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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").

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 ).^{[citation needed]}

The Western letter (expressing /kʰ/, later /x/) was adopted into the Old Italic alphabets, and its shape is also continued into the Algiz rune <ᛉ> 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.^{[citation needed]}

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

Use as a symbol

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: $\langle \phi |\psi \rangle$ . 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 ^{(m)}(z)={\frac {d^{m}}{dz^{m}}}{\frac {\Gamma '(z)}{\Gamma (z)}}

where $\Gamma (z)$ 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, $\psi _{P}(x)\,\!$ represents the return value $Y\,\!$ of a program $P\,\!$ .
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]

Unicode

U+03A8 Ψ GREEK CAPITAL LETTER PSI (Ψ)^[7]
U+03C8 ψ GREEK SMALL LETTER PSI (ψ)
U+0470 Ѱ CYRILLIC CAPITAL LETTER PSI
U+0471 ѱ CYRILLIC SMALL LETTER PSI
U+1D2A ᴪ GREEK LETTER SMALL CAPITAL PSI
U+2CAE Ⲯ COPTIC CAPITAL LETTER PSI
U+2CAF ⲯ COPTIC SMALL LETTER PSI
U+1D6BF 𝚿 MATHEMATICAL BOLD CAPITAL PSI^[a]
U+1D6D9 𝛙 MATHEMATICAL BOLD SMALL PSI
U+1D6F9 𝛹 MATHEMATICAL ITALIC CAPITAL PSI
U+1D713 𝜓 MATHEMATICAL ITALIC SMALL PSI
U+1D733 𝜳 MATHEMATICAL BOLD ITALIC CAPITAL PSI
U+1D74D 𝝍 MATHEMATICAL BOLD ITALIC SMALL PSI
U+1D76D 𝝭 MATHEMATICAL SANS-SERIF BOLD CAPITAL PSI
U+1D787 𝞇 MATHEMATICAL SANS-SERIF BOLD SMALL PSI
U+1D7A7 𝞧 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL PSI
U+1D7C1 𝟁 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL PSI

^ The MATHEMATICAL characters are used in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.

Notes and references

^ "psi". The Chambers Dictionary (9th ed.). Chambers. 2003. ISBN 0-550-10105-5.
^ Oliver, Revilo P. (1949). "The Claudian Letter Ⱶ". American Journal of Archaeology. 53 (3): 249–257. doi:10.2307/500662. JSTOR 500662. S2CID 193082268.
^ IUPAC-IUB Commission on Biochemical Nomenclature (1970). "Abbreviations and symbols for nucleic acids, polynucleotides, and their constituents". Biochemistry. 9 (20): 4022–4027. doi:10.1021/bi00822a023.
^ Although the university itself refers to its logo as a trident, not the Greek letter psi: "The IU trident—the only logo at Indiana University". Indiana University. Retrieved 2020-07-31. At IU, the trident is the only logo we use, both institution-wide and at the unit level.
^ Buchholz, W. (1986). "A new System of proof-theoretic ordinal functions" (PDF). Ann. Pure Appl. Logic. 32 (3): 195–207. doi:10.1016/0168-0072(86)90052-7.
^ Gesenius' Hebrew Grammar
^ Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)

[8] The MATHEMATICAL characters are used in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.

[1] "psi". The Chambers Dictionary (9th ed.). Chambers. 2003. ISBN 0-550-10105-5.

[oliver-2] Oliver, Revilo P. (1949). "The Claudian Letter Ⱶ". American Journal of Archaeology. 53 (3): 249–257. doi:10.2307/500662. JSTOR 500662. S2CID 193082268.

[3] IUPAC-IUB Commission on Biochemical Nomenclature (1970). "Abbreviations and symbols for nucleic acids, polynucleotides, and their constituents". Biochemistry. 9 (20): 4022–4027. doi:10.1021/bi00822a023.

[4] Although the university itself refers to its logo as a trident, not the Greek letter psi: "The IU trident—the only logo at Indiana University". Indiana University. Retrieved 2020-07-31. At IU, the trident is the only logo we use, both institution-wide and at the unit level.

[5] Buchholz, W. (1986). "A new System of proof-theoretic ordinal functions" (PDF). Ann. Pure Appl. Logic. 32 (3): 195–207. doi:10.1016/0168-0072(86)90052-7.

[6] Gesenius' Hebrew Grammar

[7] Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)

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@@ Line 1: / Line 1: @@
+{{Short description|Penultimate letter in the Greek alphabet}}
 {{about|the Greek letter|the Cyrillic letter|Psi (Cyrillic)}}
 {{pp-semi-indef}}
@@ Line 6: / Line 7: @@
 [[File:NAMA Alphabet grec.jpg|thumb|The Greek alphabet on a [[black-figure pottery]] vessel, with an archaic chickenfoot-shaped psi.]]
-'''Psi''' {{IPAc-en|'|s|aɪ}} (uppercase '''{{lang|el|Ψ}}''', lowercase '''{{lang|el|ψ}}'''; {{Lang-el|ψι}} ''psi'' {{IPA-el|ˈpsi|}}) is the 23rd letter of the [[Greek alphabet]] and has a [[Greek numerals|numeric]] value of 700. In both [[Classical Greek|Classical]] and [[Modern Greek]], the letter indicates the combination {{IPA|/ps/}} (as in English word "[[:wikt:lapse|lapse]]").
+'''Psi''' {{IPAc-en|'|(|p|)|s|aɪ|,_|"|(|p|)|s|i:}} {{respell|(P)SY|,_|(P)SEE}}<ref>{{Cite book|title=The Chambers Dictionary|publisher=Chambers|year=2003|isbn=0-550-10105-5|edition=9th|chapter=psi}}</ref> (uppercase '''{{lang|el|Ψ}}''', lowercase '''{{lang|el|ψ}}''' or '''{{lang|el|𝛙}}'''; {{Langx|el|ψι}} ''psi'' {{IPA-el|ˈpsi|}}) is the twenty-third and penultimate letter of the [[Greek alphabet]] and is associated with a [[Greek numerals|numeric]] value of 700. In both [[Classical Greek|Classical]] and [[Modern Greek]], the letter indicates the combination {{IPA|/ps/}} (as in English word "[[:wikt:lapse|lapse]]").
 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 {{IPA|/ps/}} in the Eastern alphabets, but {{IPA|/kʰ/}} in the [[Western Greek alphabet|Western alphabets]] (the sound expressed by [[Χ]] in the Eastern alphabets). In writing, the early letter appears in an angular shape ([[File:Greek Psi straight.svg|15px|]]).
+The letter's origin is uncertain. It may or may not derive from the [[Phoenician alphabet]]. It appears in the 7th century BC, expressing {{IPA|/ps/}} in the Eastern alphabets, but {{IPA|/kʰ/}} in the [[Western Greek alphabet|Western alphabets]] (the sound expressed by [[Χ]] in the Eastern alphabets). In writing, the early letter appears in an angular shape (<span style="background-color: white;">[[File:Greek Psi straight.svg|15px|]]</span>).
-There were early graphical variants that omitted the stem ("chickenfoot-shaped psi" as: [[File:Greek Psi V-shaped.svg|15px]] or [[File:Greek Chi 05.svg|15px]]).
+There were early graphical variants that omitted the stem ("chickenfoot-shaped psi" as: <span style="background-color: white;">[[File:Greek Psi V-shaped.svg|15px]]</span> or <span style="background-color: white;">[[File:Greek Chi 05.svg|15px]]</span>).{{Citation needed|date=June 2024}}
-The Western letter (expressing {{IPA|/kʰ/}}, later {{IPA|/x/}}) was adopted into the [[Old Italic alphabets]], and its shape is also continued into the [[Algiz]] rune of the [[Elder Futhark]].
+The Western letter (expressing {{IPA|/kʰ/}}, later {{IPA|/x/}}) was adopted into the [[Old Italic alphabets]], and its shape is also continued into the [[Algiz]] rune <ᛉ> of the [[Elder Futhark]].
-Psi, or its Arcadian variant {{GrGl|Psi 01}} or {{GrGl|Psi X-shaped}} 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]].<ref name="oliver">{{cite journal |last=Oliver |first=Revilo P. |year=1949 |title=The Claudian Letter Ⱶ |journal=American Journal of Archaeology |volume=53 |issue= 3|pages=249–257 |doi=10.2307/500662 |jstor= 500662}}</ref> However, it was abandoned after his death.
+Psi, or its Arcadian variant <span style="background-color: white;">{{GrGl|Psi 01}}</span> or <span style="background-color: white;">{{GrGl|Psi X-shaped}}</span> 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]].<ref name="oliver">{{cite journal |last=Oliver |first=Revilo P. |year=1949 |title=The Claudian Letter Ⱶ |journal=American Journal of Archaeology |volume=53 |issue= 3|pages=249–257 |doi=10.2307/500662 |jstor= 500662|s2cid=193082268 }}</ref> However, it was abandoned after his death.{{Citation needed|date=June 2024}}
 The classical Greek letter was adopted into the [[early Cyrillic alphabet]] as "[[psi (Cyrillic)|Ѱ]]".
@@ Line 35: / Line 36: @@
 *The planet [[Neptune]].
 *[[Indiana University]] (as a superimposed [[I]] and [[U]]).<ref>Although the university itself refers to its logo as a [[trident]], not the Greek letter psi: {{cite web |publisher=Indiana University |url=https://brand.iu.edu/design/logos-lockups/index.html |access-date=2020-07-31 |title=The IU trident—the only logo at Indiana University |quote=At IU, the trident is the only logo we use, both institution-wide and at the unit level. }}</ref>
-*A [[Sai (weapon)|sai]], the name of which is pronounced the same way.
+*A [[Sai (weapon)|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.
@@ Line 42: / Line 43: @@
 *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 [[ordinal notation|notation]] developed by Wilfried Buchholz.<ref>{{cite journal |last=Buchholz |first=W. |year=1986 |title=A new System of proof-theoretic ordinal functions |journal=Ann. Pure Appl. Logic |volume=32 |issue=3 |pages=195–207 |url=https://epub.ub.uni-muenchen.de/3841/1/3841.pdf }}</ref>
+*The [[ordinal collapsing function]] and [[ordinal notation|notation]] developed by Wilfried Buchholz.<ref>{{cite journal |last=Buchholz |first=W. |year=1986 |title=A new System of proof-theoretic ordinal functions |journal=Ann. Pure Appl. Logic |volume=32 |issue=3 |pages=195–207 |url=https://epub.ub.uni-muenchen.de/3841/1/3841.pdf |doi=10.1016/0168-0072(86)90052-7 }}</ref>
 *In [[Biblical studies]], as an abbreviation for the book of [[Psalms]].<ref>[https://en.wikisource.org/wiki/Template:GHGbible-ref Gesenius' Hebrew Grammar]</ref>
+==Unicode==
-==Character encodings==
+* {{unichar|03A8|html=}}<ref>Unicode Code Charts: [https://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]</ref>
-* Greek / Coptic Psi
+* {{unichar|03C8|html=}}
+* {{unichar|0470|html=}}
+* {{unichar|0471|html=}}
+* {{unichar|1D2A|html=}}
+* {{unichar|2CAE|html=}}
+* {{unichar|2CAF|html=}}
+* {{unichar|1D6BF|html=}}{{efn|The {{sc|mathematical}} characters are used in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
+* {{unichar|1D6D9|html=}}
+* {{unichar|1D6F9|html=}}
+* {{unichar|1D713|html=}}
+* {{unichar|1D733|html=}}
+* {{unichar|1D74D|html=}}
+* {{unichar|1D76D|html=}}
+* {{unichar|1D787|html=}}
+* {{unichar|1D7A7|html=}}
+* {{unichar|1D7C1|html=}}
+{{notelist}}
-{{charmap
-|03A8|name1=Greek Capital Letter Psi
-|03C8|name2=Greek Small Letter Psi
-|1D2A|name3=Greek Letter Small Capital Psi
-|2CAE|name4=Coptic Capital Letter Psi
-|2CAF|name5=Coptic Small Letter Psi
-|map2=DOS Greek|map2char1=96|map2char2=AF
-|map3=DOS Greek-2|map3char1=D4|map3char2=F6
-|map4=Windows 1253|map4char1=D8|map4char2=F8
-|namedref1=[[TeX]]|ref1char1=\Psi|ref1char2=\psi
-}}
-<ref>Unicode Code Charts: [https://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]</ref>
-* Cyrillic Psi
-{{charmap
-|0470|name1=Cyrillic Capital Letter Psi
-|0471|name2=Cyrillic Small Letter Psi
-}}
-* Mathematical Psi
-{{charmap
-|1D6BF|name1=Mathematical Bold<br />Capital Psi
-|1D6D9|name2=Mathematical Bold<br />Small Psi
-|1D6F9|name3=Mathematical Italic<br />Capital Psi
-|1D713|name4=Mathematical Italic<br />Small Psi
-|1D733|name5=Mathematical Bold Italic<br />Capital Psi
-|1D74D|name6=Mathematical Bold Italic<br />Small Psi
-}}
-{{charmap
-|1D76D|name1=Mathematical Sans-Serif<br />Bold Capital Psi
-|1D787|name2=Mathematical Sans-Serif<br />Bold Small Psi
-|1D7A7|name3=Mathematical Sans-Serif<br />Bold Italic Capital Psi
-|1D7C1|name4=Mathematical Sans-Serif<br />Bold Italic Small 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.
 ==See also==