The commutation rules assume the form in VS .NET

Creating PDF417 in VS .NET The commutation rules assume the form

The commutation rules assume the form
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[<J>(k), <J>(k')] = [ii(k), ii(k')] = 0 [<J>(k),ii(-k')] = i2::e- i(k-k')n
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The last sum is also 2n Ln c5(k - k' + 2nn) which means that if <J> and ii are extended as periodic functions of k this reduces to 2nc5(k - k'). By virtue of (3-13) the hermitian hamiltonian is now expressed as
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{iit(k)ii(k)
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+ <J> t(k)[m2 + 2(1 + 2(1
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cos k)] <J>(k)}
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To identify this with a sum of individual uncoupled oscillators we set
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Wk = W-k = Jm 2 ak
- cos k)
-...;4nwk
~ [wk<J>(k) + iii(k)]
~ [wk<J>t(k) iiit(k)]
(3-16)
at =
-...;4nwk
[ak' ak'] = c5(k - k')
The operators at and ak create or destroy the mode k with energy Wk' The hamiltonian H reads 1 f+1t f+1t (3-17) H ="2 -1t dk wk(atak + akat) = -1t dk Hk while the fields are expressed as
(3-18)
110 QUANTUM FIELD THEORY
The denomination of ak and al as destruction and creation operators is justified by the fact that if IE) is an eigenstate of H with energy E we have
+ [H, ak] IE) = (E - wk)ak IE) HallE) = EallE) + [H, IE) = (E + wk)alIE)
Hak IE)
Eak IE)
(3-19)
The mode k of energy Wk is interpreted in this mechanical model as a coherent quantized vibration of the lattice atoms, or phonon. Here we understand clearly the relation between particles (phonons) and fields (CPn is the displacement of the nth atom). Our physical intuition might lead us first to consider states of the crystal characterized by a wave function obtained by diagonalizing the field
We shall, rather, choose to generate the states starting from the ground state 10) and its excitations in terms of phonons. We encounter at once a difficulty since the ground-state energy, the lowest eigenvalue of the hamiltonian, is in fact infinite. Each mode k contributes an amount iWk to the zero point energy. This is in agreement with the 'uncertainty relations since each oscillator has a minimum momentum spread due to its potential energy. Since we have a continuum of modes, each slice (k, k + dk) leads to an infinite energy-a fact to be traced to the infinite size of the system. Indeed, since
<Olal = 0
(3-20)
and the last integral is meaningless since [ak> al] = b(O)!. If, however, the crystal is of finite size (- N :::;; n :::;; N) let us take periodic boundary conditions by identifying the sites nand n + p(2N + 1), realizing a circular arrangement. The wave vector k would then be restricted to values k = [2n/(2N + l)]q with q an integer and - N :::;; q :::;; N, and the zero point energy (N /2N) I ~ ~ Wq would be finite. Clearly (1/2N) I~~Wq has a finite limit as N tends to infinity, showing that the above energy is indeed proportional to the size of the system. By taking a discrete instead of a continuous model, we have introduced a Brillouin zone -n :::;; k :::;; n in momentum space, equivalent to an ultraviolet cutoff in the original model. The reader will recall that the origin of the expression "ultraviolet catastrophe" stems from the divergent contribution to the blackbody thermal radiation of high-frequency modes of the electromagnetic field. A space cutoff allows, further, an unambiguous definition of the hamiltonian operator. This is referred to by saying that we put the system in a box. We can, however, use the following device. Zero point energy is unobservable unless we destroy the crystal. In field theory, the ground state will be interpreted as the vacuum and it will be even harder to destroy! Energy exchanges with the crystal are insensitive to the choice of an origin. We declare by fiat that the ground state has zero
QUANTIZATION-FREE FIELDS
energy and we redefine the hamiltonian as
(3-21)
f-+,," dk wkalak
We shall, of course, have to make sure in relativistic theories that this procedure preserves Lorentz covariance. In this new expression the creation and annihilation operators appear in "normal order," the latter to the right of the former. This is also called Wick's ordering and is denoted by a double-dot symbol:
:i(alak + akal):
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