Page 17 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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8 The wave function
Re Ψ(x, t)
x
Figure 1.3 A standing harmonic wave at various times.
1.2 Wave packet
We now consider the formation of a composite wave as the superposition of a
continuous spectrum of plane waves with wave numbers k con®ned to a narrow
band of values. Such a composite wave Ø(x, t) is known as a wave packet and
may be expressed as
1
1
Ø(x, t) p A(k)e i(kxÿùt) dk (1:11)
2ð ÿ1
The weighting factor A(k) for each plane wave of wave number k is negligible
except when k lies within a small interval Äk. For mathematical convenience
we have included a factor (2ð) ÿ1=2 on the right-hand side of equation (1.11).
This factor merely changes the value of A(k) and has no other effect.
We note that the wave packet Ø(x, t) is the inverse Fourier transform of
A(k). The mathematical development and properties of Fourier transforms are
presented in Appendix B. Equation (1.11) has the form of equation (B.19).
According to equation (B.20), the Fourier transform A(k) is related to Ø(x, t)
by
1
1
A(k) p Ø(x, t)e ÿi(kxÿùt) dx (1:12)
2ð ÿ1
It is because of the Fourier relationships between Ø(x, t) and A(k) that the
factor (2ð) ÿ1=2 is included in equation (1.11). Although the time t appears in
the integral on the right-hand side of (1.12), the function A(k) does not depend
iùt
on t; the time dependence of Ø(x, t) cancels the factor e . We consider below
