Psi (letter)

Greek alphabet
Αα	Alpha	Νν	Nu
Ββ	Beta	Ξξ	Xi
Γγ	Gamma	Οο	Omicron
Δδ	Delta	Ππ	Pi
Εε	Epsilon	Ρρ	Rho
Ζζ	Zeta	Σσς	Sigma
Ηη	Eta	Ττ	Tau
Θθ	Theta	Υυ	Upsilon
Ιι	Iota	Φφ	Phi
Κκ	Kappa	Χχ	Chi
Λλ	Lambda	Ψψ	Psi
Μμ	Mu	Ωω	Omega
History
Archaic local variants  ·  ·  ·  ·  ·
Ligatures (ϛ, ȣ, ϗ) · Diacritics
Numerals: (6) · (90) · (900)
In other languages
Bactrian  · Coptic  · Albanian
Scientific symbols
Book  · Category · Commons

Psi (uppercase Ψ, lowercase ψ; in English pronounced /ˈpsaɪ/ or /ˈsaɪ/, after Greek ψῖ /ˈpsiː/) is the 23rd letter of the Greek alphabet and has a numeric value of 700. In both Classical and Modern Greek, the letter indicates the combination /ps/ (like in English "lapse").

In Greek loanwords in Latin and modern languages with Latin alphabets, Psi is usually transliterated as "ps". In English, due to phonotactic constraints, its pronunciation is usually simplified to /s/ at the beginning of a word.

Historically, the letter is a Greek innovation and not derived from the Phoenician alphabet. It appears in the 7th century BC, expressing /ps/ in the Eastern alphabets, but /kʰ/ in the Western alphabets (the sound expressed by Χ in the Eastern alphabets). Epigraphically, the early letter appears in an angular shape (). There were early graphical variants which omitted the stem ("chickenfoot-shaped psi" ).

The Western letter (expressing /kʰ/, and later /x/) was adopted into the Old Italic alphabets, and its shape is also continued into the Algiz rune of the Elder Futhark. The classical Greek letter was adopted into the early Cyrillic alphabet as Ѱ.

The Greek alphabet on a black figure vessel, with an archaic chickenfoot-shaped psi.

Use as a symbol

The letter psi is commonly used in physics for representing a wavefunction in quantum mechanics, particularly with the Schrödinger equation and bra-ket notation: $\langle\phi|\psi\rangle$. It is also used to represent the (generalized) positional states of a qubit in a quantum computer.

Psi is also used as the symbol for the polygamma function, defined by

$\psi_n(x) = \frac{d^{(n)}}{dx^{(n)}}\frac{\Gamma '(x)}{\Gamma (x)}\,\!$

where Γ(x) is the gamma function.

The letters Ψ or ψ can also be a symbol for:

• psychology, psychiatry, and sometimes parapsychology (involving paranormal or relating with the supernatural subjects, especially research into extrasensory perception).
• In mathematics, the reciprocal Fibonacci constant.
• Water potential in movement of water between plant cells.
• In biochemistry, it denotes the rare nucleotide pseudouridilic acid.
• Stream function in fluid mechanics defining the curve to which the flow velocity is always tangent.
• One of the dihedral angles in the backbones of proteins
• The planet Neptune
• The Schrödinger equation and throughout quantum mechanics, ψ(x) stands for the wave function
• Indiana University (as a superimposed I and U)
• Gangster Disciples
• A Sai, the name of which is pronounced the same way.
• Pharmacology, general pharmacy
• In virology the ψ site is a viral packaging signal.
• The J/ψ meson, in particle physics.
• In the computability theory, $\psi_P(x)\,\!$ represents the return value $Y\,\!$ of a program $P\,\!$.
• In the comic Monochrome, Psi is a mentally disturbed character, though her name is usually spelled out.
• In the television show Babylon 5, Ψ is the symbol of the Psi Corps.
• In circadian physiology, ψ represents the phase relationship between a zeitgeber and a biological rhythm.

See also

• Psi and phi type figurine
• Psi (Cyrillic)