Varying Forces
In physics, Work is Force times Distance (\(W = F \cdot d\)). But if the force changes as you move (like a spring), we must use an integral:
\[W = \int_a^b F(x) dx\]
Worked Examples
Example 1: Stretching a Spring
According to Hooke's Law, the force to stretch a spring is \(F = kx\). How much work is needed to stretch it from \(x=0\) to \(x=A\)?
- Integral: \(\int_0^A kx dx = [\frac{1}{2}kx^2]_0^A = \frac{1}{2}kA^2\).
- Result: \(\frac{1}{2}kA^2\). This is the Potential Energy stored in the spring.
The Bridge to Quantum Mechanics
The concept of "Work" as an integral is the foundation of the Hamiltonian Operator. In Quantum Mechanics, we don't calculate work done by forces; instead, we use the potential energy \(V(x)\) which is the result of that work. For the Quantum Harmonic Oscillator, the potential energy is exactly \(\frac{1}{2}kx^2\). The fact that this energy grows quadratically is what "traps" the quantum particle, forcing its energy to be quantized into discrete levels.