Impossibility of Monochromatic Light [duplicate]

Question

Pages 24-25 of my textbook, Optics by Hecht, says the following:

Using the above definitions we can write a number of equivalent expressions for the traveling harmonic wave: $$\psi = A\sin k(x \mp vt)$$ $$\psi = A\sin 2 \pi \left(\dfrac{x}{\lambda} \mp \dfrac{t}{\tau} \right)$$ $$\psi = A\sin 2 \pi (\kappa x \mp vt)$$ $$\psi = A\sin (kx \mp \omega t)$$ $$\psi = A\sin 2 \pi \nu \left( \dfrac{x}{v} \mp t \right)$$ ... Note that all these idealized waves are of infinite extent. That is, for any fixed value of $t$, there is no mathematical limitation on $x$, which varies from $- \infty$ to $+ \infty$. Each such wave has a single constant frequency and is therefore monochromatic or, even better, monoenergetic. Real waves are never monochromatic. Even a perfect sinusoidal generator cannot have been operating forever. Its output will unavoidably contain a range of frequencies, albeit a small one, just because the wave does not extend back to $t = - \infty$. This all waves comprise a band of frequencies, and when that band is narrow the wave is said to be quasimonochromtic.

These are the problems I'm having:

1. It does not explain how time is relevant to a wave being monochromatic; so why does the fact that physical waves do not extend back to $t = -\infty$ mean that physical monochromatic waves are an impossibility?
2. As relating to 1, how do the equations of the idealized harmonic waves imply that the wave exists for all time? Time $t$ is an independent variable of the equations, which is free to be selected by us, so I'm a little confused about this.

I have read several explanations of this, both on this website and on others, but these explanations do not seem to sufficiently clarify my two points above.

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

1.It does not explain how time is relevant to a wave being monochromatic; so why does the fact that a physical waves do not extend back to t=−∞ mean that physical monochromatic waves are an impossibility?

One has to understand what a mathematical model of a physical observable, in this case light of a certain frequency, means. It means that for the model to hold all its implications are manifest.

This means that the monochromatic model above says that if we go one kilometer away from the beam (lets suppose there is a monochromatic beam) the same intensity will be found ( lets alone what happens in time, that the beam should always exist, and we could always measure it). This does not fit our observations, because we have beams of light that start appearing , and stop appearing. BUT the model above is not useless, the mathematics leads to wave packets, which can have close enough frequency for our observations to apply "monochromaticity". Wavepackets solve the same wave equations and take away the problem of monochromaticity.

2. As relating to 1, how do the equations of the idealized harmonic waves imply that the wave exists for all time? Time t is an independent variable of the equations, which is free to be selected by us, so I'm a little confused about this.

It is a logical conclusion: if it is time independent it should be measurable at any time t, which as I argue above , is not what is observed

So we use the solutions of the wave equations so as to fit what we are really observing.

Quantum mechanics, which mathematically describes how light appears, also explains the physics of why there is always a width to the frequencies of monochromatic light: light comes from energy levels which have a width, which you will learn if you study further in physics.

Answer 2

To Question 2:

As relating to 1, how do the equations of the idealized harmonic waves imply that the wave exists for all time? Time $t$ is an independent variable of the equations, which is free to be selected by us, so I'm a little confused about this.

The equations alone do not determine the solution of the wave equation, what is important here is: In what Region are the harmonic wave equations being solved? The author wants to make a Statement about our perceived reality, so he chooses the Region to be all time and all space. And yes: A proper solution of the harmonic wave equation with only one frequency then has to exist for all time and all space. Any position / time where the wave would suddenly "stop" would be a position where the wave equation fails. Such a solution (and ONLY such a solution) is called monochromatic wave.

This should also answer question 1: In reality every wave we observe begins / ends somewhere, and as such is not a solution of the harmonic wave equation of the class said above. Implications of this are that if you had such a wave running through a spectrum analyzer with abitrary high Resolution, it would detect a range of different frequencies, stemming from the beginning and the end of the wave.