The wavefunction for the quantum harmonic oscillator is n(x) = 1/(^(1/4) n! 2n) * e^(x^2/(2^2)) * Hn(x/),

^2 = h/m , n = 0, 1, 2, . . .

Here m is the particle mass, is the oscillator frequency, and Hn(x) is the Hermite polynomial of order n. This model permits particles to travel a greater distance from its equilibrium point than classical physics would allow, and is a vital feature for understanding of solids.

(a) Give the range of particle positions that would be allowed by classical physics in terms of n and .

(b) Write, but do not attempt to evaluate, an integral expression in terms of dimensionless variable = x/ for the probability that a particle has a position within its allowed classical range given that the particle is in excitation state n. Clearly state the limits of integration.

(c) Write a short computer program that evaluates this integral numerically using the trapezoid rule for any valid excitation state. Use only basic coding language functions: arithmetic, loops, comparisons, exponentiation, and standard math functions such as exponentials, logarithms, and trigonometric functions. Attach all computer code with your assignment submission. To receive full credit, your coding must be commented and easy to understand.

(d) Report the probabilities for n = 0, 1, 2 from your program and discuss the trend as n gets large. Use a sufficient number of integration points to achieve at least three decimal places of accuracy.