Problem 30
Operate with (a) \(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\) and (b) \(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\) on the function \(A e^{-i k_{1} x} e^{-i k_{1} y} e^{-i k_{1} z}\) Under what conditions is the function an eigenfunction of one or both operators? What is the eigenvalue?
Problem 31
Form the operator \(\hat{A}^{2}\) if \(\hat{A}=x-d / d x .\) Be sure to include an arbitrary function on which the operator acts.
Problem 34
Show that the following pairs of wave functions are orthogonal over the indicated range. a. \(e^{-\alpha x^{2}}\) and \(x\left(x^{2}-1\right) e^{-\alpha x^{2}},-\infty \leq x<\infty\) where \(\alpha\) is a constant that is greater than zero b. \(\left(6 r / a_{0}-r^{2} / a_{0}^{2}\right) e^{-r / 3 a_{0}}\) and \(\left(r / a_{0}\right) e^{-r / 2 a_{0}} \cos \theta\) over the interval \(0 \leq r<\infty, 0 \leq \theta \leq \pi, 0 \leq \phi \leq 2 \pi\)
Problem 35
Express the following complex numbers in the form \(a+i b\) a. \(2 e^{3 i \pi / 2}\) b. \(4 \sqrt{3} e^{i \pi / 4}\) c. \(e^{i \pi}\) d. \(\frac{\sqrt{5}}{1+\sqrt{2}} e^{i \pi / 4}\)
Problem 36
Which of the following wave functions are eigenfunctions of the operator \(d / d x ?\) If they are eigenfunctions, what is the eigenvalue? a. \(a e^{-3 x}+b e^{-3 i x}\) b. \(\sin ^{2} x\) c. e^{-i x}\( d. \)\cos a x\( e. \)e^{-i x^{2}}$
Problem 37
Form the operator \(\hat{A}^{2}\) if \(\hat{A}=d^{2} / d y^{2}+3 y(d / d y)-5 .\) Be sure to include an arbitrary function on which the operator acts.
