By the end of this lesson, you should be able to:
- Write a complex number in Cartesian and polar form.
- Interpret magnitude and phase geometrically.
- Explain why phase can matter even when probability is unchanged.
Why complex numbers appear
Quantum amplitudes are generally complex numbers. A complex amplitude can be written in Cartesian form \(a+ib\) or polar form \(re^{i\phi}\), where \(r\) is its magnitude and \(\phi\) is its phase.
\[\alpha=a+ib=re^{i\phi}.\]
The probability associated with an amplitude is its squared magnitude, \(|\alpha|^2\). Phase does not appear directly in this probability, but phase differences become observable when amplitudes interfere.
Geometric interpretation
Use the interactive complex plane above. Change the magnitude to stretch or shrink the amplitude vector and change the phase to rotate it. Notice that a change in phase can leave the magnitude unchanged.
Euler's formula
The bridge between trigonometry and complex exponentials is
\[e^{i\phi}=\cos\phi+i\sin\phi.\]
This compact expression is fundamental in quantum gates, time evolution, interference, and the Bloch-sphere representation.
What should stay with you
- Magnitude controls probability weight.
- Relative phase controls how amplitudes later interfere.
- Euler’s formula connects phase rotations to sine and cosine.
If α changes from r to re^{iφ}, what stays unchanged?
Multiplication by a unit-magnitude phase factor rotates the amplitude without changing its magnitude.