2.4: Problems

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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 \(|f(r,\theta,\phi)|^2\)