Like it or not, you can’t get very far in physics without running into mathematics. Mathematics is the language of physics, and at the end of the day nearly every law of physics has an underlying mathematical model. These models are what make physical laws so useful, and they are the very basis for making predictions or designing new technologies.
QUANTUM QUOTE
Do not worry about your difficulties in mathematics. I can assure you mine are still greater.
—Albert Einstein
A good example with which you may already be familiar is Albert Einstein’s discovery that matter and energy are inter-convertible under certain conditions. The basic notion is easy enough to write down in words, but to make any real use of it we need a mathematical model. This is provided by Einstein’s famous formula: E = mc2. You’ve probably seen this before, and you may even know that “E” stands for energy, “m” stands for mass, and “c” refers to the speed of light. In a few strokes of a pen, this formula tells you exactly how much energy is contained within material with a given amount of mass.
Physicists take two very different paths to uncover the laws of physics and their corresponding mathematical models. The first of these is taken by a type of physicist known as a “theorist,” and is typically done at a desk, with either a pen and ink or (these days) a computer. Theorists use logic to derive formulas that they believe will describe the natural world. Often, the starting point for this process are other, well established laws of physics, but it’s not unusual for brand new models to be introduced based on intuition alone. Indeed, as we delve into the development of quantum physics, we’ll encounter many instances of such sudden insights.
The second path is taken by a type of physicist known as an “experimentalist,” and it usually takes place in a laboratory. The starting point for experimental physics is usually a set of careful observations followed by careful measurements. In most cases, experimentalists will actively perturb some physical system (e.g., shining light on a metal) and look for the way it responds. Once enough data have been collected, experimentalists will look for patterns and draw conclusions that help establish new laws of physics.
Of course, there is significant overlap between these two approaches, and the line between experimental and theoretical physics is never clear. More important, these two approaches are strongly dependent on one another. Experimentalists rely on theorists to help develop the mathematical frameworks needed to describe their observations. Conversely, theorists rely on experimentalists to confirm that the phenomena they predict are actually observed.
QUANTUM QUOTE
Physical theory without experiment is empty. Experiment without theory is blind.
—American physicist Heinz Pagels
There is a third process worth mentioning here, not least because of its importance to quantum physics. While experimental physicists are a talented bunch, there are limits to what they can actually set up in the lab. In these cases, physicists often employ a device known as a Gedankenexperiment, a German term meaning “thought experiment.”
DEFINITION
A Gedankenexperiment is used to imagine the outcome and examine the consequences of an experiment that cannot actually be conducted due to physical, resource, or societal limitations.
For example, in the eighteenth century, English physicist Isaac Newton asked what would happen if you fired a cannonball at different speeds parallel to Earth’s surface. Based on previous observations, he knew that at low speed the cannonball would fall to Earth after a short distance. He also knew that if fired a little bit faster, it would travel a little bit farther.
He then went on to reason that if you could launch a cannonball fast enough, such that the surface of Earth curved downward at the same rate that the cannonball was falling, it could actually travel all the way around to return to where it was fired! Since there was no way to actually fire a cannonball so fast, he arrived at this conclusion using logic alone. We will see many uses of the Gedankenexperiment when trying to interpret the strange consequences of quantum physics.
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