147 lines
2.3 KiB
Markdown
147 lines
2.3 KiB
Markdown
# Plane Waves, Phase Twist, and the Corkscrew Picture
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## Key Realisation
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A quantum plane wave is **not just an oscillating wave**.
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It is a wave whose **phase is continuously twisting through space**.
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This spatial twist corresponds to **motion and momentum**.
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---
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## Start with the plane wave
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$$
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\psi(x,t) = e^{i(kx-\omega t)}
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$$
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Using Euler's identity:
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$$
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e^{i\theta} = \cos\theta + i\sin\theta
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$$
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So
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$$
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\psi(x,t) = \cos(kx-\omega t) + i\sin(kx-\omega t)
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$$
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The cosine part looks like an ordinary travelling wave.
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But the deeper structure is in the **phase**.
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---
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## The phase
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The phase is
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$$
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\theta(x,t) = kx - \omega t
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$$
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As we move through space:
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$$
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\frac{\partial \theta}{\partial x} = k
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$$
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So the phase **rotates steadily as position changes**.
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At every point in space the wavefunction behaves like a **tiny rotating vector (phasor)**.
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---
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## Corkscrew picture
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If we plot:
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- x = position
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- y = cos(kx)
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- z = sin(kx)
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the curve becomes a **helix**.
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So the plane wave can be visualised as a **corkscrew twisting through space**.
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The wave isn't just going up and down — its **phase is spiralling as it moves**.
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---
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## Momentum comes from the twist
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De Broglie discovered:
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$$
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p = \hbar k
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$$
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But
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$$
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k = \frac{\partial \theta}{\partial x}
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$$
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So momentum can be written as
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$$
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p = \hbar \nabla \theta
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$$
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Meaning:
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> **Momentum is the spatial rate of phase twist.**
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The faster the corkscrew twists through space, the larger the momentum.
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---
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## Kinetic energy and curvature
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Classically:
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$$
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T = \frac{p^2}{2m}
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$$
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Substituting $p = \hbar k$:
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$$
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T = \frac{\hbar^2 k^2}{2m}
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$$
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But the second derivative of the wave gives:
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$$
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\frac{\partial^2}{\partial x^2} e^{ikx} = -k^2 e^{ikx}
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$$
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So kinetic energy acting on the wave becomes
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$$
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T = -\frac{\hbar^2}{2m} \nabla^2
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$$
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This is why the **Laplacian appears in the Schrödinger equation**.
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---
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## Final intuition
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The wavefunction contains two things:
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**Amplitude**
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- how much probability is present
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**Phase**
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- how the probability flow moves through space
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The phase twists through space like a **corkscrew**, and that twist is directly related to **momentum and motion**.
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---
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## One‑line summary
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> A quantum plane wave is a **corkscrew of phase twisting through space**, and the rate of twist determines the particle's momentum.
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