Hilbert space and inner product

In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ(x) which depends upon the position variable x, or by an alternative function ψ(p) of the momentum variable p. It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars, which are complex numbers, are the elements of the quantum Hilbert space H. (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.)

If α is any scalar (complex number), the ket corresponding to the wave function ɑψ(x) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø(x)+ψ(x) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ(x) and (p), so that αø(x)+βψ(x) and αϕ(p) + βψ(p) correspond to the same quantum state α|ψ〉+β|ø〉.