QuantumPhysics.dvi

Clearly,Dhas a well-defined action on every basis vector. But now consider itsaction on

a general vector|φ〉 ∈ H, with the decomposition of (3.18), and compute the norm of the

result. We find,

||D|φ〉||^2 =

∑ n

n^2 |cn|^2 (3.29)

If only a finite number ofcnare different from zero, then this norm will be finite. But it is

perfectly possible to have||φ〉||finite, but||D|φ〉||=∞. Take for examplecn= 1/n, which

defines a normalizable vector|φ〉, but for whichD|φ〉isnot normalizable. In the isomorphic

Hilbert spaceL^2 ([−πℓ,+πℓ]), this issue appears under a more familiar guise. A functionf

for which||f||is finite is square normalizable; the operatorDis equivalent to taking the

derivative of the functionfin the Fourier basis. But we know very well that the derivative

of a square normalizable function need not be normalizable; in fact it need not be a function

at all but could be a Diracδ-function.

There are many other facts and relations that hold generally true for operators on finite-

dimensional Hilbert spaces, but fail for infinite-dimensional ones. Astriking example has

to do with the nature of the commutator. For two finite dimensionalmatrices, we always

have tr[A,B] = 0, because trAB= trBA. But for operators in Hilbert space, this is more

tricky, as may be seen by taking the familiar position and momentum operators, for which

[x,p] =i ̄h. The rhs of this equation is really multiplied by the identity operator, whose trace

is not expected to vanish !!

Subtleties, such as this one, associated with operators acting on infinite dimensional

Hilbert spaces, will not be discussed in all generality here. Instead,we shall handle these

issues as they arise, more or less on a case by case basis. Beyond that, there is a vast mathe-

matics and mathematical-physics literature on the subject, and things get quite complicated.

3.3 Special types of operators

We now generalize and extend the role of certain special operatorsto a separable Hilbert

spaceHwhich may be of finite or of infinite dimension.

• The identity operator inH, denoted by I or IH maps every |φ〉 ∈ H into itself,

IH|φ〉=|φ〉. In an orthonormal basis{|n〉}n=1,···,N, it may be expressed as

IH=