CONSERVATION OF ENERGY

Before we go on with our investigation of the hydrogen atom, we will take a short break to discuss the idea of conservation of energy. This idea, which originated in Newtonian mechanics, survives more or less intact in our modern particle-wave picture of matter.
Physicists pay attention to the concept of energy only because energy is conserved. If energy disappears from one place, it will show up in another. We saw this in the Bohr model of hydrogen. When the electron lost energy falling down from one allowed orbit to a lower energy orbit, the energy lost by the electron was carried out by a photon.
You can store energy in an object by doing work on the object. When you lift a ball off the floor, for example, the work you did lifting the ball, the energy you supplied, is stored in a form we call gravitational potential energy. Let go of the ball and it falls to the floor, loosing its gravitational potential energy. But just before it hits the floor, it has a lot of energy of motion, what we have called kinetic energy. All the gravitational potential energy the ball had before we dropped it has been converted to kinetic energy.
After the ball hits the floor and is finally resting there, it is hard to see where the energy has gone. One place it has gone is into thermal energy, the floor and the ball are a tiny bit warmer as a result of your dropping the ball.
Another way to store energy is to compress a spring. When you release the spring you can get the energy back. For example, compress a watch spring by winding up the watch, and the energy released as the spring unwinds will run the watch for a day. We could call the energy stored in the compressed spring spring potential energy. Physicists invent all sorts of names for the various forms of energy.

One of the big surprises in physics was Einstein’s discovery of the equivalence of mass and energy, a relationship expressed by the famous equation E = mc² . In that equation, E stands for the energy of an object, m its mass, and c is the speed of light. Since the factor c² is a constant, Einstein’s equation is basically saying that mass is a form of energy. The c² is there because mass and energy were initially thought to be different quantities with different units like kilograms and joules. The c² simply converts mass units into energy units.
What is amazing is the amount of energy that is in the form of mass. If you could convert all the mass of a pencil eraser into electrical energy, and sell the electrical energy at the going rate of 10¢ per kilowatt hour, you would get about 10 million dollars for it. The problem is converting the mass to another, more useful, form of energy. If you can do the conversion, however, the results can be spectacular or terrible. Atomic and hydrogen bombs get their power from the conversion of a small fraction of their mass energy into thermal energy. The sun gets its energy by “burning” hydrogen nuclei to form helium nuclei. The energy comes from the fact that a helium nucleus has slightly less mass than the hydrogen nuclei out of which it was formed. If you have a particle at rest and start it moving, the particle gains kinetic energy. In Einstein’s view the particle at rest has energy due to its rest mass. When you start the particle moving, it gains energy, and since mass is equivalent to energy, it also gains mass. For most familiar speeds the increase in mass due to kinetic energy is very small. Even at the speeds travelled by rockets and spacecraft, the increase in mass due to kinetic energy is hardly noticeable. Only when a particle’s speed gets up near the speed of light does the increase in mass become significant.

One of the first things we discussed about the behavior of matter is that nothing can travel faster than the speed of light. You might have wondered if nature had traffic cops to enforce this speed limit. It does not need one, it uses a law of nature instead. As the speed of an object approaches the speed of light, its mass increases. The closer to the speed of light, the greater increase in mass. To push a particle up to the speed of light would give it an infinite mass and therefore require an infinite amount of energy. Since that much energy is not available, no particle is going to exceed nature’s speed limit.
This raises one question. What about photons? They are particles of light and therefore travel at the speed of light. But their energy is not infinite. It depends instead on the wavelength or color of the photon. Photons escape the rule about mass increasing with speed by starting out with no rest mass. You stop a photon and nothing is left. Photons can only exist by traveling at the speed of light.
When a particle is traveling at speeds close enough to the speed of light that its kinetic energy approaches its rest mass energy, the particle behaves differently than slowly moving particles. For example, push on a slowly moving particle and you can make the particle move faster. Push on a particle already moving at nearly the speed of light, and you merely make the particle more massive since it cannot move faster. Since the relationship between mass and energy came out of Einstein’s theory of relativity, we say that particles moving near the speed of light obey relativistic mechanics while those moving slowly are nonrelativistic. Light is always relativistic, and all automobiles on the earth are nonrelativistic.