# Quantum Mechanics 2

## Bohr Atom

Perhaps the foremost scientists of the 20th century was Niels Bohr, the first to apply Planck's quantum idea to problems in atomic physics. In the early 1900's, Bohr proposed a quantum mechanical description of the atom to replace the early model of Rutherford.
In 1913 Bohr proposed his quantized shell model of the atom to explain how electrons can have stable orbits around the nucleus. The motion of the electrons in the Rutherford model was unstable because, according to classical mechanics and electromagnetic theory, any charged particle moving on a curved path emits electromagnetic radiation; thus, the electrons would lose energy and spiral into the nucleus. To remedy the stability problem, Bohr modified the Rutherford model by requiring that the electrons move in orbits of fixed size and energy. The energy of an electron depends on the size of the orbit and is lower for smaller orbits. Radiation can occur only when the electron jumps from one orbit to another. The atom will be completely stable in the state with the smallest orbit, since there is no orbit of lower energy into which the electron can jump.

Bohr's starting point was to realize that classical mechanics by itself could never explain the atom's stability. A stable atom has a certain size so that any equation describing it must contain some fundamental constant or combination of constants with a dimension of length. The classical fundamental constants--namely, the charges and the masses of the electron and the nucleus--cannot be combined to make a length. Bohr noticed, however, that the quantum constant formulated by the German physicist Max Planck has dimensions which, when combined with the mass and charge of the electron, produce a measure of length. Numerically, the measure is close to the known size of atoms. This encouraged Bohr to use Planck's constant in searching for a theory of the atom.

Planck had introduced his constant in 1900 in a formula explaining the light radiation emitted from heated bodies. According to classical theory, comparable amounts of light energy should be produced at all frequencies. This is not only contrary to observation but also implies the absurd result that the total energy radiated by a heated body should be infinite. Planck postulated that energy can only be emitted or absorbed in discrete amounts, which he called quanta (the Latin word for "how much"). The energy quantum is related to the frequency of the light by a new fundamental constant, 'h'. When a body is heated, its radiant energy in a particular frequency range is, according to classical theory, proportional to the temperature of the body.

With Planck's hypothesis, however, the radiation can occur only in quantum amounts of energy. If the radiant energy is less than the quantum of energy, the amount of light in that frequency range will be reduced. Planck's formula correctly describes radiation from heated bodies. Planck's constant has the dimensions of action, which may be expressed as units of energy multiplied by time, units of momentum multiplied by length, or units of angular momentum. For example, Planck's constant can be written as h = 6.6x10-34 joule seconds.

Using Planck's constant, Bohr obtained an accurate formula for the energy levels of the hydrogen atom. He postulated that the angular momentum of the electron is quantized--i.e., it can have only discrete values. He assumed that otherwise electrons obey the laws of classical mechanics by traveling around the nucleus in circular orbits. Because of the quantization, the electron orbits have fixed sizes and energies. The orbits are labeled by an integer, the quantum number 'n'.

With his model, Bohr explained how electrons could jump from one orbit to another only by emitting or absorbing energy in fixed quanta. For example, if an electron jumps one orbit closer to the nucleus, it must emit energy equal to the difference of the energies of the two orbits. Conversely, when the electron jumps to a larger orbit, it must absorb a quantum of light equal in energy to the difference in orbits. The Bohr model basically assigned discrete orbits for the electron, multiples of Planck's constant, rather than allowing a continuum of energies as allowed by classical physics.

The power in the Bohr model was its ability to predict the spectra of light emitted by atoms. In particular, its ability to explain the spectral lines of atoms as the absorption and emission of photons by the electrons in quantized orbits.

Our current understanding of atomic structure was formalized by Heisenberg and Schroedinger in the mid-1920's where the discreteness of the allowed energy states emerges from more general aspects, rather than imposed as in Bohr's model. The Heisenberg/Schroedinger quantum mechanics have consistent fundamental principles, such as the wave character of matter and the incorporation of the uncertainty principle.
In principle, all of atomic and molecular physics, including the structure of atoms and their dynamics, the periodic table of elements and their chemical behavior, as well as the spectroscopic, electrical, and other physical properties of atoms and molecules, can be accounted for by quantum mechanics => fundamental science.

## de Broglie Matter Waves

Perhaps one of the key questions when Bohr offered his quantized orbits as an explanation to the UV catastrophe and spectral lines is, why does an electron follow quantized orbits?The response to this question arrived from the Ph.D. thesis of Louis de Broglie in 1923. de Broglie argued that since light can display wave and particle properties, then perhaps matter can also be a particle and a wave too.

So a photon, or a free moving electron, can be thought of as a wave packet, having both wave-like properties and also the single position and size we associate with a particle. There are some slight problems, such as the wave packet doesn't really stop at a finite distance from its peak, it also goes on for ever and ever. Does this mean an electron exists at all places in its trajectory?
de Broglie also produced a simple formula that the wavelength of a matter particle is related to the momentum of the particle. So energy is also connected to the wave property of matter.

The electron matter wave is both finite and unbounded. Only certain wavelengths will 'fit' into an orbit. If the wavelength is longer or shorter, then the ends do not connect. Thus, de Broglie explains the Bohr atom in that on certain orbits can exist to match the natural wavelength of the electron. If an electron is in some sense a wave, then in order to fit into an orbit around a nucleus, the size of the orbit must correspond to a whole number of wavelengths.

Notice also that this means the electron does not exist at one single spot in its orbit, it has a wave nature and exists at all places in the allowed orbit. Thus, a physicist speaks of allowed orbits and allowed transitions to produce particular photons (that make up the fingerprint pattern of spectral lines). And the Bohr atom really looks like the following diagram:

While de Broglie waves were difficult to accept after centuries of thinking of particles are solid things with definite size and positions, electron waves were confirmed in the laboratory by running electron beams through slits and demonstrating that interference patterns formed.
How does the de Broglie idea fit into the macroscopic world? The length of the wave diminishes in proportion to the momentum of the object. So the greater the mass of the object involved, the shorter the waves. The wavelength of a person, for example, is only one millionth of a centimeter, much to short to be measured. This is why people don't 'tunnel' through chairs when they sit down.

## Probability Fields

The idea that an electron is a wave around the atom, instead of a particle in orbit begs the question of 'where' the electron is at any particular moment. The answer, by experimentation, is that the electron can be anywhere around the atom. But 'where' is not evenly distributed. The electron as a wave has a maximum chance of being observed where the wave has the highest amplitude. Thus, the electron has the highest probability to exist at a certain orbit.
Where probability is often used in physics to describe the behavior of many objects, this is the first instance of an individual object, an electron, being assigned a probability for a Newtonian characteristic such as position. Thus, an accurate description of an electron orbit is one where we have a probability field that surrounds the nucleus, as shown aside:

For more complicated orbits, and higher electron shells, the probability field becomes distorted by other electrons and their fields, like the following example:

Thus, for the first time, the concept of existence begins to take on an elusive character at the subatomic level.

## Young Two Slit Experiment

The wave-like properties of light were demonstrated by the famous experiment first performed by Thomas Young in the early nineteenth century. In original experiment, a point source of light illuminates two narrow adjacent slits in a screen, and the image of the light that passes through the slits is observed on a second screen.

The dark and light regions are called interference fringes, the constructive and destructive interference of light waves. So the question is will matter also produce interference patterns. The answer is yes, tested by firing a stream of electrons.

However, notice that electrons do act as particles, as do photons. For example, they make a single strike on a cathode ray tube screen. So if we lower the number of electrons in the beam to, say, one per second. Does the interference pattern disappear?

The answer is no, we do see the individual electrons (and photons) strike the screen, and with time the interference pattern builds up. Notice that with such a slow rate, each photon (or electron) is not interacting with other photons to produce the interference pattern. In fact, the photons are interacting with themselves, within their own wave packets to produce interference.

But wait, what if we do this so slow that only one electron or one photon passes through the slits at a time, then what is interfering with what? i.e. there are not two waves to destructively and constructively interfere. It appears, in some strange way, that each photon or electron is interfering with itself. That its wave nature is interfering with its own wave (!).

The formation of the interference pattern requires the existence of two slits, but how can a single photon passing through one slit 'know' about the existence of the other slit? We are stuck going back to thinking of each photon as a wave that hits both slits. Or we have to think of the photon as splitting and going through each slit separately (but how does the photon know a pair of slits is coming?). The only solution is to give up the idea of a photon or an electron having location. The location of a subatomic particle is not defined until it is observed (such as striking a screen).

## Role of the Observer

The quantum world can be not be perceived directly, but rather through the use of instruments. And, so, there is a problem with the fact that the act of measuring disturbs the energy and position of subatomic particles. This is called the measurement problem.

Thus, we begin to see a strong coupling of the properties of an quantum object and and the act of measuring those properties. The question of the reality of quantum properties remains unsolved. All quantum mechanical principles must reduce to Newtonian principles at the macroscopic level (there is a continuity between quantum and Newtonian mechanics).
How does the role of the observer effect the wave and particle nature of the quantum world? One test is to return to the two slit experiment and try to determine (count) which slit the photon goes through. If the photon is a particle, then it has to go through one or the other slit. Doing this experiment results in wiping out the interference pattern. The wave nature of the light is eliminated, only the particle nature remains and particles cannot make interference patterns. Clearly the two slit experiments, for the first time in physics, indicates that there is a much deeper relationship between the observer and the phenomenon, at least at the subatomic level. This is an extreme break from the idea of an objective reality or one where the laws of Nature have a special, Platonic existence.

If the physicist looks for a particle (uses particle detectors), then a particle is found. If the physicist looks for a wave (uses a wave detector), then a wave pattern is found. A quantum entity has a dual potential nature, but its actual (observed) nature is one or the other.

## Quantum Wave Function

The wave nature of the microscopic world makes the concept of 'position' difficult for subatomic particles. Even a wave packet has some 'fuzziness' associated with it. An electron in orbit has no position to speak of, other than it is somewhere in its orbit.
To deal with this problem, quantum physics developed the tool of the quantum wave function as a mathematical description of the superpositions associated with a quantum entity at any particular moment.

The key point to the wave function is that the position of a particle is only expressed as a likelihood or probability until a measurement is made. For example, striking an electron with a photon results in a position measurement and we say that the wave function has 'collapsed' (i.e. the wave nature of the electron converted to a particle nature).

## Superposition

The fact that quantum systems, such as electrons and protons, have indeterminate aspects means they exist as possibilities rather than actualities. This gives them the property of being things that might be or might happen, rather than things that are. This is in sharp contrast to Newtonian physics where things are or are not, there is no uncertainty except those imposed by poor data or limitations of the data gathering equipment.
Further experimentation showed that reality at the quantum (microscopic) level consists of two kinds of reality, actual and potential. The actual is what we get when we see or measure a quantum entity, the potential is the state in which the object existed before it was measured. The result is that a quantum entity (a photon, electron, neutron, etc) exists in multiple possibilities of realities known as superpositions.

The superposition of possible positions for an electron can be demonstrated by the observed phenomenon called quantum tunneling.

Notice that the only explanation for quantum tunnelling is if the position of the electron is truly spread out, not just hidden or unmeasured. It's raw uncertainty allows for the wave function to penetrate the barrier. This is genuine indeterminism, not simply an unknown quantity until someone measures it.
It is important to note that the superposition of possibilities only occurs before the entity is observed. Once an observation is made (a position is measured, a mass is determined, a velocity is detected) then the superposition converts to an actual. Or, in quantum language, we say the wave function has collapsed.

The collapse of the wave function by observation is a transition from the many to the one, from possibility to actuality. The identity and existence of a quantum entities are bound up with its overall environment (this is called contextualism). Like homonyms, words that depend on the context in which they are used, quantum reality shifts its nature according to its surroundings.
In the macroscopic world ruled by classical physics, things are what they are. In the microscopic world ruled by quantum physics, there is an existential dialogue among the particle, its surroundings and the person studying it.