No. Electrons Do Not Have Orbits
Atomic structure and composition have long fascinated scientists and philosophers. The theories pertaining to this topic date as far back as 440 BC, when Democritus proposed the first known model of the atom. However, it is theorized that even his work was extrapolated from the work of past philosophers. From that day forth, various models and ideas have been proposed, throned, dethroned, and refashioned as an increasing amount of knowledge became accessible to us.
Fast forward to 1913, and we find ourselves with the most recent semi-classical model proposed by Neils Bohr. Here’s a summary of what his theory states:
- The electron revolves around the nucleus in certain stable orbits which are stationary and at discrete distances from the nucleus.
- Only those orbits can exist for which, the orbital angular momentum of the electron is an integral multiple of ℏ.
- Electrons can lose or gain energy only by jumping from one allowed orbit to another.
The second and third of these postulates are perfectly fine and there is experimental evidence to support them. However, the first postulate runs into a problem as soon as we step into the realm of quantum physics. To prove that there is a problem inherently associated with the concept of discrete orbits, I turn to the famous thought experiment of Heisenberg’s microscope.
Heisenberg’s Microscope
The reason for choosing this thought experiment lies behind the role it plays. Due to its basis in using a theoretical microscope, it serves as a bridge between classical optics & quantum mechanics and uses the former’s principles to argue in favor of the famous uncertainty principle.
If λ is the wavelength of the incident photons, then classical optics tells us that the electron will be resolved only up to an accuracy of Δx = λ/sin (ε). Further, perceiving or seeing an image is equivalent to the reflection of a photon from the object and that reflected/scattered photon hitting our eyes.
However, a photon scattered from an electron will have a Compton Shift in its wavelength that will be proportional to ℎ/λ. The exact value of this shift can never be exactly known since we are perceiving the electron across a light cone of angle ε. Hence, the exact direction of the scattered photon is spread across this cone. Therefore, we can only estimate the maximum possible accuracy in the momentum along the x direction. This is given by Δpₓ ≈ (ℎ/λ) sin (ε).
Hence, we have an expression for the relation between the maximum possible accuracies, or the uncertainties associated with the position and momentum of an electron as Δx Δpₓ ≈ ℎ. Those who have already studied basics of Modern Physics will recognize this as an estimate of the Heisenberg Uncertainty Principle.
What Does This Mean?
It means that our ability to know where the electron is, and how fast it is moving, is limited. To a hundred per cent accuracy, we can only determine one of those at an instant in time. It is impossible to figure out the exact and simultaneous values of position and momentum with zero error.
Does this uncertainty arise out of the limitations of our instruments? Could we somehow create an extremely precise measuring device which could remove any spread in the values of position and momentum that we find?
The answer is no.
In fact, the uncertainty associated with the simultaneous measurement of canonically conjugate variables is an inherent law of nature we can never go around. No matter how much we try to remove it, we can never go beyond the limit set on us by the relation Δx Δpₓ ≥½ ℏ. There is no instrument that can remove it, and if we try to decrease the uncertainty in x to the impossible value of 0, we will find that the uncertainty in pₓ simultaneously shoots up to infinitely large values. The above inequality must hold.
Back to Our Electrons
Let us now get back to the image of an electron revolving in discrete orbits around the nucleus. The problem arises with just a single word: discrete. When we say discrete orbit, we are clearly specifying its distance from the centeral nucleus, as well as the tangential velocity associated with the electron in that orbit. Further, we also do not have a delay in measurement. In the orbit, we can figure out both of these values simultaneously.
After all, the position and momentum are nothing but a description of where the object is, and how it is moving. This data is included in the r and v vectors, respectively. Here is an image:
Thus, we have ourselves a paradox of sorts, a dilemma, if you will. On one hand, the uncertainty principle forbids us from simultaneous measurement of position and momentum. On the other hand, what’s there to prevent us from calling Heisenberg’s equation invalid now?
But here’s the deal. The second of these choices is not really an option at all. As I previously mentioned, the uncertainty principle is an inherent law of nature. It holds regardless of what realm we are working in. My choice of its derivation started with classical optics only because that was a relatively simple phenomenon to discuss and understand. Indeed, the fundamental formalism of quantum mechanics may be used to derive the exact form of the uncertainty principle for any pair of variables in terms of operators and commutation relations. The generalized equation is given below:
As long as the commutator of two quantities is zero, we can determine them simultaneously. However, if the commutator takes a non-zero value, the two observables corresponding to these operators cannot be simultaneously determined with 100% accuracy. Since the uncertainty relation holds without exception, the only conclusion is that our picture of the electron orbiting the nucleus is wrong.
So, How Does the Electron Move?
With the uncertainty principle in the picture, we are already into the realm of quantum mechanics. And this is precisely what shall be our savior. Bohr’s image was semi-classical in nature and thus, only partially incorrect.
The truth is that electrons do have orbits, but only in the most rudimentary sense. These orbits aren’t discretely defined as Bohr predicted. They are fuzzy and blurred. Due to the uncertainty principle, the representation of a quantum particle in the phase space now represents more of a fuzzy ball than an ideal point. Thus, while the electron does revolve around the nucleus, its orbital distance isn’t a ‘fixed’ value. Instead, it’s a probability distribution given by the square of the wavefunction, |ψ|². It could be anywhere between a range of values, with the most probable one being equal to the radius of that orbit as predicted by Bohr.
For example, in the ground state of the Hydrogen atom, instead of orbiting the nucleus exactly at the Bohr radius a₀ = 0.529E-10 m, it has a probability of being found at smaller or larger distances. Naturally, if you try to find the most probable or the average value of r in this scenario, you will still get the answer as 0.529 Angstrom.
Hence, the picture of one circle showing the location of the orbit of the electron is incorrect. Instead, in the image below, the black circle represents the most probable value of r in the ground state of Hydrogen, while all the blue dots are some of the infinitely many possible values that r could take.
Why the Failure?
Bohr’s model fails once we probe deeper since it is a semi-classical model that treats the electron as a particle. However, the electron is a quantum particle that exhibits dual nature and thus, Bohr’s model cannot fully determine its properties. That said, for most rudimentary or fundamental applications, Bohr’s model agrees excellently with experimental results, and is even supported by the de Broglie hypothesis of matter waves.
As for the quantization of orbits, that postulate still holds since the most probable values of each of the orbits is still such that the orbital angular momentum is an integral multiple of the reduced Planck’s constant, ℏ.
