Pure Quantum State Representation Postulate

This is a foundational postulate of standard quantum mechanics, so it is not derived from more fundamental quantum-mechanical propositions within this node.

Let $\mathcal{H}$ be the complex Hilbert space associated with a quantum system.

Let $\psi\in\mathcal{H}$ be a non-zero vector.

The ray generated by $\psi$ is:

The postulate assigns one pure physical state to each ray $[\psi]$, rather than to one particular non-zero vector $\psi$.

This means that any two non-zero vectors $\phi,\psi\in\mathcal{H}$ represent the same pure quantum state exactly when they belong to the same ray:

Equivalently, they represent the same pure state when there exists a non-zero complex scalar $c\in\mathbb{C}\setminus\{0\}$ such that:

A particular vector $\psi$ is therefore a representative of the pure state, while the ray $[\psi]$ is the pure state itself.

The Complex Hilbert Space Definition supplies the space $\mathcal{H}$ in which state representatives exist.

The Ray in a Complex Vector Space Definition supplies the meaning of $[\psi]$.

Experimental Justification

The pure-state representation postulate is not derived from a more fundamental quantum-mechanical proposition. It is supported by experiments whose measured probability distributions agree with the predictions of complex Hilbert-space quantum mechanics.

Interference experiments show that changing the relative phase between components of a quantum-state representative changes observable outcome probabilities.

For example, if two state components are represented by $\psi_1$ and $\psi_2$, a relative phase change produces a representative of the form:

Changing $\theta$ changes interference probabilities, so the relative phase between components is physically significant.

By contrast, multiplying every component of the same state representative by one common non-zero complex scalar produces another vector in the same ray:

In particular, a global phase transformation has the form:

Standard quantum-mechanical measurement probabilities are unchanged by this transformation. This is why the physical pure state is represented by the ray $[\psi]$, rather than by one uniquely selected vector $\psi$.

More directly, experiments have tested predictions that distinguish the usual complex Hilbert-space formulation from a specified real-valued alternative. In one photonic Bell-like experiment with independently prepared entangled states, the measured result violated the relevant real-valued quantum-mechanics bound by $5.30$ standard deviations.

This result supports the use of complex Hilbert-space structure in standard quantum mechanics. It does not logically prove that no alternative mathematical formulation could reproduce the same observations.