please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is 2m (an + a) v(z, y) = E¼(x, y), for 0< 2, y < L. The bound- ary conditions on y are that it vanishes at the edges of the box. Verify that solutions are given by v(z, y) = 2 sin L sin ("), where n, ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as T ny show that the density of states g(k), defined such that the number of states with k| between k and k + dk is given by g(k)dk, is Ak g(k) : 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 o, = n = mk„T/2nh?. Hence show that the average energy of the particle is k,T, and explain the result.

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18. Please answer question throughly and detailed.

Consider a particle of mass m which can move freely along the x axis anywhere from x=-a/2 to x=a/2, but which is strictly prohibited from being found outside this region (technically this is the infinite square well potential). 1. Verify that the attached wavefunction is a solution of the Schrodinger's equation. Consider V=0.  2. Deduce the value of the energy E. 3. Verify the  uncertainty principle