Background of the question (see pp. 161, section 47 in Landau & Lifshitz's quantum mechanics textbook Vol3, 2nd Ed. Pergamon Press). We a following potential well U(x)≤Eforx≤a,
The WKB solutions right and left to the turning point are
ψ=C2√pexp(−1ℏ|∫xapdx|)forx>a,
$$\psi=\dfrac{C_1}{\sqrt{p}}\exp{\left(\dfrac{i}{\hbar}\int_a^x pdx\right)}+\dfrac{C_2}{\sqrt{p}}\exp{\left(-\dfrac{i}{\hbar}\int_a^x p dx\right)}\quad \text{for} \quad x
respectively. Most quantum mechanics textbooks determine the relation between C and Ci's by find the exact solution near the turning point. And then let the exact solution match with the WKB solutions.
However, in Landau & Lifshitz's quantum mechanics textbooks (vol3, section 47) they let x vary in the complex plane and pass around the turning point a from right to left through a large semicircle in the upper complex plane. Landau claims starting from +∞, when arrive at −∞ (left to a), there is phase gain π in the denominator of prefactor in Eq.(2). From this we can determine C2=C2exp(iπ4).
They also claim the first term will exponential decay along the semicircle in the upper half plane. Question is why? Can we show ℑ(∫xapdx),
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