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Physics Quantum Mechanics Level: High School

Consider a one-dimensional harmonic oscillator . For the classical oscillator , the amplitude is given by . A = Sqrt h/m omega Which is gives the turning point for the classical oscillator. This is the point at which the kinetic energy is zero and the potential energy is a maximum , ie . the particle stops and reverses direction .

a) For the ground state wave function , show that its curvature (second derivative) is zero at the classical turning point .

b) Calculate the probability that the particle will be in the classically forbidden region (beyond the turning point) by calculating the probability of finding the particle between the turning point and infinity, A<= x <=infinity for n = 0 and n = 4 .