Chapter #2 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Evaluate all of the following, and express all of your final answers in the form a + bi:(a) i (2 − 3i)(3 + 5i)(b) i/(i − 1)(c) (1 + i)30 Get solution

2. In the complex plane, there are 5 different fifth roots of 1. Determine the five values for ..., and express them in polar form. Get solution

3. Suppose that z = 1 + eiθ. Calculate z*, z2, and |z|2. Your expression for |z|2 should not contain any imaginary numbers. Get solution

4. Suppose that a complex number z has the property that z* = z. What does this indicate about z? Get solution

5. Reduce ii to a real number. Get solution

6. What is wrong with the following argument?...Therefore,... Get solution

7. Determine which of the following are linear operators, and which are not.(a) The parity operator Π[f(x)] = f(−x).(b) The translation operator T[f(x)] = f(x + 1).(c) The operator L[f(x)] = f(x) + 1. Get solution

8. Consider the identity operator I, defined by I[f(x)] = f(x).(a) Show that I is a linear operator.(b) Find the eigenfunctions and corresponding eigenvalues of I. Get solution

9. Suppose that the function f(x) is an eigenfunction of the linear operator P with eigenvalue p, and f(x) is also an eigenfunction of the linear operator Q with eigenvalue q. Show that PQ[f(x)] = QP[f(x)], where PQ[f(x)] means to first apply the operator Q to f(x), and then apply P to the result. Get solution

10. Consider the square of the derivative operator D2.(a) Show that D2 is a linear operator.(b) Find the eigenfunctions and corresponding eigenvalues of D2.(c) Give an example of an eigenfunction of D2 which is not an eigenfunction of D. Get solution

11. Let f(x) be an eigenfunction of a linear operator L with eigenvalue a. Show that cf(x) (where c is a constant) is an eigenfunction of L with eigenvalue a. Get solution

12. Consider the following operator L:...(a) Show that L is a linear operator.(b) Find the eigenfunctions of L, or show that L has no eigenfunctions. Get solution