My purpose for this blog post is to explore pedagogical approaches to explaining solutions (i.e. wavefunctions) for the time-independent Schrodinger equation and how electrons are associated with those solutions. In its present form, the blog is not polished and may have erroneous statements, so keep in mind the blog just follows my train of thought. I apologize in advance for confusion or mistakes. For clarity I will try to avoid any gobbledygook (i.e., unnecessary nomenclature).
Quantum States$^\dagger$: A Train of Thought
The first thing to discuss is that we are interested in explaining the motion and interactions of our – humans that is – notion of particles that make up matter. The main confusion is that our intuition/experience tells us to think of the world as particles, however, the whole premise of quantum mechanics is that the motion and interactions cannot be described by classical point particles but rather by "wave-like"$^\mp$ states which are solutions to quantized operators/transformations. It behooves us then to concede that with our current mathematical tools (i.e. equations of quantum mechanics) we are told that the material world, at its smallest scales, is in "wave-like" states. What does "wave-like" states even mean? It means that the mathematics used to treat the motion and interactions of what we call particles$^\ddagger$ within the theory of quantum mechanics (QM) looks similar to the mathematics used to describe the motion of waves. The difference from the classical picture of waves arises in the nature of the quantum mechanical solutions, that is, the "wave-like" states and their amplitudes exist in a complex number space.
The essence in interpreting the physical meaning of the "wave-like" state solutions, is an additional layer of struggle and falls within the philosophical interpretation of quantum physics. For example what is the meaning of the "wave-like" state if when I interact with the physical world I always observe or measure particles — this is what experiments show us. This bewildering problem still exist even though it has been nearly 100 years since the conception of QM. So why don't we know the meaning? I'm not sure. Its not to say that many great minds haven't proposed insightful ideas, but that there is no clear victor. The most widely accepted and used approach is commonly referred to as the Copenhagen interpretation, which was primarily pushed by Niels Bohr. The main premise is that it is only the magnitude of the complex numbered "wave-like" state that we can assign meaning to as a kind of probability about our [human] intuition of where particles could be. Then there are the many-worlds theory initially proposed by Hugh Everett, pilot-wave theory of David Bohm and De Broglie, and the spontaneous wavefunction collapse theory (GRW). It seems to me that the complexity of trying to explain the philosophical implications of the quantum "wave-like" state is something to admire given that it eludes some of most brilliant of minds. The general theme in physics seems to focus on QM (i.e., wavefunction) predictive description of reality not the inference of its existential meaning.
The essence in interpreting the physical meaning of the "wave-like" state solutions, is an additional layer of struggle and falls within the philosophical interpretation of quantum physics. For example what is the meaning of the "wave-like" state if when I interact with the physical world I always observe or measure particles — this is what experiments show us. This bewildering problem still exist even though it has been nearly 100 years since the conception of QM. So why don't we know the meaning? I'm not sure. Its not to say that many great minds haven't proposed insightful ideas, but that there is no clear victor. The most widely accepted and used approach is commonly referred to as the Copenhagen interpretation, which was primarily pushed by Niels Bohr. The main premise is that it is only the magnitude of the complex numbered "wave-like" state that we can assign meaning to as a kind of probability about our [human] intuition of where particles could be. Then there are the many-worlds theory initially proposed by Hugh Everett, pilot-wave theory of David Bohm and De Broglie, and the spontaneous wavefunction collapse theory (GRW). It seems to me that the complexity of trying to explain the philosophical implications of the quantum "wave-like" state is something to admire given that it eludes some of most brilliant of minds. The general theme in physics seems to focus on QM (i.e., wavefunction) predictive description of reality not the inference of its existential meaning.
Okay, lets shift back and think about some examples where waves describe the behavior of a classical physical system. The two most simplest to start with are a vibrating violin string and drum membrane. In both cases a medium (the string and drum membrane materials) acts as a field — a property given at a point in space, here the property is the elasticity of the material. It turns out that because the string and drum head are pinned down at the edges, only certain characteristic vibrations can exist, these are called modes or in mathematics eigenstates (eigen in German means self/characteristic). The characteristic vibrations are a state that the system (i.e., string or drum head) can exist in, but they do not necessarily always exist, it depends on how the system is excited (i.e. plucked or banged). In many instances the overall vibration of the string or drum head is a combination of characteristic vibrations, an analogy one could think of is how we combine the colors red and green to get yellow. We can combine characteristic vibrations to get a new vibration, this is commonly called linear superposition or combination.
So now that we've discussed a little about waves, lets revisit the "wave-like" aspect of quantum mechanics. We start by stating that the total motion and interactions of a quantum system is described by a "wave-like" state or wavefunction, $$\Psi\left(\chi_1,\chi_2,\chi_3,\cdots,\chi_N\right)$$,
where $\chi_i$ is a single state function that will depend on position, time, and the type/number of particles. When the time dependence is removed from the description of the system, that is the system is time-independent, it described by the equation:
$$\hat{H}|\Psi(\chi_i,\cdots)\rangle=E|\Psi(\chi_i,\cdots)\rangle $$
This is the Schrödinger time-independent non-relativistic wave equation which is an eigenvalue problem. Lets describe each term in the equation in more details. The term $\hat{H}$ is called the Hamiltonian operator named after William Rowan Hamiltonian and describes the dynamics, interactions, and energy components of the system. In Newtonian mechanics we describe a system in terms of the acting forces, in Hamiltonian's representation we use momentum and energy. For a quantum system we can write a generic electronic Hamiltonian as,
\begin{align}
H &= K.E. + P.E. \\
&=\frac{-\hbar}{2m_e}\sum_{i}^{N}\nabla_{i}^2+\frac{1}{2}\sum_{i,j}^N V(\mathbf{r}_i,\mathbf{r}_j)
\end{align}.
The first term representing kinetic energy (momentum) and potential energy (field interactions). In this representation the K.E. and P.E. are said to be operators — I tend to think of these as being equivalently equal to transformations — that are "quantized" in their representation. These operators act on the "wave-like" state, $\Psi$, of a system producing a representation of the system in terms of K.E. and P.E. At this point the representation is a linear combination of characteristic "wave-like" states that the system could be in.
But what happens if I ask the question "What energy is associated with a particle in a given state and where can that particle be in physical space?", more specifically what can I know about the particle that makes sense or has meaning to me, the human. See at this point we just know a spectrum of energies associated with the "wave-like" solution states, which doesn't have a clear comprehensible meaning to us/me yet. It is at this junction where the "reality" of outcomes of Q.M. falls in the the realm of philosophical physics, or in other words, subjected to interpretation. In quantum physics pedagogy the stance is we need to make an expectation or guess on what the outcome is to be. To do this we say we have to project/collapse/measure the "wave-like" states onto a known meaningful platform, a set of states, to get a understanding of where a particle might be. It turns out that this projection is best interpreted as information about the probability or probability of transition of "wave-like" state; that is with respect to the state its in and the state I know. This is typically what is understood as the expectation value or measurement collapse argument (also more commonly called the Copenhagen interpretation), which is necessary for us (humans) to relate the math or experimental results of quantum mechanics to our notion of particles. To write this out we would have:
$$ <\Psi|H|\Psi> = E <\Psi|\Psi> $$
I use this equation to try and understand the measurement outcome in quantum experiments, that is, the act of observing in an experiment is akin to enforcing a projection onto a known state/solution.
So lets recap, we mentioned that quantum systems can be treated mathematically similar to wave dynamics. We said that these quantum systems have characteristic "wave-like" solutions, e.g., eigenvalues and eigenvectors. Then an equation relating the operation on or transformation of the quantum "wave-like" state resulted in the same state multiplied by the eigenenergies, but we said it only tells us about the energy spectrum and the states the quantum system could be in. Finally, we tried to understand this by projecting or collapsing the "wave-like" solutions onto known solutions (I forgot to mention these could initial states we used prior to applying any operators or transformations).
All this discussion has so far abstracted the "wave-like" states from actual matter that occupies the states. This is akin to our string or drum which we known has vibrating solutions regardless of if its actually been plucked or banged. The next step is to understand how our notion of particles or matter is associated with the "wave-like" states. In other words, how particles or matter can occupy those solutions. It turns out that electrons, which are the basic building blocks for how we (humans that is) experience chemistry and materials, have very specific rules for how they assume a given state. Electrons are indistinguishable particles, meaning they cannot be individual tracked or resolved, but they have a unique property associated with the "wave-like" state that requires them to take on a specific intrinsic angular momentum which is called spin$^\star$. This spin characteristic gives rise to "wave-like" states such that only a single electron is allowed to acquire a give "wave-like" state of specified spin. In QM this is called the Pauli exclusion principal for fermions which was postulated by Wolfgang Pauli based on spin-statistics theorem. It wasn't till Paul Dirac's work with relativistic QM of an electron that "spin" was deduced to be a consequence of the inclusion of Einstein's special relativity, at least this is my understanding of the outcome of the Dirac Equation.
So now let me try and give analogy to re-explain this, not sure it will work out.
The Race Track (Quantum) Architect
We start our conceptual understanding of a quantum system which describes electrons in an environment of atomic nuclei.
A quantum architect is told that they will need to design a race track. The race track has several constraints in how it can be built. The first piece of information the architect is given is where the race track will be built. This will dictate the shape, number of bends and slopes of the race track. The race track terrain is our analogy for the potential energy surface created by the combinatoric interactions among the atomic nuclei and electrons.
We then tell the quantum architect how many lanes/tracks are needed to accommodate the drivers, these are our analogy for the "wave-like" states an electron can be in. There are additional pieces of information we need to tell our architect about this race track, that is, the drivers of this race track have unique characteristics. The first is that they all look identical and drive the same car, so we cant tell them apart, therefore we never know who is really in each lane. The next unique characteristic is that each driver has a preferred direction (spin) of driving, forward or reverse, and preferred lane (spatial location). The third characteristic is that each driver dislikes all other drivers, so they try to avoid each other at all costs and never use the same lane, with one exception, that is when two drivers who drive in opposite directions use the same lane/track they both usually perform better — we will take this to mean lower energy — but no lane can ever accommodate drivers who are moving in the same direction.
So lets break down our analogy. The number of lanes corresponds to the number of "wave-like" state solutions we can have which depends on the type of system we are interested in; here the system is our race track configuration based on the number of drivers, terrain, etc. The fact that all the drivers look identical and drive the same car is a characteristic of electrons, they are indistinguishable particles. The preferred driving direction of each driver corresponds to a innate property of electrons (more broadly fermions) dubbed "spin", where its value corresponds to a positive (up) or negative (down) sign due to its behavior in magnetic fields. The fact that the drivers dislike each other but can have special configurations in the lanes corresponds to the Coulomb and exchange forces, where the latter is due to spin and the Pauli exclusion principal.
With all this information the architect is ready to build our race track, but, we through in an additional caveat, that we need him/her to build it in such a way that the number and shape of the lanes can change dynamically. At this point he/her is annoyed, but says their company is so skilled that they can dynamically adjust the state of the race-track to accommodate any number of drivers. So now drivers can be added or removed from the rack track simply by the architects ability to add or remove lanes.
Okay, so we have described the quantum state of electrons using an analogy. I'm not sure how good this was but I will continually revisit it to improve. No quote for this blog given it is not final.
With all this information the architect is ready to build our race track, but, we through in an additional caveat, that we need him/her to build it in such a way that the number and shape of the lanes can change dynamically. At this point he/her is annoyed, but says their company is so skilled that they can dynamically adjust the state of the race-track to accommodate any number of drivers. So now drivers can be added or removed from the rack track simply by the architects ability to add or remove lanes.
Okay, so we have described the quantum state of electrons using an analogy. I'm not sure how good this was but I will continually revisit it to improve. No quote for this blog given it is not final.
$^\dagger$ Please comment if you find any explanation of analogy incorrect. This blog post was a result in trying to explain to my wife, who is not trained in a STEM field, the physics that gives rise to the chemical and material world around us. What inspired this was my reading of books discussing self-learning approach dubbed the Feynman technique, pioneered by the famous physicist Richard Feynman. The idea is you take a topic or idea that your familiar with but not sure your level of understanding and work through the topic/idea (e.g. Quantum States) as if you were teaching it to someone else. The purpose is to help identify areas where you may lack understanding or capability (i.e. mathematics). Since I'm always trying to improve my level of knowledge in quantum mechanics, this blog post was my attempt to do so.
$^\mp$ I have chosen to use the quoted term "wave-like" instead of the more common and probably accurate term wavefunction. My reason for doing so is at this level of discussion I don't think any additional clarity is gained by using the word wavefunction, but "wave-like" in my opinion, conveys that the wavefunction will have a functional form that is reminiscent of a wave or wave packet. Anyhow, I will use the two interchangeably throughout this blog.
$^\ddagger$ My understanding is that the leading view in quantum (field) physics is that particles are a manifestation of excitations in quantum fields. In other words, it is quantum fields that exist not particles, and when we speak of particles, say an electron, we are just talking about quantized excitation in an electron field.
$^\star$ The origin of the term spin is unfortunate because it forces a physical picture of the electron which isn't necessarily true in the framework of QM. It comes from the initial view that an electron is not a structure-less point object in space, but that it has some diameter/volume associated with it and spins about its own axis (as the earth does). To my knowledge the diameter of a electron has never been observed or measured. As mentioned in the main text, the spin is a result of special relativity which adds chirality (handedness) to the spatial part of the wavefunction, also called a spinor.
$^\star$ The origin of the term spin is unfortunate because it forces a physical picture of the electron which isn't necessarily true in the framework of QM. It comes from the initial view that an electron is not a structure-less point object in space, but that it has some diameter/volume associated with it and spins about its own axis (as the earth does). To my knowledge the diameter of a electron has never been observed or measured. As mentioned in the main text, the spin is a result of special relativity which adds chirality (handedness) to the spatial part of the wavefunction, also called a spinor.
Reuse and Attribution
