The Question
While tracing the history leading toward Bell’s theorem, a more fundamental question appeared:
Who gave us the equation
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle?\]
The important answer is that this familiar qubit expression was not introduced as a standalone equation by one particular physicist. It is the modern quantum-information form of ideas that developed gradually during the creation of quantum mechanics.
Before Qubits: Superposition Was Already Familiar
Long before quantum mechanics, physicists understood that ordinary waves could be superposed.
If two waves overlap, their amplitudes can add to form a combined wave.
Schematically:
\[\text{wave 1}+\text{wave 2}=\text{combined wave}.\]
This occurs with familiar classical waves such as water waves and light waves.
So the mathematical idea of combining waves was not originally a quantum idea.
Schrödinger and the Quantum Wavefunction
In 1926, Erwin Schrödinger developed wave mechanics and introduced an equation describing the evolution and behavior of quantum wavefunctions.
A quantum system could be represented by a wavefunction such as
\[\psi.\]
Suppose two possible solutions are
\[\psi_1\]
and
\[\psi_2.\]
A crucial property of Schrödinger’s equation is that it is linear.
Because of that linearity, combinations of solutions can also be solutions. Thus a state can have the form
\[\alpha\psi_1+\beta\psi_2.\]
This linear structure is one of the mathematical roots of quantum superposition.
From Wavefunctions to Abstract Quantum States
Quantum mechanics was being developed in several mathematical forms during the 1920s. Schrödinger developed wave mechanics, while Werner Heisenberg, Max Born, Pascual Jordan and others developed matrix-based approaches.
Paul Dirac subsequently helped formulate quantum mechanics in a more abstract vector-space language and introduced the bra-ket notation that is now standard.
A quantum state can therefore be written as
\[|\psi\rangle.\]
If the relevant quantum state space is two-dimensional and we choose two basis states
\[|0\rangle\quad\text{and}\quad|1\rangle,\]
then ordinary linear algebra tells us that any vector in that two-dimensional space can be expressed as a linear combination of those basis vectors:
\[\boxed{|\psi\rangle=\alpha|0\rangle+\beta|1\rangle}.\]
So Who Invented This Equation?
There is therefore no single historical moment in which someone simply invented the modern qubit equation
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle.\]
Instead, it emerges from several ideas coming together:
\[\text{wave superposition}\]
\[\downarrow\]
\[\text{linear quantum mechanics}\]
\[\downarrow\]
\[\text{quantum states as vectors}\]
\[\downarrow\]
\[\text{Dirac notation}\]
\[\downarrow\]
\[\text{two-dimensional quantum state space}\]
\[\downarrow\]
\[\boxed{|\psi\rangle=\alpha|0\rangle+\beta|1\rangle}.\]
The Qubit Came Much Later
Schrödinger and Dirac were not developing quantum computers or talking about qubits in the modern sense.
The language of quantum information came much later.
For a two-level quantum system used to represent quantum information, the two basis vectors are conventionally written
\[|0\rangle\quad\text{and}\quad|1\rangle.\]
The general pure qubit state therefore naturally takes the familiar form
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\]
with normalization
\[|\alpha|^2+|\beta|^2=1.\]
An Important Historical Twist
Although Schrödinger introduced the wavefunction, this did not immediately settle what the wavefunction physically meant.
That became one of the great conceptual problems of early quantum mechanics.
The next natural question is therefore:
If \(\psi\) is a wave, what exactly does that wave represent?
This question leads directly to Max Born and the probabilistic interpretation of the wavefunction. From there, the historical path eventually leads to the Einstein-Bohr debate, EPR, hidden variables and Bell’s theorem.
Note metadata
- Note type: learning-note
- Subject: learning-notes
- Source: Quantum Computing for Everyone
Related notes
- Measuring Entangled Qubits in Same and Different Bases (Post ID: 37)