Gt i MO

The a's can then be expressed in terms of the real p's and q's ipn + unqn „ -ipn + uinqn an =

sfi^n and the Hamiltonian becomes a„ =

which is a sum of independent oscillator Hamiltonians. This is confirmed by substituting (1.12) into Hamilton's equations of motion dH

OH dqn

-^nln wnich gives back the familiar equations of motion for uncoupled oscillators.


We now quantize the string by the canonical procedure: the canonical variables are made into operators which are defined by transforming the Poisson bracket relations into commutation relations [for a review of this procedure see, for example, Schiff (1968), Sec. 24], For the generalized coordinates and momenta this leads to the following commutation relations:

In what follows we will always set h = c = 1. This defines the so-called natural system of units, which is very convenient. It is important to realize that the correct factors of h and c can always be uniquely restored at the end of any calculation, if desired. These units are discussed in Prob. 1.1 at the end of this chapter.

1.3 QUANTIZATION OF THE STRING 11 From the commutation relations (1.13) we obtain

[an, 4,]


= [4. air.] = 0.

where the complex conjugate of a complex number (sometimes called a c-number) must be generalized to the Hermitian conjugate of an operator (sometimes called a q-number), and the operators an are independent of time. The time dependence is in the fields, which are also operators*:

4>{z,t) = v J] i {flBei(t"'-«"<)+flte-i(fc»'-u»t)}

where the positive frequency part, contains the sum over an (later to be identified as annihilation operators) and the negative frequency part, <j>(~\ is the sum over oj, (the creation operators). In this case <f> is Hermitian because it is associated with a physical observable (the displacement), but in general a field need not be a Hermitian operator. We will study such fields in Part in of this book.

The Hamiltonian is also an operator, and its precise form depends on the order of a* and a, which was unimportant when these were c-numbers. Perhaps the most "natural" form for H is oo 1

However, the sum over \uin gives an infinite contribution to the energy (the zero-point energy), which can be removed simply by redefining the energy. This redefinition will lead to the idea of a normal ordered product, which will be defined and discussed in Sec. 1.6 below. For now we will simply adopt the following form for H:

Note that H is the sum of the dimensionless operators a],an, each multiplied by the energy wn of the normal mode which it describes.

*To avoid singularities, we will exclude the state n = 0 from this sum. Later, when we take the limit L —* oo (the continuum limit), the sum will include states of arbitrarily small energy.


The commutation relations between a and a* also imply relations between the fields 4>. Suppose we regard 0 as a canonical coordinate. Then, the canonical momentum is [using C defined in Eq. (1.2)]

Then, generalizing the commutation relations (1.13) to a continuous field, we expect to find relations of the form


= iS(z - z')


= 0


where the 6(z - z') function is the generalization of the Kronecker 6nm which appears in (1.13). These important commutation relations are known as the canonical commutation relations, sometimes referred to as the CCR's.

To prove the relations (1.17), we use the explicit form for n:

7r(z,t)=- Y i—iujnan e'C^-""') + ^„qt e-i(knz-»nt)\ _


+ [on,aJ„] e<(fcn«-fcmz')-i(.-n-«m)t _ [4,am] * - fcm )+ .(u,„) t J

However, the functions -^e'*"2 are complete (i.e., any periodic function can be expanded in terms of them) and orthonormal, and hence

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