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The letter phi 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 } $\langle \phi |\psi \rangle$ .
The golden ratio {\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}\approx } ${\tfrac {1+{\sqrt {5}}}{2}}\approx$ 1.618033988749894848204586834… in mathematics,^[3] art, and architecture. (Its reciprocal, 1/φ, is {\displaystyle {\tfrac {{\sqrt {5}}-1}{2}}} ${\tfrac {{\sqrt {5}}-1}{2}}$ and is equal to φ − 1