Questions 1. a) (Very easy) Construct the density matrix for an electron beam that is composed of electrons either in the state spin up along z (with probability 1/4) or in the state spin down along y (with probability 3/4). b) (Very easy) Show if the ensemble is pure or mixed. c) (Very easy) Calculate langlehat{S}_{z} angle. 2. Consider a system of angular momentum l = 1. The basis of its state space is given by {|1 angle, |0 angle, |-1 angle}, which are the eigenstates of the z-component of the angular momentum operator L_{z}. Let the Hamiltonian for this system in this basis be hat{H} = hbar omega egin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 end{pmatrix} where omega is a real constant. a) (Very easy) Find the stationary states of the system and their energies. b) (Easy) At time t = 0, the system is in the state |psi(0) angle = frac{1}{sqrt{3}} { |1 angle + |0 angle - |-1 angle } Find the state vector |psi(t) angle at time t. c) (Easy) At time t the value of L_{z} is measured, find the probabilities of the various possible results. 3. Consider a one dimensional infinite-wall potential V = infty for x > L and x < 0, and V = 0 for 0 le x le L. There are two identical spin half fermions in this potential well and spin states are denoted by |uparrow angle and |downarrow angle for spin up and down, respectively. a) (Very easy) Find the eigenvalues and the corresponding wave functions for the potential well. b) (Easy) Write the full wave function for the fermions in singlet state with possible minimum energy. c) (Easy) Write the full wave function for the fermions in triplet state with possible minimum energy.
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The electron in a hydrogen atom occupies the combined spin and position state. $$ R_{21}\left(\sqrt{1 / 3} Y_{1}^{0} \chi_{+}+\sqrt{2 / 3} Y_{1}^{1} \chi_{-}\right) $$ (a) If you measured the orbital angular momentum squared $\left(L^{2}\right)$, what values might you get, and what is the probability of each? (b) Same for the $z$ component of orbital angular momentum $\left(L_{z}\right)$. (c) Same for the spin angular momentum squared $\left(S^{2}\right)$. (d) Same for the $z$ component of spin angular momentum $\left(S_{z}\right)$. Let $\mathbf{J} \equiv \mathbf{L}+\mathbf{S}$ be the total angular momentum. (e) If you measured $J^{2}$, what values might you get, and what is the probability of each? (f) Same for $J_{z}$. (g) If you measured the position of the particle, what is the probability density for finding it at $r, \theta, \phi$ ? (h) If you measured both the $z$ component of the spin and the distance from the origin (note that these are compatible observables), what is the probability density for finding the particle with spin up and at radius $r ?$
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2. Put an electron at the origin of a Cartesian coordinate system and let the spin point in the +z-direction. Treat the electron as a magnetic moment of strength m = -gμBmsẑ. Here g is the g-factor and is approximately equal to 2, μB is the Bohr Magneton, and ms is the z-component of the spin angular momentum of the electron. This is equal to ½. For the vector potentials in this problem use the Coulomb gauge. a. What is the actual value of the magnetic moment of the electron? b. What is the vector potential at the point z = 0.1nm directly above the electron, that is at z = 0.1nm? c. What is the vector potential 0.2 and 0.5 nm above the electron? d. What is the magnetic field at these three points? Use B = ∇ × A and the formula we derived for the vector potential from a magnetic dipole. You will find that using the Levi-Civita symbol simplifies calculating the derivatives substantially. This is essentially what you did in discussion section on Monday. Make sure you show your calculation. e. What is the vector potential at points z = y = 0 and x = 0.1nm , x = 0.2 nm and x = 0.5nm . Make sure you write these as vector expressions. f. What is the magnetic field at those points? g. Put another electron at the six points listed above, three on the x-axis and 3 on the z-axis. If its spin is pointed in the +z direction what are the magnetic potential energies of the system in each case? If its spin is pointed in the -z direction what are the magnetic potential energies in each case? Ignore the coulomb interaction.
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