The Greek Symbol \(\Sigma\)
Writing out \(1 + 2 + 3 + \dots + 100\) is tedious. We use the Greek letter Sigma as a shorthand for "add all terms from the bottom number to the top number."
\[\sum_{n=1}^{k} a_n\]
Worked Examples
Example 1: Reading Sigma
Evaluate \(\sum_{n=1}^{4} n^2\).
- This means \(1^2 + 2^2 + 3^2 + 4^2\).
- \(1 + 4 + 9 + 16 = 30\).
- Result: 30
Example 2: Writing in Sigma
Write \(5 + 10 + 15 + 20\) in sigma notation.
- The rule is \(5n\). The sum goes from \(n=1\) to \(n=4\).
- Result: \(\sum_{n=1}^{4} 5n\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, a particle's total wavefunction is a sum of its possible states: \(\Psi = \sum c_n \psi_n\). This Sigma notation is how we describe the Principle of Superposition. When we calculate the "Expectation Value" (the average result of a measurement), we are essentially performing a weighted sum using Sigma. It is the fundamental symbol for any system where multiple possibilities combine into one reality. If you see a \(\Sigma\) in physics, it means "the total result of all parts."