Psi (letter)
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Psi Letter

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

Psi (; uppercase ?, lowercase ?; Greek: psi ['psi]) 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/ (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 /k?/ in the Western alphabets (the sound expressed by ? 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 /k?/, 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 "?".

## 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: ${\displaystyle \langle \phi |\psi \rangle }$. 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 ? or ? can also be a symbol for:

## Character encodings

• Greek / Coptic Psi
Character Ψ ψ
Unicode name GREEK CAPITAL LETTER PSI GREEK SMALL LETTER PSI GREEK LETTER SMALL CAPITAL PSI COPTIC CAPITAL LETTER PSI COPTIC SMALL LETTER PSI
Encodings decimal hex decimal hex decimal hex decimal hex decimal hex
Unicode 936 U+03A8 968 U+03C8 7466 U+1D2A 11438 U+2CAE 11439 U+2CAF
UTF-8 206 168 CE A8 207 136 CF 88 225 180 170 E1 B4 AA 226 178 174 E2 B2 AE 226 178 175 E2 B2 AF
Numeric character reference &#936; &#x3A8; &#968; &#x3C8; &#7466; &#x1D2A; &#11438; &#x2CAE; &#11439; &#x2CAF;
Named character reference &Psi; &psi;
DOS Greek 150 96 175 AF
DOS Greek-2 212 D4 246 F6
Windows 1253 216 D8 248 F8
TeX \Psi \psi

[4]

• Cyrillic Psi
Character Ѱ ѱ
Unicode name CYRILLIC CAPITAL LETTER PSI CYRILLIC SMALL LETTER PSI
Encodings decimal hex decimal hex
Unicode 1136 U+0470 1137 U+0471
UTF-8 209 176 D1 B0 209 177 D1 B1
Numeric character reference &#1136; &#x470; &#1137; &#x471;
• Mathematical Psi
Character 𝚿 𝛙 𝛹 𝜓 𝜳 𝝍
Unicode name MATHEMATICAL BOLD
CAPITAL PSI
MATHEMATICAL BOLD
SMALL PSI
MATHEMATICAL ITALIC
CAPITAL PSI
MATHEMATICAL ITALIC
SMALL PSI
MATHEMATICAL BOLD ITALIC
CAPITAL PSI
MATHEMATICAL BOLD ITALIC
SMALL PSI
Encodings decimal hex decimal hex decimal hex decimal hex decimal hex decimal hex
Unicode 120511 U+1D6BF 120537 U+1D6D9 120569 U+1D6F9 120595 U+1D713 120627 U+1D733 120653 U+1D74D
UTF-8 240 157 154 191 F0 9D 9A BF 240 157 155 153 F0 9D 9B 99 240 157 155 185 F0 9D 9B B9 240 157 156 147 F0 9D 9C 93 240 157 156 179 F0 9D 9C B3 240 157 157 141 F0 9D 9D 8D
UTF-16 55349 57023 D835 DEBF 55349 57049 D835 DED9 55349 57081 D835 DEF9 55349 57107 D835 DF13 55349 57139 D835 DF33 55349 57165 D835 DF4D
Numeric character reference &#120511; &#x1D6BF; &#120537; &#x1D6D9; &#120569; &#x1D6F9; &#120595; &#x1D713; &#120627; &#x1D733; &#120653; &#x1D74D;
Character 𝝭 𝞇 𝞧 𝟁
Unicode name MATHEMATICAL SANS-SERIF
BOLD CAPITAL PSI
MATHEMATICAL SANS-SERIF
BOLD SMALL PSI
MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL PSI
MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL PSI
Encodings decimal hex decimal hex decimal hex decimal hex
Unicode 120685 U+1D76D 120711 U+1D787 120743 U+1D7A7 120769 U+1D7C1
UTF-8 240 157 157 173 F0 9D 9D AD 240 157 158 135 F0 9D 9E 87 240 157 158 167 F0 9D 9E A7 240 157 159 129 F0 9D 9F 81
UTF-16 55349 57197 D835 DF6D 55349 57223 D835 DF87 55349 57255 D835 DFA7 55349 57281 D835 DFC1
Numeric character reference &#120685; &#x1D76D; &#120711; &#x1D787; &#120743; &#x1D7A7; &#120769; &#x1D7C1;

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.

## Notes and references

1. ^ 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.
2. ^ Buchholz, 1986 (Ann. Pure Appl. Logic)
3. ^ Gesenius' Hebrew Grammar
4. ^ Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)