Question 705051

SCQHARD

The Hamiltonian of a particle of mass mm is given by H=p22m+V(x)H = \frac{p^2}{2m} + V(x), with V(x)={−αxfor x≤0βxfor x>0V(x) = \begin{cases} -\alpha x & \text{for } x \le 0 \\ \beta x & \text{for } x > 0 \end{cases} where α,β\alpha, \beta are positive constants. The nthn^{th} energy eigenvalue EnE_n obtained using WKB approximation is En3/2=32(ℏ22m)1/2π(n−12)f(α,β)E_n^{3/2} = \frac{3}{2} (\frac{\hbar^2}{2m})^{1/2} \pi (n - \frac{1}{2}) f(\alpha, \beta). The function f(α,β)f(\alpha, \beta) is

(A)

αβ2(α2+β2)\frac{\alpha\beta}{2(\alpha^2+\beta^2)}

(B)

αβα+β\frac{\alpha\beta}{\alpha+\beta}

(C)

α+β4\frac{\alpha+\beta}{4}

(D)

12α2+β22\frac{1}{2}\sqrt{\frac{\alpha^2+\beta^2}{2}}

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