2 1. PROLOGUE
that any attempt to permute the physicists’ meanings of
and would threaten
We shall, however, employ the standard mathematical notation for the norm
on the Hilbert space: u = u|u
We also denote the transpose of a complex
matrix A by
; for real matrices we shall generally use
Now, some more subtle matters. In mathematicians’ dialect, u|v is the inner
product of two vectors u and v in the Hilbert space H. Moreover, if u ∈ H, the
map φu(v) = u|v is a bounded linear functional on H; the correspondence u ↔ φu
gives a conjugate-linear identification of H with its dual H , which mathematicians
generally take for granted without employing any special notation for it. Physicists,
on the other hand, distinguish between elements of H and elements of H , which
they respectively call ket vectors and bra vectors and denote by symbols of the
form |u and u|. If |u is a ket vector (what mathematicians might call simply u),
the corresponding bra vector u| is the linear functional denoted by φu above. The
action of the bra vector (linear functional) u| on the ket vector (element of H) |v
is the inner product (i.e., “bracket” or “bra-ket”) u|v .
In this system “u” can be any sort of convenient label to identify the vector:
either a name for the vector itself, like the mathematicians’ u, or an index or set
of indices that specify the vector within an indexed family. For example, if one
is working with an operator with a set of multiplicity-one eigenvalues, one might
denote a unit eigenvector for the eigenvalue λ by |λ , or an eigenvector for the nth
eigenvalue simply by |n . Likewise, joint eigenvectors for a pair of operators with
eigenvalues λn and μj might be denoted by |n, j ; and so forth. (In most situations
the ambiguity of a scalar factor of modulus one in the choice of eigenvector is of
no importance, for reasons that will become clear in Chapter 3.) Mathematicians
may find this convention uncomfortably informal, but its virtue lies in its flexibility
and its ability to strip away inessential symbols. Mathematicians would typically
denote the nth eigenvector by something like un,
but the symbol u is just a place-
holder; it is the n that carries the useful information, and the physicists’ notation
gives it the starring role it deserves.
Next, let A be a linear operator1 on H: in mathematicians’ notation, A maps
a vector v to Av; and in physicists’ notation, it maps |v to A|v : no surprises here.
But the physicists’ notation for the scalar product of A|v with the bra vector u|
is u|A|v :
physicists’ u|A|v = mathematicians’ u|Av .
Now, u|A|v can also be considered as the scalar product of a bra u|A with the
ket |v , and in this way A defines a linear operator on bra vectors. This operator
is what the mathematicians call the adjoint of A when H is identified with H — a
point that can lead to some confusion if one does not remain alert. Recalling that
the adjoint of A is denoted by A† and the conjugate of a ∈ C is denoted by a∗, we
On the theory that flexibility and concision are more important than consistency,
we shall feel free to denote elements of the Hilbert space by either u or |u , and we
shall write either u|A|v or u|Av as convenience dictates.
need not be bounded. The notion of adjoint for unbounded operators involves some
technicalities, but they are beside the point here.