What Does “i” Do in Schrödinger’s Equation?

What Does “i” Do Here?

The imaginary unit 𝑖 is not just a mathematical trick — it plays a critical role:

✔ 1. It Encodes Oscillations (Wave Behavior)

• Quantum particles are wave-like.
• The evolution of waves over time is naturally described by complex exponentials:

• These oscillations are at the heart of interference, superposition, and tunneling in quantum mechanics.

➡ Without i, the wave-like time evolution wouldn’t be possible.

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✔ 2. It Ensures Unitary Evolution

• In quantum mechanics, probabilities must be conserved over time.
• Time evolution via e−iHt/ℏ is unitary, meaning the total probability remains 1.
• If “i” were missing, the equation would predict growing or shrinking probabilities (non-physical).

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✔ 3. It Connects Time Evolution to Energy

• “i” helps define how a quantum system changes with time:

This complex exponential links energy (E) with how fast the wavefunction rotates in time.

• The rotation is a phase change—and interference depends on these phases.

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Analogy: Real-Life Example — Spinning Arrow

Think of the wave function as a spinning arrow (vector) on a clock face:

• The length of the arrow represents probability amplitude.
• The angle it makes (the phase) determines how it interferes with other arrows.
• The rotation of this arrow over time is driven by i.

No “i” → no rotation → no phase change → no interference patterns → no quantum magic.

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Table: Comparison of Equations With and Without “i”

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Graphical Representation: Oscillation vs. Exponential Decay

Let’s compare:

1. With “i”:

Produces oscillating (cyclic) motion:

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2. Without “i”:

Produces growth or decay – not suitable for describing quantum systems:

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Scientific Examples

▶ Electron in a Hydrogen Atom

• The electron’s behavior is described by a wave function with a phase that evolves with time.
• The interference of these phases gives rise to quantized energy levels.

▶ Quantum Tunneling

• The wave function extends into a region where classical particles cannot go.
• The complex phase (thanks to “i”) allows part of the wave to "leak through", explaining how tunneling occurs.

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Summary in Simple Terms

• The imaginary unit 𝑖 is not about imaginary things — it's essential to describing real quantum behavior.
• It allows waves to evolve, probabilities to stay consistent, and interference to happen.
• Without “i”, quantum mechanics would lose its core features and fail to describe the microscopic world.