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Scientific claim

Superposition Principle for Pure Quantum States

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Claim

Let H\mathcal H be the complex Hilbert space associated with a quantum system, and let ∣ψ1⟩,∣ψ2⟩∈H|\psi_1\rangle,|\psi_2\rangle\in\mathcal H represent possible pure states. For any α,β∈C\alpha,\beta\in\mathbb C such that

∣ψ⟩=α∣ψ1⟩+β∣ψ2⟩≠0,|\psi\rangle=\alpha|\psi_1\rangle+\beta|\psi_2\rangle\neq0,

the ray

[ψ]={c∣ψ⟩:c∈C, c≠0}[\psi]=\{c|\psi\rangle:c\in\mathbb C,\ c\neq0\}

also represents a possible pure state of the system.

Theoretical: Supported

Assumptions

No assumptions recorded.

Proof

Because H\mathcal H is a vector space, it is closed under linear combinations. Therefore, if

∣ψ1⟩,∣ψ2⟩∈H|\psi_1\rangle,|\psi_2\rangle\in\mathcal H

and α,β∈C\alpha,\beta\in\mathbb C, then

∣ψ⟩=α∣ψ1⟩+β∣ψ2⟩∈H.|\psi\rangle=\alpha|\psi_1\rangle+\beta|\psi_2\rangle\in\mathcal H.

If ∣ψ⟩≠0|\psi\rangle\neq0, it determines the ray

[ψ]={c∣ψ⟩:c∈C, c≠0}.[\psi]=\{c|\psi\rangle:c\in\mathbb C,\ c\neq0\}.

By the Pure Quantum State Representation Postulate, every ray in the system’s Hilbert space represents a possible pure quantum state. Therefore, [ψ][\psi] represents a possible pure state.

If a normalized representative is required, define

∣ψ~⟩=∣ψ⟩∥ψ∥.|\tilde\psi\rangle=\frac{|\psi\rangle}{\|\psi\|}.

Then

⟨ψ~∣ψ~⟩=⟨ψ∣ψ⟩∥ψ∥2=1.\langle\tilde\psi|\tilde\psi\rangle = \frac{\langle\psi|\psi\rangle}{\|\psi\|^2} =1.

Thus, every nonzero linear combination of possible pure-state vectors determines another possible pure quantum state.

This is a mathematical consequence of the Hilbert-space structure and the Pure Quantum State Representation Postulate, rather than an independently proven physical theorem. Its applicability to physical systems is supported experimentally by quantum interference.

Examples

  1. Spin-12\frac12 superposition

    Let ∣↑⟩|\uparrow\rangle and ∣↓⟩|\downarrow\rangle represent spin-up and spin-down along the zz-axis. Then

    ∣+⟩=∣↑⟩+∣↓⟩2|+\rangle= \frac{|\uparrow\rangle+|\downarrow\rangle}{\sqrt2}

    represents a possible pure state.

    This is not a classical 5050–5050 mixture. Its components have a definite relative phase and can produce interference in measurements made along other axes.

  2. Two-path superposition

    If ∣L⟩|L\rangle and ∣R⟩|R\rangle represent two distinct paths available to a quantum system, then

    ∣ψ⟩=∣L⟩+eiϕ∣R⟩2|\psi\rangle= \frac{|L\rangle+e^{i\phi}|R\rangle}{\sqrt2}

    represents a coherent pure-state superposition, where ϕ∈R\phi\in\mathbb R is the relative phase.

    Changing ϕ\phi changes the interference pattern observed when the two paths are recombined.

Evidence

01
Quantum Superposition: How Qubits Live in Many States at OnceMarin Ivezic · 2017-05-10 · supporting
02
Understanding Quantum Mechanics #2: Superposition and EntanglementSabine Hossenfelder · 2020-05-15 · supporting

Related Concepts

No dependency connections recorded.