# Psi (Greek)

Psi (/(p)s/; uppercase Ψ, lowercase ψ; Greek: ψι psi [ˈpsi]) 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 /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. 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: ${\displaystyle \langle \phi |\psi \rangle }$. 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:

## Character encodings

• Greek / Coptic Psi
CharacterΨψ
Unicode nameGREEK CAPITAL LETTER PSIGREEK SMALL LETTER PSIGREEK LETTER SMALL CAPITAL PSICOPTIC CAPITAL LETTER PSICOPTIC SMALL LETTER PSI
Encodingsdecimalhexdecimalhexdecimalhexdecimalhexdecimalhex
Unicode936U+03A8968U+03C87466U+1D2A11438U+2CAE11439U+2CAF
UTF-8206 168CE A8207 136CF 88225 180 170E1 B4 AA226 178 174E2 B2 AE226 178 175E2 B2 AF
Numeric character reference&#936;&#x3A8;&#968;&#x3C8;&#7466;&#x1D2A;&#11438;&#x2CAE;&#11439;&#x2CAF;
Named character reference&Psi;&psi;
DOS Greek15096175AF
DOS Greek-2212D4246F6
Windows 1253216D8248F8
TeX\Psi\psi

[4]

• Cyrillic Psi
CharacterѰѱ
Unicode nameCYRILLIC CAPITAL LETTER PSICYRILLIC SMALL LETTER PSI
Encodingsdecimalhexdecimalhex
Unicode1136U+04701137U+0471
UTF-8209 176D1 B0209 177D1 B1
Numeric character reference&#1136;&#x470;&#1137;&#x471;
• Mathematical Psi
Character𝚿𝛙𝛹𝜓𝜳𝝍
Unicode nameMATHEMATICAL 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
Encodingsdecimalhexdecimalhexdecimalhexdecimalhexdecimalhexdecimalhex
Unicode120511U+1D6BF120537U+1D6D9120569U+1D6F9120595U+1D713120627U+1D733120653U+1D74D
UTF-8240 157 154 191F0 9D 9A BF240 157 155 153F0 9D 9B 99240 157 155 185F0 9D 9B B9240 157 156 147F0 9D 9C 93240 157 156 179F0 9D 9C B3240 157 157 141F0 9D 9D 8D
UTF-1655349 57023D835 DEBF55349 57049D835 DED955349 57081D835 DEF955349 57107D835 DF1355349 57139D835 DF3355349 57165D835 DF4D
Numeric character reference&#120511;&#x1D6BF;&#120537;&#x1D6D9;&#120569;&#x1D6F9;&#120595;&#x1D713;&#120627;&#x1D733;&#120653;&#x1D74D;
Character𝝭𝞇𝞧𝟁
Unicode nameMATHEMATICAL 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
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode120685U+1D76D120711U+1D787120743U+1D7A7120769U+1D7C1
UTF-8240 157 157 173F0 9D 9D AD240 157 158 135F0 9D 9E 87240 157 158 167F0 9D 9E A7240 157 159 129F0 9D 9F 81
UTF-1655349 57197D835 DF6D55349 57223D835 DF8755349 57255D835 DFA755349 57281D835 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.