— a* am e~'(fc"""fcm)z+'('*'"_ü'">)t — a^ an e*(fc"-fc"»)z_*(ü;n_ü'm)t^ = {L-fl«a- e_2iWnt + fc-«anflU e2^-4 - 2fcnat a„} .

However, the first two terms sum to zero, because they are odd when n is changed to -n (recall kn = — k-n but = w_n). Hence which expresses the total momentum as a vector sum of the momentum of each phonon (kn) times the number of phonons with that momentum (a^a,,). [The full vector character of the momentum operator is only partially illustrated by this one-dimensional example, where Pz has only a 2 component and all kn must be in the ¿-direction so that kn can be only positive or negative.] We see that the momentum operator is precisely what we would expect from the interpretation of phonons as particles with energy wn and momentum kn.

1.7 THE CLASSICAL LIMIT: FIELD-PARTICLE DUALITY

The Fock states are the quantum mechanical eigenstates of the Hamiltonian. What do these have to do with the classical vibrational states of a string? What is the classical limit? Before giving a full answer to these questions, we make two preliminary observations.

First, note that a state with a definite number of quanta corresponds to a case where the average field is zero, but otherwise the field is completely unknown. To show this, consider a state with n\ quanta of type 1: |ni). Then, for quanta of any type m (including m = 1)

so that the average field is zero,

However, the average of the square of the field is not zero. In fact,

However, the average of the square of the field is not zero. In fact, where (m|oiaj|ni) = 0 = (nj|a|a]|ni) has been used. Next, note that

(nilaiajlnx) = 6i3 + (m|ajai|ni) = + niSnSji (n^atajlri!) = ni6a<5ji and hence

because |z| diverges. Hence the uncertainty in <p, A<f>, is

and <j) is completely uncertain, beyond knowing that (<fi) = 0. [To define the field so that A4> / oo, we may "smear" it, introducing

<P(f) = J dz f(z)<t>(z,t) , where f{z) is strongly peaked in the neighborhood of a point 2 = z0, and very small elsewhere; see Prob. 1.5 at the end of the chapter.]

For our second observation we note that no state behaves like a classical wave for all z and t* This would require that the field <p and its "velocity" n commute, and the CCR's (1.17) show that this is not the case. Another way to see this is to rewrite 0 as a sum of traveling waves,

<£(z, t) = V y^ 1 {An cos (knz - LJnt) + Bn sin (knz - wnt)} , where or, dropping the n

The operators A and B must be simultaneously diagonalized in order that 4>{z, t) have a definite value for all 2 and t. But this is impossible, because A and B are non-commuting operators:

[A, B] = i [(a + af) , (a - a*)] = -2i and hence cannot be simultaneously diagonalized (i.e., cannot both have definite values). Furthermore, the above commutator implies an uncertainty relation

These two results give limitations on our ability to define the field and show that it cannot be defined exactly. However, states do exist in which A and B have a very small fractional uncertainty. Such states correspond to a classical field as much

* Thanks to Charles Sommerfield and Alan Chodos for clarification of this point.

as is possible in quantum mechanics. Since an optimization of (1.26) requires that AA ~ AB ~ 1, small fractional uncertainty in the values of (A) and (B) is possible only if (A) and (B) are both very large. However, these quantities are related to the average number of quanta through the relation

and hence such states must have a large average number of quanta (M). If we parameterize (A) and (B) by

and if (M) —► oc, then the fractional uncertainty in (A) and (B) goes like A A 1

oo and the fractional uncertainty in A and B is small and the average field (<j>) is well-defined in both amplitude and phase (except for exceptional cases where sin 6 or cos S = 0).

An example of a class of states with this property is the coherent states, which are the eigenfunctions of the annihilation operator a. These states can be written

where C is a normalization constant, and K is the complex eigenvalue corresponding to the eigenvector | K)

J2L jsn 00 Kn a\K) = C £ ~7=¡a\n) = C £ 4=p/¡¡|n - 1> ^ Vn! ^ Vn!

We will parameterize the eigenvalue K by

K — V~N eia and normalize the state

which implies that

It is worth noting that the operation of the creation operator on the coherent state is equivalent to differentiating the state with respect to the complex number K,

at|K) = C £ "rfaV> = C £ ^VnTT|n + 1) vn! ^ vn!

Using these remarkable results, we can quickly calculate (A), (B), A A, and AB. First, using (Ay = (K\K*,

(K\at\K) = K* (AT|at2|/fc) - K*2 (K\cta\K) = \K\2 = N

and hence

(A2) = (a2 + a*2 + aaf + a! a} = 2Re(K2) + 1 + 2\K\2

= 2N(cos2a + 1) + 1 = 4JVcos2 a + 1 (B2) = - (a2 + a*2 - aat - a+a) = -2i*eA2 + 1 + 2(A)2 = 27V(1 — cos 2a) +1 = 4N sin2 a +1 .

Therefore

A A = ((A2) - (A)2) = (4N cos2 a + 1 - 4 N cos2 a)1/2 = 1

AB = ((B2) - (B)2) = (4JV sin2 a + 1 - 4N sin2 a)1/2 = 1

and the fractional uncertainty in A and B does indeed approach 0 if N —► 00. Furthermore,

so that N is indeed the average number of phonons, but the uncertainty in the number of phonons also approaches zero as N —► oo:

So far our considerations have been limited to a specific frequency. To obtain a well-defined field, we must construct a coherent state for each frequency. Hence the general state is of the form and there is a field-particle uncertainty relation, or complementarity principle. If AAf = 0, then A<j) = oo, while if A<p is small, AM must be large.

One of the most fundamental problems in physics is the determination of the time evolution of physical observables. In the language of quantum mechanics, this problem is solved by finding an operator from which it is possible to calculate how matrix elements of quantum mechanical operators evolve in time. We close 'his chapter with an introductory discussion of how this is done in field theory. We will return to this issue several times in later chapters, but our development will always be very similar to the one presented here.

The time translation operator can be found from the Hamiltonian, which describes how states evolve over an infinitesimal period of time. In field theory, this property of the Hamiltonian is described mathematically by the following relations: _

.dj>(z,t) 1 dt | |

[H,7T(z,t)] = |
.dir{z,t) 1 dt ■ |

These fundamental relations are sufficient to establish H as the generator of time translations and to permit the construction of the operator for finite time translations (for more discussion, see Chapter 8).

To prove the above relations for the one-dimensional string, we ignore the fact that H is normal ordered, since the only difference between a regular product and a normal ordered product is a c-number, which commutes with <p and -k. Then

Now this must hold for any operator 4>, and assuming that these operators are a mathematically complete set, so that any operator on the space of Fock states can be expanded in terms of them, the combination H - can commute with all <p only if it is a multiple of the identity (this is an application of Schur's Lemma), giving

where Eq is an arbitrary constant. Hence dU

For H independent of time, this gives

U(t,to) — exp {—i{H — Eo)(t — ¿o)] , (1-36)

where the normalization of the exponential is fixed by the initial condition

This result assumes H is independent of time but can be generalized to cases where H depends on the time, which is normally the case when interactions are included. This will be discussed in Chapter 3.

If we choose E0 to be the ground state expectation value of H, then H - E0 has a zero ground state expectation value, and that is equivalent to using the normal ordered form for H and taking Eq = 0. With this choice (which we made in the previous sections).

The form (1.15) for <j> satisfies this condition (see Prob. 1.4).

In the next chapter we apply these ideas to the quantization of the electromagnetic field.

1.1 In this book we are using natural units in which h = c = 1. This means that length (L) and time (t) have the same dimensions and that mass (m) has the dimensions of an inverse length.

(a) Using the Fermi (/) as the fundamental unit of length, where If = 10"15 meters, find:

• The radius of the first Bohr orbit of hydrogen.

• The energy of the ground state of hydrogen.

(b) Repeat part (a) using the MeV as the fundamental unit of energy. Find a conversion factor between / and MeV.

(c) An expression in natural units can always be converted uniquely into an expression in ordinary units (L,t,m) by inserting h and c in the correct places. Give an argument describing precisely how to do this for any expression and give some examples showing the correctness of your argument.

1.2 The momentum operator of the string is rL dz d(t> d<j>

Prove that this is the generator of translation in the ¿-direction. In particular, prove that

[P\n(z,t)}=i 1.3 Consider the Lagrangian density dz Ö7r(z, t)

dz at dtp dz

where (p = 4>{z,t) is a generalized coordinate.

(a) Find the momentum conjugate to 4>.

(b) Find the equations of motion for the fields and the solutions. Use periodic boundary conditions.

(c) Suppose the field is expanded in normal modes

= Yl°n {an^n(-2,t) + al(p*n{z,t)} , n where an satisfy the commutation relations

Find the coefficients cn which will insure that the CCR's assume the standard form

(d) Find the Hamiltonian density, and express the Hamiltonian in terms of the number operators aj,an.

(e) What is the physical significance of this field?

PROBLEMS 27

1.4 The Hamiltonian is the generator of time translation. This means that emt-t0)^Z! to) e-imt-t0) = t) _

Prove that this relation holds for the one-dimensional string.

1.5 [Taken from Sakurai (1967).] Consider a three-dimensional scalar field like that introduced in Prob. 3 above:

(b) To make the fields more regular, we smear the fields by averaging them over a small region of space. Suppose we define the average field in the neighborhood of the origin by where kn ■ x = wnt -kn r and |an, a*, = 6nn>, cun = y'm2 + (a) These fields are singular operators. Show that

Find the precise result if m = 0.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 2

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