# 2.4: Problems

• • Contributed by Marcia Levitus

Note: Always express angles in radians (e.g. $$\pi/2$$, not $$90^{\circ}$$). When expressing complex numbers in Cartesian form always finish your work until you can express them as $$a+bi$$. For example, if you obtain $$\frac{2}{1+i}$$, multiply and divide the denominator by its complex conjugate to obtain $$1-i$$.

Remember: No calculators allowed!

Problem $$\PageIndex{1}$$

Given $$z_1=1+i$$, $$z_2=1-i$$ and $$z_3=3e^{i \pi/2}$$, obtain:

• $$z_1 z_2$$
• $$z_1^2$$
• $$2z_1-3z_2$$
• $$|z_1|$$
• $$2z_1-3z_2^*$$
• $$\frac{z_1}{z_2}$$
• Express $$z_2$$ as a complex exponential
• $$|z_3|$$
• $$z_1+z_3$$, and express the result in cartesian form
• Display the three numbers in the same plot (real part in the $$x$$-axis and imaginary part in the $$y$$-axis)

Problem $$\PageIndex{2}$$

The following family of functions are encountered in quantum mechanics:

$\Phi_m(\phi)=\frac{1}{\sqrt{2 \pi}}e^{i m \phi}, m= 0, \pm 1,\pm 2, \pm 3 \dots, 0 \le \phi \le 2\pi$

Notice the difference between $$\Phi$$ (the name of the function), and $$\phi$$ (the independent variable). The definition above defines a family of functions (one function for each value of $$m$$). For example, for $$m=2$$:

$\Phi_2(\phi)=\frac{1}{\sqrt{2 \pi}}e^{2i \phi},$

and for $$m=-2$$:

$\Phi_{-2}(\phi)=\frac{1}{\sqrt{2 \pi}}e^{-2i \phi},$

• Obtain $$|\Phi_m(\phi)|^2$$
• Calculate $$\int_0 ^{2\pi}|\Phi_m(\phi)|^2 \mathrm{d}\phi$$
• Calculate $$\int_0 ^{2\pi}\Phi_m(\phi)\Phi_n^*(\phi) \mathrm{d}\phi$$ for $$m \neq n$$
• Calculate $$\int_0 ^{2\pi}\Phi_m(\phi) \mathrm{d}\phi$$ for $$m = 0$$
• Calculate $$\int_0 ^{2\pi}\Phi_m(\phi) \mathrm{d}\phi$$ for $$m \neq 0$$

Problem $$\PageIndex{3}$$

Given the function

$f(r,\theta,\phi)=4 r e^{-2r/3} \sin{\theta}e^{-2i\phi/5}$

Write down an expression for $$|f(r,\theta,\phi)|^2$$