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In quantum mechanics, bra–ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product on a complex vector space) of two states is denoted by :, consisting of a left part, called the bra , and a right part, , called the ket . The notation was introduced in 1939 by Paul Dirac〔 〕 and is also known as Dirac notation, though the notation has precursors in Grassmann's use of the notation for his inner products nearly 100 years earlier. Bra–ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics – including a large portion of modern physics – is usually explained with the help of bra-ket notation. Part of the appeal of the notation is the abstract representation-independence it encodes, together with its versatility in producing a specific representation (e.g. , or , or eigenfunction base) without much ado, or excessive reliance on the nature of the linear spaces involved. The overlap expression is typically interpreted as the probability amplitude for the state to collapse into the state . ==Vector spaces== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bra–ket notation」の詳細全文を読む スポンサード リンク
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