The failure of F=ma : Part 1
The need for Quantum theory
Are you new to the world of Quantum Mechanics or perhaps struggling to make sense of the equations that seem like a random ensemble of Greek alphabets? Don’t worry! It’s natural to study something quite rigorously without giving much thought to “why” we’re studying it. I have hence, compiled a few of many reasons why the great minds of the last century felt a need to come up with a new formalism to explain the physics of “small” objects. Why were physicists unsatisfied with Newton’s findings, despite F = ma having driven all of physics for the past 300 years? Let’s look back at the birth of quantum mechanics.
On a separate note, the following link to my article on classical mechanics concerns readers who aren’t familiar with the Lagrangian and Hamiltonian formalism of classical mechanics and serves as a recap for those who have studied it before. In a nutshell, the Lagrangian and Hamiltonian formalism are mathematical models used to find the variables of state as a function of time. The state of a system in classical mechanics is described by the Lagrangian, itself. The theory of subatomic particles typically prefers the use momentum as a state variable over velocities, putting the Hamiltonian at an equal footing as the Lagrangian in classical mechanics. One would be inclined to think that the Hamiltonian is what we would call the state of the system but that would be wrong, to say the least. It turns out, that the difference between the physics of “big and small” objects is the loss of this determinism of finding the complete set of variables, when we speak of the latter. I will return to this notion in a bit.
Measurement
Measurement is a process of determining a physical property of a system, such as its mass, position, momentum and energy, to name few. Of course, there are infinitely many methods one can opt from to carry out this process of measurement, however only a few that are not only efficient, but also that produce values of physical quantities that are very close to the actual values.
Having used the word “small” quite loosely earlier, a natural question may emerge in the mind of the reader. “How small is small? and for that matter, “how big is big?”. The terms small and big are relative for objects we can observe with the naked eye, but we must do better than that when we deal with the world of “small objects”.
In general, it suffices to assume that an object is big enough if the disturbances that accompany our method of measurement can be ignored. When weighing our body weights the first thing in the morning, one doesn’t heed to the weight of the clothes they’re wearing or perhaps the weight of the food they consumed the night before. This is quite natural, since we can consider their contributions negligible and yet have a satisfactory idea about our body weight. This in turn gives us quite accurate insights into our health. Of course the process of measuring our body weight can be improved by making improvements in our measurement techniques by taking the above into consideration.
In a similar way, we can define an object to be small if the disturbances accompanying our method of measurement can cause significant variations in the property of the object we’re trying to measure. After decades of development of the theory of quantum mechanics, it’s safe to say that an electron is a “small” object given that even the stray beams of light around it can effect its physical properties like its position, momentum and energy, to name a few.
Having the notion of Measurement at hand, one can immediately define the Apparatus. An apparatus is a system that interacts with the system being studied. The influences on the apparatus can be easily analyzed to deduce the properties of the system being studied. For instance, the velocity of a ball can be determined by analyzing how much it compresses a fixed spring placed in its path, as show in the figure that follows.
Since it is quite easy to figure out how much a spring compresses after collision with a ball, given the information about the physical properties of the spring, one can simply solve the energy conservation equation from classical mechanics to calculate how fast the ball was going when the experiment was carried out. In this example, the ball is the system, the spring is the apparatus and conservation of energy is the principle of measurement.
The concept of an apparatus becomes more profound when realized in the subatomic domain. One can only observe the effects of, let alone just observe, subatomic particles by analyzing the way they interact with macroscopic objects. This is simply because our perception of change is limited only to changes suffered by large objects, where large is again defined as explained earlier.
That said, our apparatus, that measures physical properties of objects in the subatomic scale, should be such that the changes it suffers caused by subatomic events are perceivable by our senses. This could be, say, in the form or light or sound. Additionally, given the extreme sensitivity of subatomic objects it is critical, in the very least, to keep track of the magnitude of disturbances involved in the interaction between the apparatus and the subatomic particle, that may result from the carelessness of the experimenter.
The Uncertainty Principle
However, the fact of the matter remains that the concept of Loss of Determinism is so deeply etched into the roots of quantum mechanics, to an extent that there is a principle, that goes by the name of Heisenberg’s Uncertainty Principle, that quantifies our limit of knowledge of system of small objects, where “small” refers to the objects that fit the definition from above.
The principle concerns two specific physical properties of particles, known to us even in classical mechanics — position and momentum. In fact it applies to any two canonically conjugate quantities. It states that,
“The position and momentum of a subatomic particle cannot be measured simultaneously to an arbitrary precision.”
We need to concretely define what we mean by “simultaneous” and “arbitrary”. By simultaneous we simply mean that our apparatus integrates technology that can measure the two properties, position and momentum, at the same mathematical instant of time. And what we mean by the precision of our apparatus being arbitrary, is that it simply gives us the true value of the position and momentum. A maximum precision device, if you will. The use of these fancy words is just a means to an end, since now the chances of the apparatus or perhaps the experimenter being at fault are eliminated, by considering a hypothetically perfect apparatus with infinite precision and unparalleled timing. And even then, says the principle, we are unable to determine the exact values of position and momentum of a subatomic particle, at the same instant of time.
From the laws of classical mechanics, we know that knowing the position and momenta of the constituent particles of our system forms the so called state of our system. We now seem to be stuck in this vicious loop that “the state is all we must know to predict how the system has been evolving and how it will continue to evolve, but nature prevents us from knowing both of them at the same time.” So does this mean we will never determine what our state is and how our system will evolve?
It turns out the answer to this big question lies in changing our definition of what the state of a system really means. Quantum mechanics does exactly that. The state of our system is no longer just the Lagrangian or the Hamiltonian of our system. This is because the “path” or the series of events our system goes through loses its meaning, as it is no longer possible to tell for sure what the state will look like at a later time. Instead the information about our system is encoded in a new type of mathematical object that we call the state of our system.
Summary
Heisenberg’s view so disturbed Einstein, to such an extent, that it led him to believe the existence of “hidden variables”. This is a notion that, if we can’t determine the position and momentum simultaneously there must simply be other physical a properties of the system, the existence of which we don’t know of. When these properties are unveiled as functions of time, they provide just the right amount of information to figure out the evolution of our system. He was eventually proved wrong.
To end this discussion, it is safe to say that, “all that we know, is that we can’t know all”. This uncertainty of knowledge of the variables of state, has got nothing to do with the lack of technology, measurement techniques or even faults in the part of the experimenter and his apparatus. Nature, herself, restricts us from knowing everything that is needed, to say with certainty how a system will evolve through time.
