5-37. We can use the harmonic-oscillator wave function to illustrate tunneling, a strictly quantum-mechancial property. The probability that the displacement of a harmonic os- cillator in its ground state lies between x and x + dx is given by 1/2 P(x) dx = vo(x) dx = e ax2 dx (1) The energy of the oscillator in its ground state is (h/2)(k/u)/2. Show that the greatest displacement that this oscillator can have classically is its amplitude h 71/2 A = ( ku ) 1/2 of 1/2 (2) According to equation 1, however, there is a nonzero probability that the displacement of the oscillator will exceed this classical value and tunnel into the classically forbidden region. Show that this probability is given by dz (3) The integral here cannot be evaluated in closed form but occurs so frequently in a number of different fields (such as the kinetic theory of gases and statistics) that it is well tabulated under the name complementary error function, erfc(x), which is defined as erfc(x) = 2 1/ 2 e- dz (4) By referring to tables, it can be found that the probability that the displacement of the molecule will exceed its classical amplitude is 0.16