2. Verify that: a. b. G), + 6), (²), P S (), = C), (3) (start from the fundamental property relation). = 0
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- 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 - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L 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 ("", ""), 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 9(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 oq = ng = mk„T/2nh?. Hence show that the average…53. Please answer question throughly and detailed.
- 1. The Hamiltonian of the qubit in the standard basis is given by H = X⁰⁰ - X¹1 - ¡Xº¹ + ix¹⁰ (in units of eV). Find the possible values of the qubit energy E, and E₁ (in eV). Give the answer in decimals with accuracy to 3 significant figures.Consider the sheet formed by the intersection of the curves: x = 0, x = 4, y = 0, y = 3 [=] cm, with a variable density of mass per unit area ρ(x,y) = xy [=] g/cm2 . Write and evaluate multiple integrals to calculate the following: a. The area of the sheet [=] cm2 . b. The mass of the sheet [=] g. c. The shell moments about the x & y axes (Mx & My) [=] g∙cm. d. The position of the center of mass of the sheet ( , ) [=] cm.