Introduction to Quantum Mechanics · Problem 9.9

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · wkb-overlap-region-harmonic-oscillator

Problem

Consider a particle of mass

m

in the

n

th stationary state of the harmonic oscillator (angular frequency

\omega

). (a) Find the turning point,

x_2

. (b) How far (

d

) could you go above the turning point before the error in the linearized potential (Equation 9.33, but with the turning point at

x_2

) reaches 1%? That is, if

\frac{V(x_2+d) - V_\text{lin}(x_2+d)}{V(x_2)} = 0.01,

what is

d

? (c) The asymptotic form of

\text{Ai}(z)

is accurate to 1% as long as

z \geq 5

. For the

d

in part (b), determine the smallest

n

such that

\alpha d \geq 5

. (For any

n

larger than this there exists an overlap region in which the linearized potential is good to 1% and the large-

z

form of the Airy function is good to 1%.)