At

as was done in the second line of (1.10). The condition that (j>(z, t) is real means that the coefficient of 0* must be the complex conjugate of the coefficient of <f>n.

Equation (1.11) shows that each normal mode behaves as an independent simple harmonic oscillator (SHO) satisfying the equation

To quantize the field, it is only necessary to quantize these oscillators.

Before doing this, however, we evaluate the energy in terms of the dynamical variables a„(t). Using the "orthogonality" relations (1.8) and (1.9), which can be written

/ dz a* (0)$* (z, t)am(0)</>m(z, t) = 6„,m |an(0)|2 = fi„,m |an(t)|2 Jo rL

/ dz an{0)(j)n(z, t)am(0)<f>m(z,t) = 6_n,man(0)a_„(0) e'2^1 Jo

= 5-n,m0n(t)0-n(t) i we obtain l l fL , d4> d<t>

= 2 tf E t2c" I I2 +cnc-„àn(i)à-„(i) + c„c_n<(i)à*_n(i)]

2 Jo 9z 9z

= 2 S fc" t2c" I ,2 +Cr,c-„an(t)a_„(t) + cnc_„<(i)aln(i)] .

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