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For a given frequency w = ck, describing the electromagnetic radiation field by two scalar potentials, we can reduce the general three-dimensional problem of finding the natural modes of an open cavity to two scalar problems [84]: 1. Solving the Helmholtz equation (9.92)

(9.93)

Preferences window. To access dozens of Preferences in QuickBooks, follow these steps: 1. Go to Edit Preferences. A Preferences window, such as the one shown in Figure 12-1, appears.

over the cavity walls and Sommerfeld's radiation condition (9.63) at infinity, which amounts to 15 e ikr lim u(r) = g((},cjJ)-. (9.94)

(9.95)

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Click to restore defaults Figure12-1: All Preferences are changed from this window, which changes based on the icons accessed and whether you select the My Preferences or Company Preferences tab.

on s -

(9.96)

over the cavity walls and the asymptotic condition lim v(r)

2. Scroll to the type of Preferences you want to access using the scroll bar on the left of the Preferences window.

0 1995 1996 1997 1998 1999 2000 2001 2002

(9.97)

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The radiation condition implies that a linear combination of natural modes can only reproduce outgoing fields. Thus, it does not form a complete set capable of representing every conceivable quantum vacuum fluctuation which can also have incoming fields from infinity. To get a complete set of modes for the quantized radiation field, we must change the asymptotic boundary condition at infinity to allow for incoming and outgoing waves together, as in a scattering problem. The resulting modes will form a continuum. They are the modes ofthe universe, which were determined in Section 2.5.2 using precisely such a scattering formulation. There it was convenient to work with plane waves. Here, however, it is more convenient to formulate the asymptotic boundary condition in terms of spherical waves. So in place of Sommerfeld's radiation condition, we demand that [607] lim If>(r) = XI((}, '1')-r

QuickBooks offers hundreds of Preferences that are organized by program area. It s helpful to have an overview of the Preference features you can use to customize QuickBooks to meet the needs of your company. Some of the following Preferences are discussed later in this chapter. Other Preferences, such as the ones for nance charges, charts and graphs, Missing Send Forms, Service Connections, and Spelling Preferences, are self-explanatory. Accounting allows you to use account numbers, specify which subaccounts to show on your chart of accounts, use class tracking, maintain an audit trail of transactions affecting an account, automatically assign general journal entry numbers to accounts, warn you when you are posting a transaction to Retained Earnings, and set a password or establish a closing date.

+ XR((}, '1')-,

(9.98)

FIGURE 7.29 Main sources of corn imports in the EU, 1995 2002.

Checking allows you to print account names on vouchers, change the check date when it is printed, start with the payee eld on each check, display warnings about duplicate check numbers, auto ll payee account numbers in the memo portion of checks, set default accounts to use, and activate online banking payee aliasing Desktop View allows you to select options for customizing the appearance of your QuickBooks desktop Finance Charge you can specify your annual interest rate, minimum nance charge, and grace period, specify a nance charge account to debit, indicate how to assess overdue nance charges, and mark nance charge invoices to be printed General these Preferences allow you to specify additional warnings or sounds you may want to accompany transactions, and indicate the format for displaying numbers in QuickBooks elds.

where If> is u in case 1 and v in case 2. The first term on the right-hand side of (9.98) represents an incoming wave from infinity, and the second term, the scattered outgoing wave. So XI and XR are related by the scattering matrix S (Le., XR = SXI). So there lies the problem of using natural modes in a quantum theory. The Sommerfeld radiation condition satisfied by natural modes is only part of the asymptotic boundary condition at infinity (9.98) that defines the set of all possible solutions for the radiation field. To make a complete set out of natural modes, we must supplement them with other modes that, to put it crudely, satisfy the remaining required asymptotic boundary conditions (9.98). This is, as far as I know, still an open problem in

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