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- 3. For free particles in two dimensions, what is density of states (DOS) in low speed limit (=p²/2m), and in high speed limit (=pc)?1. Given the following probability density function: p(x) = Ae¬^(x-a)². 2. A particle of mass, m, has the wavefunction given by: Þ(x,t) = Ce-a[(mx²/h) + it] . 3. In a few sentences, explain why it is impossible to calculate (p) in the first problem, whereas in the second problem this is straightforward. Highlight the key concepts that differentiate these problems.The infrared spectrum of 75Br19F consists of an intense line at 380. cm-1. Calculate its force constant, k, in units of N/m. (You can use the example as a sanity check.)
- 3. Prove the function y (x, y,=) = A Sin( L Sin( -)Sin( :) L, L, x, satisfies the Schrodinger's equation for a quantum dot with a width of Lx, Ly, and Lz, and the particle energies are, n. E, = 2m L L L with n,,n.,n, = 1,2,3,...neon (Ne) has in the same excited state: 18²282p30* 10 electrous. Fou create (Ne) atoms all p৮.dর हहा Hund's ruk apply to each energy level t the meon. eleitrons 1. calculate the spin , orbital and total anqular momenta of f a the cap) level oम (रंर) b. the (ad) fevel.4. Use the variational principle to estimate the ground state energy of a particle in the potential (∞0 x < 0 U(x) = \cx x≥0 Take xe-bx as a trial function.
- 4.a. Show that the most probable value of r ( H-atom radius) in la state is a. b: Write wavefunction of the 2s orbital W2,(r,0.6).2. Let the spin-1/2 particle be in an eigenstate of Sz, with eigenvalue 1/2. Compute the new state of the system when the system is rotated by an angle of 60◦. (a)about the z- axis (b)about the x- axis4. For the potential if 0 a. calculate the product of the uncertainties O„0, for the second excited state V3(x) : sin a a
- 4. Normalize the following wavefunctions 4 55 (a) v(x) = sin (#2); =sin(); for a particle in a 1D box of length L. (b) (2) = xe-z|2 (c) (x) = e(x²/a²)+(ikz) 5. In a region of space, a particle with mass m and with zero energy has a time- independent wave-function (x) = Ae-2/12, where A and L are constants. Use your knowledge of the Schrödinger equation to determine the potential energy V(x) of the particle. Plot the potential function? What is the minimum potential energy for the particle, if it is an electron and L = 1 fm? Is this potential repulsive or attractive?Q4A Photon lneident upon a bystogo, s adom ejects pm electoon uith! a KiE of 10?ev. f the ejected electron was in the fiast excited Stoute, Calculate the ene8 Lhoton. Whot K.E Woudvy cocrgy of have been Imparted to Om electron im State. Jrbund