14:01
University Physics with Modern Physics
The WKB approximation (see Challenge Problem 40.57$)$ can be used to calculate the energy levels for a harmonic oscillator. In this approximation, the energy levels are the solutions to the equation
$$
\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad n=1,2,3, \dots
$$
Here $E$ is the energy, $U(x)$ is the potential-energy function, and $x=a$ and $x=b$ are the classical turning points (the points at which $E$ is equal to the potential energy, so the Newtonian kinetic
energy would be zero). (a) Determine the classical turning points for a harmonic oscillator with energy $E$ and force constant $k$ ' (b) Carry out the integral in the WKB approximation and show that the energy levels in this approximation are $E_{n}=\hbar \omega,$ where $\omega=\sqrt{k^{\prime}} / m$ and $n=1,2,3, \ldots . .(\text { Hint: Recall that } \hbar=h / 2 \pi$ A useful standard integral is
$$
\int \sqrt{A^{2}-x^{2}} d x=\frac{1}{2}\left[x \sqrt{A^{2}-x^{2}}+A^{2} \arcsin \left(\frac{x}{|A|}\right)\right]
$$
where arcsin denotes the inverse sine function. Note that the inte- grand is even, so the integral from $-x$ to $x$ is equal to twice the integral from 0 to $x$ . (c) How do the approximate energy levels
found in part (b) compare with the true energy levels given by Eq. $(40.26) ?$ Does the WKB approximation give an underestimate or an overestimate of the energy levels?