2.4: Problems
- Page ID
- 106809
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!
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)
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 \(m \neq 0\)
Given the function
\[f(r,\theta,\phi)=4 r e^{-2r/3} \sin{\theta}e^{-2i\phi/5}\]
Write down an expression for \(|f(r,\theta,\phi)|^2\)