not useful in the classically forbidden region$E<V$ because the complex TISE then doesn't separate into 2 real equations.

Alternatively, consider the approach of Ref. 1 (which happens to be problem 8.2 in Ref. 2). Here the wavefunction is assumed to be on the semiclassical form

$$ \psi~=~\exp\left(\frac{i}{\hbar}\sigma\right), \tag{46.1} $$

where

$$ \sigma~=~\sum_{n=0}^{\infty}\left(\frac{\hbar}{i}\right)^n\sigma_n \tag{46.3}$$

is a complex power series in Planck's constant. The leading coefficient satisfies

$$ \sigma_0~=~\pm \int p \mathrm{d}x, \qquad p=\sqrt{2m(E-V)},\tag{46.5}$$

In eq. (46.5) the momentum $p$ is imaginary in the classically forbidden region $E<V$. The next-to-leading coefficient

$$ \sigma_1 ~=~-\frac{1}{2}{\rm Ln}(p) \tag{46.8}$$

is given in terms of a complex logarithm.

References:

1. L.D. Landau & E.M. Lifshitz, QM, Vol. 3, 2nd & 3rd ed, 1981; $\S46$.

2. D. Griffiths, Intro to QM, 1995; problem 8.2.