When the signs of the wave amplitudes align, they exhibit constructive interference, and when they oppose each other, they exhibit destructive interference.
Two waves of the same amplitude and period constructively or destructively interfere depending on their relative phase.
Note that they completely cancel each other when they have a relative phase of ϕ = π
The Wave Function
In quantum physics, particles are mathematically described by wave functions:
Particle in Infinite Square Well:
Particle in Harmonic Oscillator Potential:
Here’s where things get interesting: A particle can actually interfere with ITSELF. Let’s see how this happens.
According to Huygens Principle, every point on a wavefront is a source of wavelets.
When a particle passes through a narrow slit, its wavefront propagates radially on the other side of the screen
If we place a screen with two more slits in front of the propagating wave, we get two more wave fronts.
At each point on the back screen, the resulting amplitude is the superposition of the waves propagating from both slits.
The waves travel different distances to get to any point on the back screen, so by the time they get there they are shifted by some phase ϕ.
At points where the two wave-fronts differ by a phase of π, they completely destructively interfere, resulting in 0 probability of observing the particle there. This patently non-classical behavior was one of the first indications that a new model (quantum mechanics) was needed to model certain physical phenomena!
Investigate single slit diffraction where the slit admits a continuum of wave-fronts