By Antonio Fasano, Mario Primicerio

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N } will produce a positive determinant (the required volume), while swapping any of those vectors will only change the determinant’s sign. This then gives rise to the idea of a signed volume of an ordered set of vectors, being the determinant of the matrix of those vectors laid out in the row order in which they have been speciﬁed. 2 Determinants, Handedness, and the n-Dimensional Cross Product The sign-changing behaviour of the determinant when its rows or columns are swapped makes it perfect for specifying the handedness of a set of ordered, linearly independent vectors in Rn .

If A is to represent the action of a measurement). The hermiticity implies that f |Ag = Af |g , or b b f ∗ (x) c g (x) dx = a c∗ f ∗ (x) g(x) dx . 135) by parts to write b − a b f ∗ (x) c g(x) dx = a c∗ f ∗ (x) g(x) dx . 8 Frequency and Wavenumber 47 Since f ∗ = f ∗ , we infer that c∗ = −c, so that c must be pure imaginary: c = ia for some real a. Hence ia d/dx is hermitian. Finally, because a whole number power and any real multiple of a hermitian operator are also hermitian (the proofs are straightforward), in particular d2 /dx2 must also be hermitian.

0 42 2 A Trip Down Linear Lane Two new products result immediately from making these identiﬁcations. 112) where δij is the usual Kronecker delta function. The second, outer product, comes from multiplying a ket with a bra: outer product: |j i| = a matrix of zeroes . 113) A ﬁrst use for this bracket formalism comes by way of matrix multiplication. Remember that the product AB of matrices A and B is deﬁned in the usual way as a matrix of dot products of each row with each column: (AB)ij ≡ Aik Bkj .