Explore the spatial confinement and quantization of harmonics on a clamped stretched string, modeling the 1D quantum potential analogy.
A standing wave on a string fixed at both ends (length L) is a classic mechanical example of confinement. Clamping the endpoints forces the string displacement to be zero at x=0 and x=L. These boundary conditions restrict the wave to specific, discrete resonant frequencies (normal modes or eigenmodes).
In ,Quantum mechanics, this confinement has a close mathematical analogue in a particle (such as an electron) trapped in an infinite one-dimensional potential well ("particle in a box"). Like the string, the electron's probability amplitude (wave function) ψ(x) must vanish at the walls of the well. These boundary conditions permit only discrete wave functions and corresponding quantized energy levels.
| Mode (n) | Node Count | Wavelength (λ) | Frequency Ratio |
|---|---|---|---|
| 1st (Fundamental) | 0 (Endpoints excluded) | 2 L | f1 (1.0x) |
| 2nd Harmonic | 1 (At L/2) | L | 2 • f1 (2.0x) |
| 3rd Harmonic | 2 (At L/3, 2L/3) | 2L / 3 | 3 • f1 (3.0x) |
| 4th Harmonic | 3 (At L/4, L/2, 3L/4) | L / 2 | 4 • f1 (4.0x) |