Quantum Physics

Dirac Notation: Introduction

See previous posts about Quantum Physics

Dirac notation (or bra-ket notation) was developed by English physicist Paul Dirac to help describe the quantum effects that were being observed in experiments of the time. Alongside Erwin Schrödinger, Dirac earned the 1933 Nobel Prize in Physics for “the discovery of new productive forms of atomic theory,” which was contributed to in part by his development of this notation.

At the fundamental level, Dirac notation is primarily comprised of state vectors and basis vectors.


Consider a quantum system (e.g. a particle) which has a state ψ. In Dirac’s notation, this is represented by the “ket” vector, or the state vector

which can represent either a discrete set of possible states or a continuous wavefunction.

In a discrete set, the state of the system is defined by a column vector representing each possible state the system may take

where each component ψi is a complex number. In a continuous set, the state is represented by a function

which essentially acts just like any complex function we’ve met previously in mathematics. 

It is worth noting here that the terms “state” and “wavefunction” are almost entirely analogous in describing quantum systems – the former describing quantised (previously mentioned, discrete) states and the latter describing a continuous state-distribution (i.e. a function).

Whether the wavefunction exists in reality or is simply a philosophical concept is a topic of debate in quantum theory, with some seeing it as a conceptual property of abstract mathematical space and others seeing it as a physical reality. Regardless of perception, the results of manipulating the wavefunction corroborate well with observations and—since the days of Dirac, Schrödinger and Einstein—the discussion is left more to small-talk at cocktail parties than real scientific debate.


Now we can introduce the basis vector of ψ, denoted by a “bra”

This can represent the conjugate transposition of either the set of states or the continuous wavefunction represented by ψ. In general,

This is known as the co-vector to the state vector | ψ ⟩. For a discrete set of states, the co-vector to | ψ ⟩—the basis vector—is

where * denotes the complex conjugate. This represents the conjugate transpose of the state vector. For a continuous function the basis vector to | ψ ⟩ is

where ψ is, again, a complex function and ψ* is its conjugate.

Alone, these states and basis vectors mean very little about our system. Dirac’s bra-ket notation can be best understood through analysis of the Hilbert space, which we’ll look at next time to examine the intrinsic mathematical properties implied within this notation.

Hertha Ayrton (1854-1923) was an English inventor, mathematician, physicist, and engineer. Throughout her life she registered 26 patents for inventions, most of them mathematical dividers or arc lamps and electrodes. For her work on electric arcs and sand ripples, she was awarded the Hughes Medal by the Royal Society, the first woman and so far one of the only two to be honoured in this way.

She attended Girton College in Cambridge, but was not awarded a degree on account of her gender. Later, she became the first woman to read a paper before the Royal Society, but was not eligible as a Fellow because of her status as a married woman. She helped found the International Federation of University Women in 1919.

Dyson is an English-American theoretical physicist and mathematician, known for his work in quantum electrodynamics, solid-state physics, astronomy and nuclear engineering. He is also known for his speculative work on extraterrestrial civilizations.