Introduction to Quantum Mechanics

Figure 1. The wave model simultaneously existing on different levels.

Today, we will introduce you to the quantum world, or the world of small particles, by explaining the two main theories of quantum chemistry that started it all:  the wave model, and the Heisenberg Uncertainty principle.  These are the basic terms of quantum mechanics, and we will explore how it can be explained using these terms.

When discussion the quantum world, we use a type of infamous analysis known as Quantum Mechanics. There are a couple of main properties of this analysis that are key to understanding the world of the microscopic. First off, every physical system is associated and can be represented within a Hilbert Space. A Hilbert Space is quite simply a place with some dimension that has some underlying properties. For example, our three dimensional space can be considered a Hilbert Space. In addition, this Hilbert Space has unit vectors that correspond to every possible state of the system. States of a system are also usually called “wave functions”. We usually denote wave functions with a ket and variable inside of it (seen in Fig. 3).

Figure 2. Copenhagen Wave Function Collapse: After taking a measurement, the wave function which initially has several superstates collapsed into a single state.

As stated before, the Hilbert Space must have some properties, and one of these is the fact that they are measurement operators. In a sense, they are equivalent to mathematical representations of a physical measurement or observables. For example, if one were to observe the speed of a particle or even an atom in a Hilbert Space, this would be the same as applying a “measurement operator” to the particle. The final main property is that measurement operators and Hilbert Spaces can have complex representations, meaning that some of the mathematical representations can have complex numbers in them. Also, an important property of any measurement taken in the quantum world is that it will always return ONLY one of the possible states that the particle can be in. This is known as the wave function collapse which was proposed by Copenhagen.

Now, back in the 1900s Max Planck stated specifically that the energies of electrons are completely quantized, which means that they only can exist within one energy level. For example, in an atom there are usually rings of electrons that surround the nucleus, the center of the atom that contains both protons (positively charged particles) and neutrons (neutrally charged particles). Planck was merely stating that an electron could not exist in more than exactly one of these rings. However, recent evidence of the past 50 years has proven otherwise– that electrons and subatomic particles can exist in something that is known as a superposition of states, which is directly related to quantum mechanics. A superposition of states means that the subatomic particles can in a sense “simultaneously” exist among several different energy levels at the same time.

Figure 3. If there was a cat in a box, it could be either dead or alive. However, until you take a “measurement” or open up the box, there’s no way of knowing which state the cat is in. Therefore, it simultaneously exists in both states. The coefficients of 1/√2 are due to the equal probability of both states.

We see that this idea seems to be very counterintuitive. How can an electron supposedly be in two different locations at the same time? However, this seemingly contradictory idea has led to the basis of quantum chemistry today. The foundation of quantum mechanics began with the wave model, which described the position of electrons not as mere rings as previously assumed. Instead, because electrons can be represented as a superposition of states, they have certain so-called “probabilities” of existing within each energy level. Therefore, the movement of electrons can be described as “clouds” moving in orbitals, where their positions are represented by probability distributions rather than specific points. For example, say we look at the oxygen atom. Traditionally the electron configuration of an oxygen atom can be represented as electrons occupying specific energy levels in the following form:

Figure 4. The electron configuration diagram of a Hydrogen model.

Figure 5. The electron probability distribution diagram of oxygen.

This electron configuration implies that any electrons can only be in one specific orbital or “ring”. However, by properties of quantum mechanics these electrons are actually existing among all of these “rings” but with certain probabilities, as pictured in Figure 5.

Another one of the most famous theorems that started it all was the Heisenberg Uncertainty Principle.  It states that given a specific particle , one cannot know both the momentum and the position at a given time. This theorem can easily be explained by quantum mechanical means and the information provided here today. As stated before the wave function collapse is a phenomenon where if  any measurement is taken in the quantum world, it will always return ONLY one of the possible states that the particle can be in. Thus, when we try to determine the momentum or position of a particle, hence a “measurement”, it will only return one state, and therefore we can not have any hope of learning about the other states, such as a particle’s position or momentum.  This video provides an overview of the Heisenberg Uncertainty Principle.

All in all, quantum mechanics is clearly an interesting and yet extremely counter intuitive idea. However, it’s clear that many of the properties of the quantum world has allowed us to explain many of the most fundamental theories in Chemistry itself. In our next post, we will look further into quantum mechanics, specifically the mathematics behind quantum mechanics and hopefully be able to prove some more theorems that help us describe our world. Tune in next time and we hope that you enjoyed this first blog post!

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One thought on “Introduction to Quantum Mechanics

  1. Pingback: Quantum mechanics | Mande's Blog

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