The strange world of quantum mechanics and quantum chemistry (v1.0)

 

Most of this blog is summarization of lectures by Erica W Carlson of Perdue University.

We are all familiar with Newtonian physics. Laws like “a body at rest continues to be at rest and a body in motion continues to be in motion in a straight line unless a force is applied.” These feel intuitively obvious and matches our everyday experiences with macro-objects and the macro world we operate in. Einstein’s laws of special and general relativity which generalizes and extends Newtonian physics are amazing at first glance, but they also apply to macro-objects. However, in the world of the exceedingly small like the atom or smaller, things and laws are vastly different and governed by quantum mechanics. Quantum mechanics can also be applied to much larger objects, but quantum effects rapidly decrease as the mass of objects gets bigger and bigger and becomes almost unmeasurable.

No one so far has explained why the strange world of quantum mechanics is the way it is. Scientists continue to work on an explanation. But it has been verified by numerous experiments and no violations are known to date. In this blog, I will mostly introduce you to some of the key rules in the strange world of quantum mechanics and focus on how it underpins chemistry. Quantum mechanics also underpins thermodynamics, particle physics, black body radiation, quantum computers, electric conduction, super conductivity, color vision, fluorescent and gas discharge lights, magnets, and tunneling among other things.

The first strange rule in quantum mechanics is that a particle behaves as both a wave and a particle. For example, shooting electrons at two closely spaced slits in a barrier will result in a wave interference pattern on the other side and register as such on a recording wall behind that is remarkably like what a wave going through the two slits would make. It is as if the electron went through both slits just like a wave does. However, observation or measurement of the position, momentum, or energy of the particle collapses the wave, and it becomes a particle and stays so for some time. For example, if we had a detector of some sort that measured which slit the electron went through, the wave behavior disappears, and it behaves as a particle that hits the wall just like a tennis ball. This is called the wave particle duality. Even light behaves as a photon particle or a light wave. It is extremely strange and non-intuitive. The observer changed the behavior!! Until observed, it is in a superposition quantum state and can have multiple positions or momentum at the same time. All we can talk about is the probability of location or momentum but cannot measure since the wave will collapse. But the probability wave function tells us everything we can know.

To understand how strange quantum mechanics is, here is a thought experiment called Schrodinger's cat. Suppose I have a sealed box. In it there is a radioactive atom. There is also a Geiger counter in it. If the Geiger counter gets a beep, a hammer descends and breaks a jar with a poisonous gas. There is also a cat in there. The gas will kill the cat. If you use quantum mechanics, the cat is both dead and alive at the same time in superposition until you open the box when it reverts to one state of dead or alive. As you can see quantum effects can be confusing!! In fact, similar effects have been noticed in super conductivity (a quantum phenomenon) where current is flowing in both directions at the same time!!!

The second strange rule of quantum mechanics is the Heisenberg uncertainty principle. The uncertainty of the position times the uncertainty of the momentum of a particle >= planks constant/2. (Delta X) * (Delta P) = h/2. The value of planks constant is 6.626 * 10 ** -34 J Hz. This is an exceedingly small number. Momentum is mass times velocity. So, it can be rewritten as (delta X) (Delta V) = h/2m where m is mass. As you can see quantum effects will disappear as mass increases by a lot. This is also extremely non intuitive.

The probabilistic nature is fundamental and intrinsic to the system and not due to a lack of knowledge or measurement technique or approach, like in statistical analysis where we cannot practically and individually measure the state of every entity to draw conclusions because there are far too many.

Waves which are bounded like a vibrating clothesline anchored on two ends, a flute, or a drum (described by Bessel functions) or a guitar string, form standing waves with harmonics. You have seen depiction of one-dimensional standing waves. There are zero, one, two or more stationary points with segments between vibrating up and down. You cannot have 1 and 1/2 stationary points. Waves have a wavelength and frequency. Smaller wavelengths mean higher energy and higher frequency/speed. A bounded wave has discrete integer number of troughs and peaks and stationary points. A two-dimensional drum is similar and is closer to the three dimensions in which an electron operates, so we will use it a lot as an analogy. For electrons around an atom, the position, speed/frequency, angular momentum, and the energy are quantized and can only take discrete values. The energy of a stable state is stationary. This is the reason we call the discipline quantum mechanics. Everything for bounded particles (for example in an atom) is quantized. In a three-dimensional electron around an atom, which is spinning, the spin is also quantized, and it is said to be spinning up or down. An electron in an atom is a bounded particle and forms three dimensional standing waves around the nucleus. The wave is described by Schrödinger wave equation. It describes a wave in 4 dimensions and is complex. Why does it have an imaginary part? What i does here is simplify a system of two real differential equations into a more compact form. The complex numbers should be thought of more as a two-dimensional vector space over the real numbers rather than as some sort of mystical imaginary thing. Complex numbers are especially good at simplifying two dimensional equations when there is some sort of symmetry involved, which we have here. Most equations in physics involve some sort of symmetry, and they have their own examples of special notation (such as dot products, cross products, and spherical coordinates) that allow several equations to be folded into 1, so imaginary numbers aren’t particularly special here. Finally, the imaginary factor corresponds to a 90-degree rotation of some sort.

The nucleus of an atom consists of protons and neutrons. Protons are positively charged. Since an atom (unless it is an ion) is charge neutral, there is an equal number of electrons to protons. Since the nucleus is thousands of times heavier than an electron, it is the electron that spins around and the nucleus mostly stationary. The electron does not stick to the proton (although it is attracted to it) because of Heisenberg’s uncertainty principle. Its momentum will have to be extremely spread out if it is stationary on the proton. It finds a balance by being close. The energy of an electron far away is zero electron volts. The energy of an electron in a wave near a nucleus is negative electron volts. The electron would settle into the lowest energy state it can. That is one reason an electron around an atom doesn't just fly away.

The lowest energy state is the simplest standing wave and is called 1s. It is a stable stationary state. This is like a drum with the center bouncing up and down and the edges stationary but in three dimensions. Since spin is quantized and can be up or down up to two quantum states are possible in 1s. So up to two electrons can fit in 1s. That is hydrogen and helium with one electron and two electrons.

This leads us to Pauli’s exclusion principle. No two electrons can be doing the same thing at the same time. So only one electron can be in any state with a given spin. An electrons spin is quantized and can only be up or down. Fermi Dirac statistics is applicable to the identical, indistinguishable particles of half-integral spin called fermions. An electron is a fermion. These particles obey Pauli Exclusion Principle. Fermi Dirac statistics is invaluable in the study of electrons.

The next lowest energy state is a stable stationary state and is a standing wave like the drum with center moving up and down, a ring around that that is stationary, and the part closest to the edge moving up and down in the opposite phase. However, for electrons it is in three dimensions and is called 2s. With up and down spin, two electrons can fit in 2s. So that is Lithium and Beryllium with 3 and 4 electrons.

Similarly, we have 3s and 4s and 5s and 6s which are all higher and higher energy states which is like a drum wave pattern with 2,3,4,5, 6, etc. circular stationary points and parts in between going up and down in opposite phase. That covers the first two columns of the periodic table.

After lithium and beryllium there is boron, carbon, nitrogen, oxygen, fluorine, and neon (6 elements). There correspond to harmonics which are radial and called 2p, 3p, 4p, 5p, etc. 2p is like a drum where the stationary part is a line through the center. The left and right part go up and down in opposite phase. These are stable stationary state. But since it is in three dimensions, there are three orientations called Px, Py and Pz. Along with up down spin that gives six slots and fits the six elements. This set of six constitutes the last six columns of the periodic table. The rows for those six columns are the 2p, 3p, 4p, 5p, etc states and correspond to adding 2,3,4,5, etc circular stationary points to the basic 2p geometry.

In the center of the periodic table are a column of ten elements which are referred to as transition metals. The rows form the successive 3d, 4d, 5d, etc states. The geometry of these standing waves in three dimensions has five geometric orientations (z ** 2, X ** 2 – y ** 2, xy, yz, zx), and are like radial stationary part of the drum with 2 lines through the center at right angles. The parts between go up and down in opposite phase. The rows for those ten columns are the 3d, 4d, 5d, etc states and correspond to adding 2,3,4,5, etc circular/spherical stationary points to the basic 3d geometry. This is a stable stationary state.

The rare earth of the periodic table is a column of 14 elements where the first two rows are called the lanthanide, and actinide. These are the 4f, 5f, etc. states that are even more complicated. It is like the radial stationary part of the drum being three lines symmetrical through the center with the parts in between going up and down in opposite phase. It has 7 geometries in three dimensions. But with that many electrons, electrons start interfering with each other and so there are some twists. Also, actinides have unstable nuclei and are often radioactive.

In the direction of increasing energy, the states are 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f. The lower energy states are populated with electrons before higher energy states. In general, the energy difference between the lower energy states is larger than the energy difference between higher energy states. If you combine multiple stationary states of different frequency/energy (example s and p) for an electron, it is unstable and will emit radiation like an antenna and the electron sinks to the lower stationary state.

You may ask. This does not look like the electron is orbiting the atom at all. How can that be? Where is the angular orbital momentum? The answer is that the wave function is complex. An imaginary number is a number multiplied by squire root of -1. To see where angular momentum is hiding in this, compute for example 2P(x) + i * 2P(y) where P(y) is rotated 90 degrees relative to P(x). Adding waves is called superposition. This looks like a spinning donut. Similarly, for 3d(x**2 - y**2) + i * 3d( xy) and other combinations of orbitals at a given energy. They all look like spinning donuts. This shows up the angular momentum which is quantized. The angular momentum of the S orbital is 0. The angular momentum of the P orbital is 1 * h bar. The orbital momentum of the D orbital is 2 * h bar. The orbital momentum of the F orbital is 3 * h bar. This angular momentum represents a current. As you can see it is quantized. The first two columns of the periodic table (the s orbitals) have an angular momentum of l = 0. The right 6 columns of the periodic table have an angular momentum of l = 1. The center 10 columns of the periodic table have an angular momentum of l = 2. The 14 columns of the rare earth elements have an angular momentum of l = 3. h bar is planks constant / (2 * pi) = 1.0546 * 10 ** -34 J s.

Both the spin of an electron and the orbital motion of an electron around a nucleus result in magnetic fields. The magnetic fields of electrons cancel each other if an orbital is full (for example 2 electrons of opposite spin in an s orbital or 6 electrons half with opposite spins in a p orbital, etc). The orbitals of the lowest energy are filled before ones with higher. It is only atoms with partially filled outermost orbitals that result in a net magnetic field (example is iron, nickel, or cobalt for the d orbitals or neodymium for the f orbital). Another condition must be met for magnets. This is alignment of the atoms along an axis. This condition also turns out to be quantum mechanical in nature and depends on electrons repelling each other and Pauli’s exclusion principle.

As you can see it is 3-dimensional geometry along with quantization due to quantum mechanics and standing waves that gives rise to the periodic table and magnetic properties.

Chemical bonds between elements are formed only with the outermost shell of electrons (highest energy). These electrons form new standing wave hybridized shapes around the combined nucleii and the possible shapes depend on the geometry of the molecule. For hydrogen, lithium, etc in the first column it is 1 in the S state that can bond. For carbon, etc in its column it is 2 electrons in the 2p state that can bond and also the 2 in the 2s state. For diamond, the carbon bonds in a symmetric 3D tetrahedra with 4 other carbon atoms. This forms a very stable setup and explains why diamond is so hard. For graphene, the carbon bonds in a flat plane with three carbon atoms and forms sheets. The fourth perpendicular 2p electron floats across the whole sheet. This explains why graphene acts as a lubricant and used in pencils. For helium it is 2 in the 1S state, but since that completes both possible states for the 1s wave, it is very stable and does not react with most things. These bonding leads to molecules, and the shape of the hybrid orbitals sets up the properties. In chemistry, the bonds most mentioned are sigma bonds and pi bonds. With the versatile carbon, hydrogen, and oxygen, we get organic chemistry and biochemistry and finally us humans. It is all because of quantum mechanics!!!! There is no chemistry without quantum mechanics and Pauli’s exclusion principle is a key part of it.

A very strange part of quantum mechanics is quantum tunneling. This says that there is a nonzero probability that an electron in front of a barrier tunnels through the barrier and appears magically on the other side. This depends on the thickness of the barrier. This phenomenon is used in many places including flash memory in computers and tunneling field effect transistors (TFET). This is directly tied to the wave nature of the electron.

The strangest part of quantum mechanics is quantum entanglement. When two particles (say a proton and electron) become entangled, they stay connected even if they are separated by vast distances!! For example, if a researcher measures the spin of one particle, the measured spin of the other particle that is entangled in the other side of the universe will be correlated. The beauty of entanglement is that just knowing the state of one particle automatically tells you something about its companion, even when they are far apart. And entanglement can occur across millions of particles. This is astonishing!! The study of entanglement and its applications is considered the most important themes of this century!! Quantum superposition and quantum entanglement is the foundation of quantum computers. As Moore's law of computing grinds on, at some point, quantum computers may be the only way to keep it going. It will ultimately stop when we run bang into the plank's constant or the speed of light.

One huge paradox of quantum mechanics is instantaneous changes through large distances implies travel much faster than the speed of light. This instantaneous change happens not only in entanglement case but also the complete wave collapse to a particle on measurement across the whole length of the wave. Doesn’t this violate relativity? It has been shown that although relativity says no energy or information can travel faster than light, in these two cases, it has been shown that neither occurs, so it is not in violation of relativity. Also, it has been shown that there is no hidden local variable between in the two entangled particles that causes this behavior (although it has not been proven that there is not a hidden nonlocal variable). So, the paradox of how this is possible remains and remains to be explained.