Psi /ˈ(p)saɪ, ˈ(p)siː/ (P)SY, (P)SEE^[1] (uppercase Ψ, lowercase ψ or 𝛙; Greek: ψι psi [ˈpsi]) is the twenty-third and penultimate letter of the Greek alphabet and is associated with a numeric value of 700. In both Classical and Modern Greek, the letter indicates the combination /ps/ (as in English word "lapse").

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For Greek loanwords in Latin and modern languages with Latin alphabets, psi is usually transliterated as "ps".

The letter's origin is uncertain. It may or may not derive 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). In writing, the early letter appears in an angular shape (). There were early graphical variants that omitted the stem ("chickenfoot-shaped psi" as: or ).

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

Psi, or its Arcadian variant or was adopted in the Latin alphabet in the form of "Antisigma" (Ↄ, ↃC, or 𐌟) during the reign of Emperor Claudius as one of the three Claudian letters.^[2] However, it was abandoned after his death.

The classical Greek letter was adopted into the early Cyrillic alphabet as "Ѱ".

The letter psi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation: ${\displaystyle \langle \phi |\psi \rangle }$. 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

${\displaystyle \psi ^{(m)}(z)={\frac {d^{m}}{dz^{m}}}{\frac {\Gamma '(z)}{\Gamma (z)}}}$

where ${\displaystyle \Gamma (z)}$ 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, the division polynomials, and the supergolden ratio.
• Water potential in movement of water between plant cells.
• In biochemistry, it denotes pseudouridine, an uncommon nucleoside.^[3]
• 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.
• Indiana University (as a superimposed I and U).^[4]
• A sai, the name of which is pronounced the same way in English.
• Pharmacology, general pharmacy.
• In virology the ψ site is a viral packaging signal.
• The J/ψ meson, in particle physics.
• In the computability theory, ${\displaystyle \psi _{P}(x)\,\!}$ represents the return value ${\displaystyle Y\,\!}$ of a program ${\displaystyle P\,\!}$.
• In circadian physiology, ψ represents the phase relationship between a zeitgeber and a biological rhythm.
• In building, to represent an adjustment to a U-value, accounting for thermal bridge effects.
• The ordinal collapsing function and notation developed by Wilfried Buchholz.^[5]
• In Biblical studies, as an abbreviation for the book of Psalms.^[6]

• Greek / Coptic Psi

^[7]

• Cyrillic Psi

• Mathematical Psi

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.

Look up Ψ or ψ in Wiktionary, the free dictionary.

• Psi and phi type figurine
• Psi (Cyrillic)