The Hard Truth: Without Math Even 'Basic Concepts' In "Popular" Physics Books Are Meaningless

 

                           Seyda Ipek solving equation in her Carleton Univ. physics class

Among the most irritating remarks in any alleged “popular” physics book was undoubtedly Stephen Hawking’s in the Acknowledgements to The Brief History Of Time.  Therein he informs us that someone told him that "each equation included in the book would halve his sales.”  Well he did include one, the famous Einstein equation  E = m c² , but it hardly halved his sales.  This is given it was on The London Sunday Times bestseller list for no less than 237 weeks.  

All of which discloses that this circulating trope - still alive- amounts to a load of semi-erudite twaddle. And mostly emanating from nincompoops who either don't know physics, or are led astray by authors who claim the math just obscures the physics.  Not so with Physics Today (May, p. 51) reviewer Seyda Ipek in her review of Sean Carroll's book The Biggest Ideas In The Universe: Space, Time and Motion.  

As she rightly points out: "Carroll is aware that you need to understand the math to truly understand the physics that underpins our greatest brainchildren.  He diligently explains infinitesimal changes, for example, so that readers understand why Isaac Newton had to invent calculus to explain planetary orbits.

Over the course of the book readers go from learning how to take a derivative to gaining an appreciation of the metric tensor that describes spacetime.  That is also the path from Newtonian gravity to Einstein's general theory of relativity."

Is Carroll mad in his popularizing expectations? Or just optimistic to the point of skirting with physics fantasyland? What about reviewer Ipek? Is she grounded in her take or in La-La land?  I suspect she likely knows, perhaps even better than anti-math (in physics) curmudgeon ('Lost in Math')  Sabine Hossenfelder what is needed regarding communication of critical physics to the public.  As Prof. Ipek notes (ibid.):  

"Even physics majors in my classes are sometimes of the mindset if they just get the basic concepts they can understand topics as challenging as quantum mechanics."

But I can assert here this is preposterous given that a number of peculiarities of QM emerge in the mathematics behind it,and are next to impossible to grasp without it. For example, the issue of why complex numbers are needed to describe the wave function, e.g. 

Does Quantum Mechanics Really Need Complex Numbers? Yes!

And also why specialized statistics are needed to address the quantum measurement problem, e.g

Is There A 'Quantum Measurement Problem'? No - Provided One Adopts A Statistical Perspective

But there are vast areas even beyond these where math is essential for the student to grasp the concept.  For example, how a lower (kinetic) energy particle can surmount a higher energy barrier potential, i.e.

                                                                      

Consider a deBroglie (matter) wave arising from a single proton (p+) of form:

U(x) ~ sin(kx)

Where x is the particle's linear displacement (e.g. in 1-D) and k, the wave number vector(k= 2π/l), where l  denotes the wavelength. Though the associated kinetic energy K < V (the barrier "height") the wavefunction is *non-zero* within the barrier, e.g.

U(x_b)~ exp(- cx)

So using the math of wave mechanics  (and how a particle can be both wave and particle) we can visualize the particle of lesser energy K moving from the left side of the Energy-axis by"tunneling" through to the right side where it acquires the new wave function, U(x) ~sin (kx + φ), and φ denotes a phase angle.mNote that if the barrier is not too much higher than the incident particle energy, and if the mass is small, then tunneling is significant.

All the basic concept assimilation in the world won't enable the student to grasp the underpinning unless one also has the math under control. This is why Prof. Ipek's last point in her review is especially cogent, i.e.

"I don't want to make it sound like The Biggest Ideas In The Universe'  is solely a book about math.  But it's just so refreshing to read a popular science book that doesn't hide the math.  Instead of shying away from the nitty gritty calculations that led to the biggest leaps in our physical understanding of the universe."

Still it may well be that contrarian Hossenfelder thinks Sean Carroll's book is "awash" in excess math computations and he may well pay the price in lost sales.  But at least those who will read his book will be well rewarded by a superior understanding of the most revolutionary concepts in modern physics.

See Also:

• Why Teaching Quantum Mechanics Remains No Easy Task

And:

• How Math Drives Our Understanding Of The Universe