Psi (letter)

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

Psi (uppercase Ψ, lowercase ψ; Greek: Ψι 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". In English, the letter is pronounced /ˈsaɪ/ or sometimes /ˈpsaɪ/. (In Greek, it is pronounced [ˈpsiː].)

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: $\langle\phi|\psi\rangle$. 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 and the division polynomials.
• 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
• 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 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^[1]

Character Encodings

• Greek / Coptic Psi

Character	Ψ		ψ		ᴪ		Ⲯ		ⲯ
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	Ψ	Ψ	ψ	ψ	ᴪ	ᴪ	Ⲯ	Ⲯ	ⲯ	ⲯ
Named character reference	Ψ		ψ
DOS Greek	150	96	175	AF
DOS Greek-2	212	D4	246	F6
Windows 1253	216	D8	248	F8
TeX	\Psi		\psi

^[2]

• 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	Ѱ	Ѱ	ѱ	ѱ

• 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	𝚿	𝚿	𝛙	𝛙	𝛹	𝛹	𝜓	𝜓	𝜳	𝜳	𝝍	𝝍

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	𝝭	𝝭	𝞇	𝞇	𝞧	𝞧	𝟁	𝟁

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

• Psi and phi type figurine
• Psi (Cyrillic)

Notes and references

1. Buchholz, 1986 (Ann. Pure Appl. Logic)
2. Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)