Physics 751 Homework #4

Due Friday October 3, 11:00 am.

1. (a) Define the derivative of the delta function by

Sketch this function for small D. What is the value of ?

(b) What function has the delta function for its derivative? Explain briefly.

2. On the interval (-1, 1) the polynomials 1, x, x², x³, … are a basis. Use the Gram-Schmidt orthogonalization procedure to find the first four members of an orthonormal basis constructed from the polynomials.

3. Suppose we define the inner product for real functions defined on the infinite line to be:

Starting with the polynomials 1, x, x², x³, … , construct the first four members of an orthonormal basis.

4. Solve the time-independent Schrödinger equation in one dimension for an attractive delta function potential V(x) = ld(x) to find the energy of a bound state. Can there be more than one bound state? Explain.

5. For a one-dimensional general time-dependent solution of Schrödinger’s equation, prove:

(a) How would the result change if the limits of integration were finite? Can you write an equation for that case?

(b)

(c)