Chapter 8: The WKB Approximation

Use equation 8.22 calculate the approximate transmission probability for a particle of energy E that encounters a finite square barrier of height V0 > E and width 2a. Compare your answer with the exact result to which it should reduce in the WKB regime T << 1.

Calculate the lifetimes of ${\mathit{U}}^{\mathbf{238}}$and$\mathit{P}{\mathit{o}}^{\mathbf{212}}$, using Equations8.25and 8.28 . Hint: The density of nuclear matter is relatively constant (i.e., the same for all nuclei), so${\left(r_1\right)}^3$is proportional to (the number of neutrons plus protons). Empirically,

${\mathbf{r}}_{\mathbf{1}}\mathbf{=}\left(1.07\mathrm{fm}\right){\mathbf{A}}^{\mathbf{1}\mathbf{/}\mathbf{3}}$

Consider the quantum mechanical analog to the classical problem of a ball (mass m) bouncing elastically on the floor.
(a) What is the potential energy, as a function of height x above the floor? (For negative x, the potential is infinite x - the ball can't get there at all.)
(b) Solve the Schrödinger equation for this potential, expressing your answer in terms of the appropriate Airy function (note that Bi(z) blows up for large z, and must therefore be rejected). Don’t bother to normalize $\mathbf{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.
(c) Using $\mathbf{g}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{80}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{}$and $\mathbf{}\mathbf{m}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{100}\mathbf{}\mathbf{kg}$ , find the first four allowed energies, in joules, correct to three significant digits. Hint: See Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover, New York (1970), page 478; the notation is defined on page 450.
(d) What is the ground state energy, in ,eV of an electron in this gravitational field? How high off the ground is this electron, on the average? Hint: Use the virial theorem to determine $\mathbf{<}\mathbf{x}\mathbf{>}$ .

Analyze the bouncing ball (Problem 8.5) using the WKB approximation.
(a) Find the allowed energies,${\mathit{E}}_{\mathbf{n}}$ , in terms of , and .
(b) Now put in the particular values given in Problem8.5 (c), and compare the WKB approximation to the first four energies with the "exact" results.
(c) About how large would the quantum number n have to be to give the ball an average height of, say, 1 meter above the ground?

Use the WKB approximation to find the allowed energies of the harmonic oscillator.

Consider a particle of massm in the n th stationary state of the harmonic oscillator (angular frequency $\mathit{\omega }$ ).

(a) Find the turning point, x2 .
(b) How far (d) could you go above the turning point before the error in the linearized potential reaches 1%? That is, if $\frac{\mathbf{V}\left(x_2+d\right)\mathbf{-}{\mathbf{V}}_{\mathbf{Iin}}\left(x_2+d\right)}{\mathbf{V}\left(x_2\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}$ what is ?
(c) The asymptotic form of Ai(z) is accurate to 1% as long as localid="1656047781997" $\mathbf{z}\mathbf{\ge }\mathbf{5}$. For the din part (b), determine the smallest nsuch that$\mathit{\alpha }\mathit{d}\mathbf{\ge }\mathbf{5}$ . (For any n larger than this there exists an overlap region in which the liberalized potential is good to 1% and the large-z form of the Airy function is good to 1% .)

Derive the connection formulas at a downward-sloping turning point, and confirm equation8.50.

$\mathit{\psi }\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}\frac{D\text{'}}{\sqrt{\left|p\right(x\left)\right|}}exp\left[-\frac{1}{h}{\int }_x^{x_1}\left|p\right(x\text{'}\left)\right|dx\text{'}\right].ifx<x_1\\ \frac{2D\text{'}}{\sqrt{p\left(x\right)}}sin\left[-\frac{1}{h}{\int }_{x_1}^{x}p(x\text{'})dx\text{'}+\frac{\pi }{4}\right].ifx<x_1\end{array}$