The Scale of the Universe
Quantum Mechanics deals with the incredibly small. Writing out "0.00000000000000000000000000000000066" for Planck's constant is impossible. We use Scientific Notation to make these numbers manageable.
Format: \(a \times 10^n\), where \(1 \leq a < 10\).
Calculating with Notation
- Multiplication: Multiply the numbers, add the 10-exponents.
- Division: Divide the numbers, subtract the 10-exponents.
Worked Examples
Example 1: Converting Small Numbers
Write \(0.0005\) in scientific notation.
- Move the decimal 4 spots right to get 5.0.
- Since we moved right, the exponent is negative.
- Result: \(5.0 \times 10^{-4}\)
Example 2: Multiplying Constants
Calculate \((2 \times 10^8) \times (3 \times 10^{-5})\).
- Numbers: \(2 \times 3 = 6\).
- Exponents: \(8 + (-5) = 3\).
- Result: \(6 \times 10^3 = 6000\)
Example 3: Division
Calculate \(\frac{8 \times 10^4}{2 \times 10^{-2}}\).
- Numbers: \(8 / 2 = 4\).
- Exponents: \(4 - (-2) = 6\).
- Result: \(4 \times 10^6\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, we constantly use three fundamental constants:
- Planck's constant: \(h \approx 6.63 \times 10^{-34} \text{ J}\cdot\text{s}\)
- Mass of an electron: \(m_e \approx 9.11 \times 10^{-31} \text{ kg}\)
- Speed of light: \(c \approx 3.00 \times 10^8 \text{ m/s}\)