Tag: Solutions

Problem 3.7 (a) Suppose that and are two eigenfunctions of an operator , with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of , with eigenvalue q.(b) Check that and are eigenfunctions of the operator , with the same eigenvalue. Construct two linear combinations of f…

Problem 3.6 Consider the operator , where (as in Example 3.1) is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is hermitian? Find its eigenfunctions and eigenvalues. What is the spectrum of ? Is the spectrum degenerate? Solution: Problem 3.6 Solution (Download)

Problem 3.4 (a) Show that the sum of two hermitian operators is hermitian.(b) Suppose is hermitian, and is a complex number. Under what condition (on ) is hermitian?(c) When is the product of two hermitian operators hermitian?(d) Show that the position operator and the Hamiltonian operator are hermitian. Solution: Problem 3.4 Solution (Download)

Problem 3.5 (a) Find the hermitian conjugates of x, i, and .(b) Show that (note the reversed order), and for a complex number c.(c) Construct the hermitian conjugate of (Equation 2.48). Solution: Problem 3.5 Solution (Download)

Problem 3.3 Show that if for all h (in Hilbert space), then for all f and g (i.e. the two definitions of “hermitian”—Equations 3.16 and 3.17—are equivalent). Hint: First let , and then let . Solution: Problem 3.3 Solution (Download)

Problem 3.2 (a) For what range of is the function in Hilbert space, on the interval ? Assume is real, but not necessarily positive.(b) For the specific case , is in this Hilbert space? What about ? How about ? Solution: Problem 3.2 Solution (Download)

Problem 3.1 (a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). Hint: The main point is to show that the sum of two square-integrable functions is itself square-integrable. Use Equation 3.7. Is the set of all normalized functions a vector space?(b) Show that the…