Bra-ket notation is a mathematical notation used in quantum mechanics to represent vectors and operators. The "bra" and "ket" symbols (< > and | >) are used to represent vectors and their dual vectors, respectively. This notation is useful for simplifying complex algebraic expressions in quantum mechanics.
In bra-ket notation, addition and subtraction are performed similarly to regular algebra. To add or subtract two bra-ket expressions, you simply add or subtract the corresponding coefficients and keep the same ket or bra symbol. For example, < |a> + < |b> = < |a + b> and < |a> - < |b> = < |a - b>.
A bra vector is represented by the symbol < > and is the dual vector of a ket vector, represented by the symbol | >. In other words, a bra vector is a row vector while a ket vector is a column vector. This distinction is important in certain algebraic operations in quantum mechanics.
In bra-ket notation, multiplication is performed using the inner product or dot product, denoted by < , >. To multiply a bra vector < |a> with a ket vector |b>, you would take the inner product < |a> < , > |b> = . This operation combines the two vectors into a scalar value.
The terms "bra" and "ket" originated from the German words "bracket" and "ketten", which mean "bracket" and "chain" respectively. This terminology was chosen to represent the dual nature of the bra and ket vectors, as well as their relationship in algebraic operations. The bracket symbol < > also resembles the notation used in linear algebra for representing vectors.