a) Is the function y(x) = sin(ax/L) acceptable as a wave function in the range 0 < x < L. b) Are the functions y(x) = e-2x and y(x) = e-8x orthogonal? Justify your answer.

[p* L₂ da 0 HO H + →

(a) Consider a wavefunction of the form: =1+x-1

P7B.1 Imagine a particle confined to move on the circumference of a circle ('a particle on a ring'), such that its position can be described by an angle & in the range 0 to 2π. Find the normalizing factor for the wavefunctions: (a) e" and (b) eim, where m, is an integer.

Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinx

4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?

from x =0 tox = L. (b) What are the Si units of this unnormalized fur 2. (a) Determine the normalization constant for the particle in a box atypical wave function b which equais NyaL - 2 in the box from x-C tox=L and equals zero outside the box. You'll need to solve the integral below. (b) Explain how this function does (or does not) satisty the boum conditions for a particle in a box. 1= *dz here if the narticle is an electron, the sphere has a radlus

E7D.5(b) Calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L for the case where the wavefunction is (i) ₁, (ii) V₂. You may make the same approximation as in Exercise E7D.5(a). pr E7 po pr

9. The ground-state wavefunction for a particle confined to a one-dimensional box of length Lis 1/2 TCX L L Suppose the box is 10.0 nm long. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm. sin en

Which of the following functions are eigenfunctions of the momentum operator? a)eikx, b)eax2, c)x, d)x2, e)ax+b, d)sin(x+3a)Give the corresponding eigenvalue if appropriate.