Prior to 1925, quantum theory consisted of a collection of principles without an overarching mathematical framework. Furthermore, there were lingering questions within the physics community regarding wave-particle duality and the departure from pre-1900 physics. The field of quantum mechanics emerged to provide a more complete and consistent description of physics at scales where quantum effects are dominant.
Quantum Mechanics & Probability
According to Newtonian mechanics, objects can be described as having a position and velocity that are both well-defined at any instant in time. Furthermore, if we know the forces acting on an object, we can use Newton’s laws of motion to predict the path it will follow and where it will be at any future instant. This physical view of the world is often called “deterministic” because it implies that objects will follow predictable, predetermined paths.
In contrast, quantum mechanics describes the sub-microscopic world as being “probabilistic”. Quantum theory cannot predict the result of a quantum measurement any more than you can predict the outcome of a die roll. However, just as we can describe how often a particular outcome will come up across a large number of die rolls (i.e., one-sixth of the time on a fair, six-sided die), we can make predictions about the outcomes of physical experiments if many experiments are conducted. The fact that measurements of the same quantity could lead to different outcomes was a radical departure from the deterministic view of the universe.
Does this mean that Newtonian physics, which can be used to describe the motion of everything from sand grains to planetary orbits, is wrong? Physicists say no. For most visible objects, Newton’s laws of motion operate in close agreement with experimental results. However, when we examine the world at the scale of the atom and the electron, Newtonian physics no longer provides an accurate description.
There are other ways that quantum mechanics tends to challenge our everyday intuition about the universe. You may be familiar with the “Schrödinger’s cat” thought experiment, which is usually described as follows:
A cat is placed into a sealed box along with a vial of poison. Using radioactive decay or some other means, a mechanism is constructed within the box such that there is exactly a 50% probability that the poison will be released and the cat will die. Applying a quantum perspective to this scenario, we cannot know the outcome until the box is opened and a “measurement” is made. Therefore, one could say the cat must be both “alive” and “dead” at the same time. Schrödinger posed this scenario in 1935 to express his skepticism of Bohr and Heisenberg’s proposition that a quantum system could simultaneously exist in multiple “states” prior to being measured. Although a cat cannot be described probabilistically, quantum-scale particles such as electrons can. The Schrödinger’s cat scenario serves as a common way for physicists to illustrate and compare ways of thinking about quantum uncertainty.
A cat is placed into a sealed box along with a vial of poison. Using radioactive decay or some other means, a mechanism is constructed within the box such that there is exactly a 50% probability that the poison will be released and the cat will die. Applying a quantum perspective to this scenario, we cannot know the outcome until the box is opened and a “measurement” is made. Therefore, one could say the cat must be both “alive” and “dead” at the same time. Schrödinger posed this scenario in 1935 to express his skepticism of Bohr and Heisenberg’s proposition that a quantum system could simultaneously exist in multiple “states” prior to being measured. Although a cat cannot be described probabilistically, quantum-scale particles such as electrons can. The Schrödinger’s cat scenario serves as a common way for physicists to illustrate and compare ways of thinking about quantum uncertainty.
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Heisenberg Uncertainty Principle
The measurement of any quantity, be it length, mass, or time, involves some amount of uncertainty— for instance, limitations in instrument precision or environmental effects. For this reason, it is standard procedure in laboratory settings to repeat a measurement multiple times in order to obtain an average value as well as uncertainty. At the macroscopic scale, uncertainty in measurement tends not to be a consequence of the scientific theory which is being tested. However, according to the theory of quantum mechanics, there is a fundamental limit to how precise our measurements can be.
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To visualize this, suppose you wanted to measure the position of an object. In order to see where the object is, light would need to reflect from it and return to your eye. That is, at least one photon must collide with the object and return to a detector. For a macroscopic object such as a bowling ball, the effect of a single photon collision would be virtually imperceptible. However, for an object governed by quantum mechanics (such as an electron), the scattering of a single photon is enough to appreciably change its position. Essentially, at the quantum scale, measurements are altered by the very act of trying to measure them.
In 1927, Werner Heisenberg introduced the Heisenberg uncertainty principle, which states that we cannot simultaneously measure both the position and the momentum of an object with absolute precision. Furthermore, the more accurately we are able to measure position, the less accurately we are able to measure momentum, and vice versa.
Although it has a great deal of importance at the quantum scale, the uncertainty principle has virtually no bearing on our experience in the macroscopic world. Nevertheless, we can imagine what it might be like to live in a world governed by quantum rules. In George Gamow’s 1940 book Mr. Tompkins in Wonderland, a bank teller explores a “quantum jungle” in which Planck’s constant is many orders of magnitude larger than its normal value. As a result, he sees an elephant with “fuzzy” skin due to uncertainty in its position, as well as a single gazelle “spreading out” into an apparent herd as it diffracts through a bamboo grove.
Modern Quantum Model
Under the current model of the atom, commonly referred to as the Schrödinger model, electrons exist as waves that occupy the three-dimensional space surrounding the nucleus. In this way, the Schrödinger model is consistent with the quantum mechanical principle that electrons do not follow fixed orbits around the nucleus as in the planetary model of the atom. The Schrödinger model is often visualized as a nucleus surrounded by a “cloud” of electrons (Figure 14). Within this cloud, there are regions in which electrons may be detected with varying probability.
The Bohr model included a single quantum number n that indexed the energy level of the atom. (n=the energy level) Electrons with the same value for n are said to occupy the same electron “shell.” We now know there to be three additional quantum numbers: the orbital quantum number l, the orbital magnetic quantum number ml, and the spin quantum number ms. Bohr’s original quantum number n is now known as the principal quantum number. A wealth of experimental evidence indicates the values of these four quantum numbers (n, l, ml and ms) are sufficient to describe the state of an electron in any atom.
The orbital quantum number l can take non-negative integer values from 0 through n - 1 (i.e. for the n = 3 state, l can take the values 0, 1 or 2, but not any value larger than 2). Electrons with the same value for n and l are said to occupy the same “subshell,” and each value of l has a corresponding subshell symbol: s, p, d, or f for l = 0, 1, 2, 3 respectively. We generally refer to a specific subshell using its principal quantum number and subshell symbol, for instance 3p for the n = 3, n = l subshell. In chemistry, the electron configuration of an atom is specified by using superscripts to indicate the number of electrons in each subshell, for example
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The Pauli exclusion principle, proposed by Wolfgang Pauli in 1925, states that no two electrons in an atom can have the same exact set of values for the quantum numbers n, l, ml, and ms. As an electron is added to an atom, it will quickly move to occupy the lowest energy state that is not currently taken by another electron. In other words, the electrons will stack in progressively higher orbitals rather than all falling to the lowest possible state.
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This “stacking” property of electrons helps explain some of the chemical properties observed within families of elements. For instance, noble gases like helium and neon have filled electron shells, and therefore do not react or combine easily with other elements. In contrast, alkali metals such as lithium and sodium have an unpaired electron and tend to be highly reactive. Figure 15 shows the electron configurations for neon, sodium, and oxygen.