Question 4.P.12: Calculate the probability of finding a particle in the class...

The classical turning points are defined by E_{n} =V(x_{n} ) or by \hbar \omega (n+\frac{1}{2} )=\frac{1}{2}m\omega ^{2} x^{2}_{n}; that is, x_{n}=\pm \sqrt{\hbar /(m \omega )} \sqrt{2n+1}. Thus, the probability of finding a particle in the classically forbidden region for a state \psi _{n}(x) is

P_{n} =\int_{-\infty }^{-|x_{n}|}\left|\psi _{n}(x)\right| ^{2} dx+\int_{|x_{n}|}^{+\infty }\left|\psi _{n}(x)\right| ^{2}dx=2 \int_{|x_{n}|}^{+\infty }\left|\psi _{n}(x)\right| ^{2}dx,       (4.297)

where \psi _{n}(x) is given in (4.172),

\psi _{n}(x)=\frac{1}{\sqrt{\pi }2^{n}n!x_{0}} e^{-x^{2}/2 x^{2}_{0}} H_{n}\left(\frac{x}{x_{0} } \right) .           (4.172)

\psi _{n}(x)=1/\sqrt{\sqrt{\pi }2^{n}n!x_{0} } e^{-x^{2} /2x^{2}_{0}}H_{n}(x/x_{0}), where x_{0} is given by x_{0}=\sqrt{\hbar /(m\omega )}. Using the change of variable y=x/x_{0}, we can rewrite P_{n} as

P_{n}=\frac{2}{\sqrt{\pi }2^{n}n! } \int_{\sqrt{2n+1} }^{+\infty }e^{-y^{2}}H^{2}_{n}(x) dy,                      (4.298)

where the Hermite polynomials H_{n}(y) are listed in (4.120).

H_{0}(y)=1,          H_{1}(y)=2y,

H_{2}(y)=4y^{2} -2,          H_{3}(y)=8y^{2}-12y,

H_{4}(y)=16y^{4}-48y^{2}-12y+12,         H_{5}(y)=32y^{5}-160y^{3}+120y.

The integral in (4.298) can be evaluated only numerically. Using the numerical values

\int_{1}^{\infty } e^{-y^{2}}dy=0.1394,           \int_{\sqrt{3} }^{\infty }y^{2}e^{-y^{2}}dy=0.0495,            (4.299)

\int_{\sqrt{5} }^{\infty }(4y^{2}-2)^{2} e^{-y^{2}}dy=0.6740,           \int_{\sqrt{7} }^{\infty }(8y^{3}-12y)^{2}e^{-y^{2}}dy=3.6363,                      (4.300)

\int_{\sqrt{9} }^{\infty }(16y^{4}-48y^{2}+12)^{2}e^{-y^{2}}dy=26.86,                  (4.301)

we obtain

P_{0}=0.1573,           P_{1}=0.1116,           P_{2}=0.095 069,        (4.302)

P_{3}=0.085 48,               P_{4}=0.078 93.        (4.303)

This shows that the probability decreases as n increases, so it would be very small for very large values of n. It is therefore unlikely to find the particle in the classically forbidden region when the particle is in a very highly excited state. This is what we expect, since the classical approximation is recovered in the limit of high values of n.