Quantum Superposition

Equivalent forms

|\psi\rangle = \sum_n c_n|\psi_n\rangle

\psi(x) = c_1\psi_1(x) + c_2\psi_2(x)

Linearity of the wave equation is quietly responsible for interference, entanglement, and quantum computing.

Symbol	Name	SI unit	Dimension	Range
$psi$	Superposed amplitudeoutput Combined wavefunction c1psi1 + c2psi2	m^-1/2	L^-1/2	-2 – 2
$c1$	Weight 1 Amplitude of the first component state	1	dimensionless	0 – 1
$phase$	Relative phase Phase difference between the two components	rad	dimensionless	0 – 6.28318

Unit systems

Symbol	SI	CGS	Imperial
$psi$	m^-1/2	cm^-1/2	ft^-1/2
$c1$	1	1	1
$phase$	rad	rad	rad

Where it holds

Holds for any linear quantum system; superposition of macroscopically distinct states is fragile and destroyed by decoherence.

Dimensional analysis

psi has units m^-1/2 (1D)

c_n (dimensionless) times psi_n (m^-1/2)

Weights dimensionless, so the sum keeps wavefunction units.

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Discovery

Paul Dirac · 1930

Dirac elevated superposition to the first postulate of his 1930 textbook, noting that its consequences 'cannot be explained in terms of classical ideas' — the interference of a system with itself is the heart of quantum theory.

Try this

How can one particle be in two states until you look?

Because the Schrodinger equation is linear, any sum of solutions is also a solution. A quantum system can occupy a weighted combination of distinct states at once, and interference between the pieces produces effects no single state could.

Research status: stable

Real-world applications

Qubit states (|0>+|1>) that make quantum parallelism possible
Interferometric sensors (atom interferometers, gravitational-wave detectors)
Coherent control of chemical reactions via shaped laser pulses

Common misconceptions

Superposition is not 'we don't know which state' — it is a genuine both-at-once until measured
The relative phase is physical: it controls interference even though it hides in a single-basis population
Macroscopic superpositions are allowed in principle but decohere almost instantly

Experimental verification

Single-photon and single-electron double-slit experiments show interference fringes built up one particle at a time — each particle superposes both paths. Superconducting qubits are routinely prepared in (|0>+|1>)/sqrt(2) and their coherence verified by Ramsey fringes.

Derivation

The time-dependent Schrodinger equation i*hbar d|psi>/

dt = H|\psi

> is linear: if |psi_1> and |psi_2> are solutions, so is c_1|psi_1> + c_2|psi_2> for any complex constants.

Normalization fixes |

c_1|^2 + |c_2|^2 = 1

(for orthonormal components).

Measuring in the {|psi_1>,|psi_2>} basis yields outcome 1 with probability |c_1|^2.

The relative phase between c_1 and c_2 is unobservable in a single-basis population measurement but governs interference in any rotated basis.

Limiting cases

c_{2} = 0

⟶ |psi>

= |\psi _1

>One weight vanishes — the state is a single basis state, no interference.

equal weights, phase

= 0

⟶ constructive interferenceIn-phase components reinforce where they overlap.

equal weights, phase

= pi

⟶ destructive interferenceOpposite phase cancels the overlap region.

What if…

What if you measure a superposition?

It collapses to one component with probability |c_n|^2; the other terms vanish from the description.

What if the two states are far apart macroscopically?

Coupling to the environment decoheres the superposition into a classical mixture almost instantly — why cats look either alive or dead.

1

Equal-weight qubit

Given ·

c1:: 0.707
c2:: 0.707

Find · measurement probabilities

Steps

| $c_{1}|^2 = 0.707^2 = 0.5$
| $c_{2}|^2 = 1 - 0.5 = 0.5$
Fair coin in the measurement basis

Result · |

c_{1}|^2 = |c_{2}|^2 = 0.5

, so each outcome is equally likely — a balanced superposition.

2

Phase-flip interference

Given ·

c1:: 0.707
phase:: pi

Find · amplitude at overlap

Steps

$\psi = 0.707 \psi 1 - 0.707 \psi 2$
Where $\psi 1 = \psi 2$ the sum is zero
Node from destructive interference

Result · With a pi phase the two equal components subtract; the overlap amplitude vanishes (destructive interference).

Born Rule→Time-Dependent Schrödinger Equation→quantum entanglementDirac Bra-Ket Notation→