As we've discussed, there are several derivations and formulations of quantum mechanics, all of which work to produce the same results—all producing, for example, the same set of states for the electron in the hydrogen atom. All of these approaches accurately describe the observable properties of our quantum world. What differs between them is the physical and philosophical view of the world that they suggest and the functional ease of their use. Even within a single approach there is to this day a range of variously disputed interpretations and their associated physical implications, and those will be revealed as we move on with our chronological narrative.

Despite the various formulations and interpretations, there is a set of essential features that make up quantum mechanics, a working description that has survived the controversy of the theory's arrival. It will be helpful to have this set of features in mind as we continue our chronological narrative, and so I summarize these features briefly as follows. I do so using Schrödinger's formalism, since Schrödinger's wave mechanics with Born's probability interpretation has emerged as the most commonly used working approach for quantum mechanics (with conceptual strategies and rules later laid out by Feynman). We will learn in the next chapter how this approach came to be accepted, and of any deviations from this basic framework.

I first list below the essential features of quantum mechanics in the Schrödinger context. Then I explain what each feature means, often with reference to what we have already learned of the electron in the hydrogen atom. To begin, I briefly review what is meant when we discuss the “state” of an object.

What I present in this chapter is a bit formal, but the chapter isn't long, and knowing these features will be helpful toward keeping track of the prevailing ideas through the historical events that follow.

In the classical approximate description of our quantum world, we define an object's state as the combination of its position, mass, speed, and direction. Depending on its position, how heavy it is, and how fast it is moving, we say that it has a certain energy. We can change the state of the object smoothly by increasing its speed a little bit, or a lot, or by moving it from one place to another through all of the positions nearby to those farther away or in between. We can continuously change its energy by increasing its speed, little by little or even rapidly.

In our actual quantum world, an object's state is defined in much the same way. Schrödinger's equation and its evolving wavefunction solutions describe precisely the movement and properties of objects, whether applied to an isolated particle, isolated pairs or groups of interacting particles, or the entire interacting complex of particles and bodies in our universe. However, when one object is in some way bound to the vicinity of another, there is no continuum of states and energies. In these instances, only particular separate and discrete (meaning not continuously connected) states and energies are possible, as we found, for example, for the bound states of the electron in the hydrogen atom. No states or energies are available in between these particular states and energies. We say that the states and energies are quantized. This runs counter to our classically based intuition, where we expect a seamless continuum of states and energies. (Even the planets orbiting the sun have quantized, discrete energies. But the energy levels are so close together that the change from one energy level to the next would appear to be continuous.)

Quantum mechanics has been proven to accurately describe the workings of this quantum world without exception, from the subatomic particles that make up the constituents of the atom to the most massive components of our galaxies. I list its essential features as follows.1

1. Allowed States: Each possible state of an object is described by a complex mathematical function, called a wavefunction, a solution to Schrödinger's equation for that object. The object can usually only be observed in (i.e., occupy) one of these states.
2. Properties: Each wavefunction contains information on the possible physically observable properties of the object (its location, its velocity, its spin, etc.), if it is in the state described by that wavefunction.
3. Overall Wavefunction: Every object or system in the universe is represented by the set of wavefunctions describing its states, and these may be combined into an overall wavefunction.
4. Probability: The probability of the object exhibiting a particular property is calculable for each state from the wavefunction for that state. And the probability of an object being in that particular state is calculable from the overall wavefunction.
5. Measurement: Measuring or observing the state or property of an object determines absolutely the state and property of that object at the instant of measurement. (Upon measurement or observation, a new set of wavefunctions and probabilities is created, and these may change and evolve with the ongoing progression of time.)

Regarding (1), an object can be in any one of the allowed states represented by these wavefunction solutions to the Schrödinger's equation for that object, but in no others. (We recall that Schrödinger's solutions provide the only allowed states for the electron bound to and surrounding the nucleus in the hydrogen atom. It is the transition of the electron from a higher energy state to a lower energy state that releases the quantum of energy [the photon] having precisely the energy [and associated color] given up by the electron in its transition. The distinct colors of light that are observed in the transitions of the electron from one bound state to another correspond precisely to the differences in the discrete energies of the states involved.)

Regarding (2), mathematical operations on the wavefunction can be used to calculate the properties of the object. Properties are more or less likely according to probabilities. (For example, the electron's position is given in terms of probabilities as described in (4) below.)

Regarding (3), everything can be represented by a Schrödinger's equation, and everything is described by the resulting collection of its wavefunction solutions. The scope of inclusion depends on the size of the isolated system being considered. Included may be societies, complex machines, systems of particles, a baseball, a grain of sand, the atom, and so on. Some wavefunctions evolve with time, explaining the way that the objects represented by them can move and spread in space. Some overall wavefunctions represent a collection of stationary spatial states, as we saw for the electron in the hydrogen atom.

Regarding (4), which state an object may occupy and what properties it may have are governed by probability. The probability of movement or events can be determined from the evolving and changing wavefunctions describing the evolution of the states. For example, the probable trajectories of a baseball are given by the evolution of the wavefunctions for the baseball. It is highly, highly, highly probable that large objects such as the ball will follow very nearly the path that would be projected from classical Newtonian physics. But for small submicroscopic objects, the probabilities can be far different from what might be classically expected. (As we have seen, the electron's possible position in the hydrogen atom is described in terms of cloudlike representations of probability with strange symmetries. These have been displayed in Figure 3.8 for five of the electron's lowest energy states. For each state, the probability of the electron being at any particular location is given by the magnitude of the wavefunction at that location, as indicated by the brightness of the cloud. The brighter regions in the clouds indicate a higher probability of the electron being in those regions. The state that an electron may occupy is itself determined by probabilities inherent in the overall wavefunction.)

Regarding (5), in the classical view of physics, the observer is passive. An object is in a particular state and has particular properties. We just observe them, we don't affect them. The quantum world (our actual world) is different. The act of measurement or observation forces but does not determine a particular outcome out of a set of possible outcomes, one state and its properties out of alternative possible contenders. If the same measurement could be made on the same object under the same conditions again and again, the resulting state and properties would each time likely be different and more or less likely to occur according to the probabilities inherent in the wavefunction.

The world works in probabilistic fashion. Quantum mechanics describes this world and operates according to the features listed above. Quantum mechanics is both a framework for understanding and a practical tool for invention. We scientists and engineers accept this quantum view, often approximating very nearly the quantum results for large objects using the simpler-to-work-with classical physics. But what quantum mechanics means, what it says about nature and the role of physics itself, is profound. What follows is a history of the formulation of these ideas and our understanding of them. It all started with Einstein in 1916.