The time-independent Schrodinger equation is an eigenvalue problem. Consider a mass m in one dimension under the influence of a restoring force, and thus a parabolic potential energy function of the form U(x) = (1/2) m omega^2 x^2 (the 1D harmonic oscillator). (a) Write down the time-independent Schrodinger equation. (b) Show explicitly that psi(x) = A exp(-m omega x^2 / (2 hbar)) is a solution (eigenfunction) of the time-independent Schrodinger equation, and find the corresponding energy E. Here A is a normalization constant that does not depend on x; we do not need its explicit form.
Consider a photon with initial energy E_gamma undergoing Compton scattering with an electron initially at rest. After the collision, the photon is scattered at an angle theta relative to its initial direction, and the electron recoils at an angle phi. (a) Derive the relation between the energy of the incident photon and the scattered photon. (b) Find the relation between the angle of the recoiling electron and the incident photon. (c) Determine the kinetic energy of the recoiling electron.
(1) The Hamiltonian operator for a two-state system is given by H = a(|1><1| - |2><2| + |1><2| + |2><1|), where a is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets. (2) Consider the space of polynomials of degree less than or equal to n. Any polynomial p(x) = a_0 + a_1 x + ... + a_n x^n can be written as a vector (a_0, a_1, ..., a_n), where the a_i are complex numbers; this is the representation of p(x) in the basis x^0, x^1, ..., x^n. What is the matrix representation of the operator d/dx in this basis?
A 51-foot wire running from the top of a tent pole to the ground makes an angle of 56 degrees with the ground. If the length of the tent pole is 44 feet, find the distance from the bottom of the tent pole to the point where the wire is fastened to the ground. (There may be more than one possible answer; give all of them, rounded to the nearest whole number.)
Prove the following theorem. If A and B are sets, then the intersection of the power sets of A and B is a subset of the power set of the intersection of A and B; that is, P(A) intersect P(B) is a subset of P(A intersect B).
A tournament is a directed graph in which every pair of vertices has exactly one edge between them (in one direction). The name comes from thinking of the graph as a competition: if the edge between x and y is directed towards y, then x beat y in their match. Prove that any n-vertex tournament contains a vertex of out-degree at least (n - 1)/2.
A (possibly infinite) graph is called a tree if it is connected and acyclic. For k >= 1, a tree is called k-regular if every vertex has exactly k neighbors. Prove that, given a vertex v in a (q + 1)-regular tree and a positive integer i, there are exactly (q + 1) q^(i-1) vertices at distance i from v.
An R-module is called cyclic if it is generated by a single element. (a) Show that an R-module is cyclic if and only if it is isomorphic to R/I for some ideal I. (b) Show that the ideal I in part (a) is unique if R is commutative. (c) Let F be a field, and let M_n(F) be the ring of n x n matrices with coefficients in F. The vector space F^n, considered as a space of column vectors, is a left module over M_n(F) via matrix multiplication. Prove that it is a cyclic module, and find a left ideal I of M_n(F) such that M_n(F)/I is isomorphic to F^n. Is there more than one ideal with this property?
Let V be a complex vector space with ordered basis B = {e1, e2, e3, e4}. Consider the linear transformation T such that T(e1) = e2, T(e2) = e3, T(e3) = e4, T(e4) = e1. (a) Find the matrix representation of T with respect to B. (b) Find the eigenvalues of T. (c) Find the eigenvectors of T. (d) Show that the eigenvectors of T form a basis C of V. (e) Find the transition matrix P_{B,C}.
Consider the matrix A with rows (1, 6) and (3, -2). (a) Find all eigenvalues and corresponding eigenvectors of A. (b) Find an invertible matrix U and a diagonal matrix D such that A = U D U^(-1), and use this to compute A^123 applied to the vector (-1, 1).
