Mechanica C

CHAPTER 3 Mechanica C

FORMALISM

3.1 HILBERTSPACE
In the last two chapters we have stumbled on a number of interesting properties of
simple quantum systems. Some of these are “accidental” features of specific poten­
tials (the even spacing of energy levels for the harmonic oscillator, for example),
but others seem to be more general, and it would be nice to prove them once and
for all(the uncertainty principle, for instance, and the orthogonality of stationary
states). The purpose of this chapter is to recast the theory in a more powerful form,
with that in mind. There is not much here that is genuinely n e w ; the idea, rather,
is to make coherent sense of what we have already discovered in particular cases.
Quantum theory is based on two constructs: w ave functions a nd operators. T he
state of a system is represented by its wave function, observables are represented
by operators. Mathematically, wave functions satisfy the defining conditions for
abstract vectors, and operators act on them as linear transformations. S o the
natural language of quantum mechanics is linear algebra.1
But it is not, I suspect, a form of linear algebra with which you areimmediately
familiar. In an N -dimensional space it is simplest to represent a vector, |a ), by the
//-tuple of its components, {a„}, w ith respect to a specified orthonormal basis:

( a\ \
ai_
[3.1]
\ a NJ
' i f you have never studied linear algebra, you should read the Appendix before continuing.

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C hapter 3

Formalism
The inner product, {a\f$), o f two vectors (generalizingthe dot product in three
dimensions) is the complex number,
(a|/J) = a * h \

H
------- \-a*NbN .

[3.2]

Linear transformations, T , are represented by m atrices ( with respect to the specified
basis), which act on vectors (to produce new vectors) by the ordinary rules of matrix
multiplication:
I hi

1 12

hi
1/5) = T \a)

b = Ta =

• ■■

t\N \

hN

/ a\\

cn
[3.3]VjVI f Nl ■ ■ tNN / \ O N /
■
But the “vectors” we encounter in quantum mechanics are (for the most part)
f unctions, and they live in i nfinite -dimensional spaces. For them the //-tuple/m atrix
notation is awkward, at best, and manipulations that are well-behaved in the finite­
dimensional case can be problematic. (The underlying reason is that whereas the
f inite s um in Equation 3.2always exists, an infinite s um— or an integral— may not
converge, in which case the inner product does not exist, and any argument involving
inner products is immediately suspect.) So even though most of the terminology and
notation should be familiar, it pays to approach this subject with caution.
The collection of a ll f unctions of .v constitutes a vector space, but for our
purposes it ismuch too large. To represent a possible physical state, the wave
function ^ must be n ormalized :

j

\ V \2 d x = 1.

The set of all square-integrable functions, on a specified interval,2
fib
f ( x ) such that
/ |/( .v )|2 clx < o o,
[3.4]
Ja
constitutes a (much smaller) vector space (see Problem 3.1(a)). Mathematicians
call it L jia . b)\ p hysicists call it H ilbert space.3 Inquantum mechanics, then,
[3.5]

2 For us. the limits ( a and b ) w ill almost always be ± o o . but we might as well keep things more
g eneral for the moment.
■^Technically, a Hilbert space is a com plete inner product space, and the collection o f squareintegrable functions is only o ne example o f a Hilbert space— indeed, every finite-dimensional vector
s pace is trivially a Hilbert space. Butsince L j is the arena o f quantum mechanics, it's what physicists
g enerally m ean w hen they say '“
Hilbert spacc." By the way. the word com plete here means that any
C auchy sequence o f functions in Hilbert space converges to a function that is also in the space: it has no
“ holes” in it. just as the set o f all real numbers has no holes (by contrast, the space o f all p olynom ials ,...