4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite squarewell of length L.Construct the state vector and determine the energy for the ground-state of thetwo-particle system assuming the particles are non-interacting.i.ii.Determine the first excited state energy of the two-particle system and give all itsstate vectors, again assuming the particles are non-interacting. What is thedegeneracy of the first excited state?ji.Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where kis positive and small. Discuss quantitatively how the ground and first-excited stateenergies differ from the case for non-interacting particles.5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus ofradius ry. Assume electric charge +Ze is uniformly distributed throughout the volume ofthe nucleus. Ignoring spin, use first order non-degenerate perturbation theory and thehydrogenic wave functions adapted to the one-electron atom to determine the dependenceof the ground state energy of the atom on ry.

For fermions, in the ground state, one particle will be spin up and the other particle will be spin…

Answered: 4. Consider two indistinguishable…

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7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -
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5. Obtain the the expression for the density of states for a system of free electrons in a three-dimensional infinite square well.
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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- axis
1) Make use of a translation operator and prove Bloch's theorem in the form : y (F+R)=e¹ky (7). An alternative equivalent form for Bloch's theorem is that the wavefunction has the form 7)=eku (F) where u (F) is lattice periodic. By substituting this into the Schrodinger equation explain the origin of energy bands.
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2. Spinors and rotations (a) For a function f(ø) that can be expanded in a Taylor series, show that f(o +4) = e²L±p/hf(6) where p is an arbitrary angle. For this reason, Lz/ħ is called the generator of rotations about the z-axis. More generally, L.ân/ħ is the generator of rotations about the direction în, in the sense that exp(iL.ng/h) effects a rotation through angle y (in the right hand sense) about the axis în. In the case of spin, the generator of rotations is S.în/ħ. In particular, for spin 1/2., X = ei(oî)g/2x tells us how spinors rotate. (b) Construct the (2x2) matrix representing rotation by 180° about the x-axis and show it converts spin up (x+) spin down (x-) as you would expect. (c) Construct the matrix representing the rotation by 90° about the y-axis, and check what it does to X+· (d) Construct the matrix representing rotation by 360° about the z-axis. Discuss the meaning of the answer. (e) Show that ei(o-î)e/2 = cos(p/2) + i(în · o) sin(4/2)
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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.