Page 12 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.1 Wave motion 3

v of the wave is then the product of í cycles per second and ë, the length of
each cycle
v ˆ íë
and ø(x, t) may be written as

x
ø(x, t) ˆ cos 2ð ÿ ít
ë
It is convenient to introduce the wave number k, de®ned as
2ð
k (1:1)
ë
and the angular frequency ù, de®ned as
ù 2ðí (1:2)
Thus, the velocity v becomes v ˆ ù=k and the wave ø(x, t) takes the form
ø(x, t) ˆ cos(kx ÿ ùt)
The harmonic wave may also be described by the sine function
ø(x, t) ˆ sin(kx ÿ ùt)
The representation of ø(x, t) by the sine function is completely equivalent to
the cosine-function representation; the only difference is a shift by ë=4 in the
value of x when t ˆ 0. Moreover, any linear combination of sine and cosine
representations is also an equivalent description of the simple harmonic wave.
The most general representation of the harmonic wave is the complex function
ø(x, t) ˆ cos(kx ÿ ùt) ‡ i sin(kx ÿ ùt) ˆ e i(kxÿùt) (1:3)
p 
where i equals ÿ1 and equation (A.31) from Appendix A has been intro-
duced. The real part, cos(kx ÿ ùt), and the imaginary part, sin(kx ÿ ùt), of the
complex wave, (1.3), may be readily obtained by the relations
1

Re [e i(kxÿùt) ] ˆ cos(kx ÿ ùt) ˆ [ø(x, t) ‡ ø (x, t)]
2
1

Im [e i(kxÿùt) ] ˆ sin(kx ÿ ùt) ˆ [ø(x, t) ÿ ø (x, t)]
2i

where ø (x, t) is the complex conjugate of ø(x, t)

ø (x, t) ˆ cos(kx ÿ ùt) ÿ i sin(kx ÿ ùt) ˆ e ÿi(kxÿùt)

The function ø (x, t) also represents a harmonic wave moving in the positive
x-direction.
The functions exp[i(kx ‡ ùt)] and exp[ÿi(kx ‡ ùt)] represent harmonic
waves moving in the negative x-direction. The quantity (kx ‡ ùt) is equal to
k(x ‡ vt)or k(x ‡ x 0 ). After an elapsed time t, the value of the shifted
harmonic wave at any point x corresponds to the value at the point x ‡ x 0 at
time t ˆ 0. Thus, the harmonic wave has moved in the negative x-direction.