Complex Phase of ψ

The wavefunction $\psi(x, t)$ is a complex-valued function. That means at every point $x$ and time $t$ it has a modulus $|\psi|$ and a phase $\phi$ such that

$\psi(x, t) = |\psi(x, t)|\, e^{i\phi(x, t)}.$

The Born rule says that probability densities come from the modulus squared:

$P(x, t) = |\psi(x, t)|^2.$

Because squaring the modulus discards all phase information, probabilities are entirely blind to phase. You can multiply the whole wavefunction by any constant phase factor $e^{i\theta}$ (a global phase) and no measurable probability anywhere in the universe changes — before or after any measurement.

Global phase is unobservable

Suppose two wavefunctions differ only by a global phase:

$\psi'(x) = e^{i\theta}\,\psi(x), \qquad \theta \in \mathbb{R}.$

Since $|e^{i\theta}| = 1$ for any real $\theta$ , we have

$|\psi'(x)|^2 = |e^{i\theta}|^2\,|\psi(x)|^2 = |\psi(x)|^2.$

The probability distributions are identical, so no experiment can distinguish $\psi'$ from $\psi$ . Global phase is therefore a redundancy in the description — physicists say $\psi$ and $\psi'$ represent the same physical state.

Relative phase is physical

Things change completely when phases differ between components of a superposition. Consider a state that is an equal superposition of two spatial configurations labeled by index $n = 1$ and $n = 2$ :

\psi' = \frac{1}{\sqrt{2}}\left(\psi_1 + e^{i\delta}\,\psi_2\right).$$ The phase $\delta$ appears only on $\psi_2$, so it is a **relative phase** between the two terms. Computing the probability density at a point where both $\psi_1$ and $\psi_2$ are nonzero: $$|\psi|^2 = \tfrac{1}{2}\bigl(|\psi_1|^2 + 2\operatorname{Re}(\psi_1^*\psi_2) + |\psi_2|^2\bigr),$$ $$|\psi'|^2 = \tfrac{1}{2}\bigl(|\psi_1|^2 + 2\operatorname{Re}(\psi_1^*\,e^{i\delta}\,\psi_2) + |\psi_2|^2\bigr).$$ The cross term $\operatorname{Re}(\psi_1^*\,e^{i\delta}\,\psi_2)$ depends on $\delta$. Changing $\delta$ shifts the pattern of constructive and destructive interference and therefore shifts observable probabilities. Relative phase is **directly measurable** through interference. <Callout type="tip"> A handy rule of thumb: if you can change a phase by choosing which component to shift, it is a relative phase and it matters. If shifting it is equivalent to multiplying the entire wavefunction by the same constant, it is global and it does not matter. </Callout> ## Where phase shows up in practice **Double-slit interference.** In the double-slit experiment the wavefunction at the screen is a superposition of contributions from each slit. The relative phase between the two paths depends on the path-length difference. At points where the two contributions are in phase ($\delta = 0, 2\pi, \ldots$) the probability is four times larger than from either slit alone — constructive interference. Where they are out of phase ($\delta = \pi, 3\pi, \ldots$) the probability drops to zero — destructive interference. The entire fringe pattern is a direct read-out of relative phase. **Time evolution.** The Schrödinger equation causes each energy eigenstate $\psi_n$ to accumulate phase at the rate $E_n / \hbar$: $$\psi_n(x, t) = \psi_n(x, 0)\,e^{-iE_n t/\hbar}.$$ For a single energy eigenstate this is a global phase and produces no observable effect — the probability density $|\psi_n|^2$ is constant in time (a **stationary state**). But if the initial state is a superposition of two eigenstates with different energies $E_1$ and $E_2$, the relative phase between them oscillates at the **Bohr frequency** $\omega_{12} = (E_2 - E_1)/\hbar$, causing the probability density to oscillate in time. This oscillation is physical and observable. **Spin and qubits.** The same logic applies to two-level systems. A qubit state $\alpha|0\rangle + \beta|1\rangle$ has an overall global phase (unobservable) and a relative phase between the $|0\rangle$ and $|1\rangle$ amplitudes (observable via interference on the Bloch sphere). The Hadamard and phase gates specifically manipulate this relative phase to carry out quantum algorithms. ## Summary | Phase type | Definition | Observable? | |---|---|---| | Global | Same factor $e^{i\theta}$ on all amplitudes | No | | Relative | Different phases on different components | Yes, through interference | The Born rule uses $|\psi|^2$, which looks like it erases all phase — and it does erase global phase. But relative phase is encoded in how the complex amplitudes add before squaring, which is precisely where quantum interference lives. Phase is the carrier of quantum coherence, and understanding it is essential for everything from interference experiments to quantum gate design.