Quantum superposition

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Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.

An example is a qubit used in quantum information processing. A qubit state is most generally a superposition of the basis states ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$:

${\displaystyle |\Psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle ,}$

where ${\displaystyle |\Psi \rangle }$ is the quantum state of the qubit, and ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$ denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes ${\displaystyle c_{0}}$ and ${\displaystyle c_{1}}$ that both are complex numbers. Here ${\displaystyle |0\rangle }$ corresponds to the classical 0 bit, and ${\displaystyle |1\rangle }$ to the classical 1 bit. The probabilities of measuring the system in the ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$ state are given by ${\displaystyle |c_{0}|^{2}}$ and ${\displaystyle |c_{1}|^{2}}$ respectively (see the Born rule). Before the measurement occurs the qubit is in a superposition of both states.

The interference fringes in the double-slit experiment provide another example of the superposition principle.