Linear algebra application
The vectors notated with right-
hand angle brackets like|...〉
are the ones that we could
represent as column vectors
(p. 965) if the vector space is
finite-dimensional. Left-hand
angle brackets are like row vec-
tors. A row vector multiplied
by a column vector is a way
of notating an inner product,
which is the same idea as a no-
tation like〈...|...〉. To turn a
column vector into a row vec-
tor, we transpose it and take
complex conjugates of its ele-
ments. This is analogous to
the rule of taking complex con-
jugates when converting back
and forth between left-hand an-
gle brackets (“bras”) and right-
hand ones (“kets”). In both
contexts, the basic reason for
the complex conjugation is that
we want the inner product of a
vector with itself to be a posi-
tive real number.in which case the wavefunction might look like Ψ(x 1 ,x 2 ). The vari-
ables that the wavefunction depends on may be either continuous,
like position and momentum, or discrete, like spin or angular mo-
mentum. Given all of these possibilities, we need to figure out an
appropriate generalization of the integral overxthat we originally
used to define our normalization condition. To provide for flexibility
and generality, we will start by simply defining a new notation that
looks like this:
〈Ψ|Ψ〉= 1.
In the case where Ψ is a function ofxalone, the angle brackets
〈...|...〉basically mean just an integral overx, and we think of the
〈...|part as automatically implying the complex conjugation of the
thing inside it. The operation〈...|...〉is called theinner product.
Because negative probabilities don’t make sense, we require that
the inner product of a wavefunction with itself always be positive,
〈u|u〉≥0.This makes it similar to the dot product used with vectors in Eu-
clidean geometry.
In the case of Euclidean geometry, the ability to add vectors
and measure their lengths automatically gives us a way to judge
the similarity of two vectors. For example, if|u|= 1,|v|= 1, and
|u+v|= 2, then we conclude thatuandvare in the same direction.
On the other hand, if|u|= 1,|v|= 1, and|u+v|=
√
2, then we can
tell thatuandvare perpendicular, which makes them as different
as two unit-length vectors can be. More generally, (u+v)·(u+v) =
|u|^2 +|v|^2 +2u·v, because the dot product is linear, so we can see that
the information about how similaruandvare is all contained in
their dot productu·v. Making the analogy with quantum mechanics,
we expect that since we can define normalization of wavefunctions,
we should automatically get, “for free,” a way of measuring how
similar two states are.
With this motivation, we assume that there is an inner product
on wavefunctions that has properties analogous to those of the dot
product. We assume linearity, so that ifu,v, andware wavefunc-
tions, then
〈u|αv+βw〉=α〈u|v〉+β〈u|w〉
and〈αu+βv|w〉=α∗〈u|w〉+β∗〈v|w〉.
In the second equation, we need to take the complex conjugatesα∗
andβ∗, for if we omitted the conjugation, then when〈u|u〉 = 1
we would have〈iu|iu〉=−1, describing a negative probability. For
similar reasons, we require that
〈u|v〉=〈v|u〉∗
Section 14.6 The underlying structure of quantum mechanics, part 2 981