IGNOU MPH 8 2026 - Determine the WKB approximation for the bound state energy of a

Question

Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:

$V(x) = \begin{cases} a|x| & x \geq 0 \\ 2a|x| & x < 0 \end{cases}$

19 Jan 2026

Answer :

Word Count : 441

The potential is piecewise linear with a slope discontinuity at (x=0): [ V(x) = \begin{cases} a|x| & x \ge 0 [2mm] 2a|x| & x < 0 \end{cases} ] We want the bound state energy using the WKB approximation. For a one-dimensional potential, the WKB quantization condition is: [ \int_{x_1}^{x_2} \sqrt{2m \big(E - V(x)\big)} , dx = \left(n + \frac{1}{2}\right)\pi \hbar ] where (x_1) and (x_2) are classical turning points defined by (E = V(x)). --- ### Step 1: Find classical turning points For (x \ge 0): (V(x) = a x \implies x_2 = \frac{E}{a}) For (x < 0): (V(x) = 2a |x| = -2a x \implies x_1 = -\frac{E}{2a}) So the particle is confined between (x_1 = -\frac{E}{2a}) and (x_2 = \frac{E}{a}). --- ### Step 2: Set up the WKB integral [ \int_{x_1}^{x_2} \sqrt{2m(E - V(x))} , dx = \int_{-E/(2a)}^{0} \sqrt{2m(E - 2a|x|)} , dx + \int_{0}^{E/a} \sqrt{2m(E - a x)} , dx ] --- ### Step 3: Evaluate

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