Quantum Mechanical Model of the Hydrogen Atom

Introduction

In this report, main topics are reviewed and discussed about the differences between classical and quantum physics, focusing on how atoms and particles behave in these states. In the year 1923, Niels Bohr attempted to explain the chemical periodicity in terms of electronic structure. According to Eric R. Scerri, Bohr’s method was called the ‘aufbau principle’ of which consists of building successive atoms by the addition of an electron to the previous atom. Bohr himself, developed a technique to utilise the periodic table and a process of where he used two quantum numbers, of where $n$

the main quantum number and $k$

the azimunthal quantum number. These numbers themselves, emerge from the quantum conditions and have the purpose to identify the stationary states of the system. Electrons themselves, in these stationary states do not radiate energy unless undergoing a transition in between stationary states. The introduction of stationary states to atomic physics was Bohr’s main contribution to the quantum theory of atoms. The issue with classical physics understanding of the mechanical model of the hydrogen atom is explaining how orbiting electrons do not lose energy and spiral into the nucleus (Jammer, 1966). Additionally, Bohr postulated that electrons exist in stationary, non-radiating states and only emit radiation on undergoing a transition between states (Bohr 1913)

1. Differences between Classical and Quantum Physics

As defined by ESALQ (2017), they state the key differences between classical and quantum physics. “Classic physics is casual; complete knowledge of the past allows computation of the future. Likewise, complete knowledge of the future allows precise computation of the past.” Hence, they’re defining Classical physics as being the more typically accepted version, less ‘crazy’, more simplistic and easier to practice. However, they also elaborate onto how quantum physics considers objects to be neither particles nor waves; they are a unique combination of both but not the same at the same time. If given complete knowledge of the past, quantum physics suggest we can only make probabilistic predictions of the future. An example of the differences between classical and quantum is often referenced from ESALQ (2017), an example of where two bombs with identical fuses that would explode at the same time. Quantum physics, two absolutely identical radioactive atoms can and generally will explode at very different times. Two identical atoms of uranium-238 will, they will experience radioactive decay that is separated over billions of years, despite them being identical.

There is a rule that physicist often use in order to separate classical physics apart from quantum, if Planck’s constant appears in the equations, it is deemed quantum physics. If it doesn’t, it’s deemed as classical physics. Physicists relish in the idea that quantum physics is the correct theory, even though it is not a completed one and is constantly updated and worked on. Classical physics can be derived from quantum physics given the constraint that some of the quantum properties are hidden. That fact is called the “correspondence principle” and shall be discussed further on in the report.

Drawing from the Supplementary information provided by Esa Jaatinen, Wave-particle duality has many consequences, one of which is the uncertainty principle, of which restricts the ability to simultaneously know the particle-like (i.e. position) and wave-like (wavelength) aspects of the object to an arbitrary level of precision. Hesienberg was one of the first to demonstrate this but it also follows naturally from Fourier analysis as will be shown below;

Consider the two waves shown in figure 6.

Figure 6. A. Wave with a well defined wavelength but poorly defined location. B. A

highly localized wave with a poorly defined wavelength.

The uncertainty principle arises when an attempt to measure an entity’s position and momentum. Newton’s laws of nature lets us determine a particle’s position and velocity if we know the forces acting upon the body or the potential that the body exists in. Drawing an example from how two identical charged particles with the same macroscopic mass, m, that come together (interact) and then move apart as displayed in Figure 7. With the classical picture it is possible to simultaneously know the position and velocity of each of the particles at any time during the encounter. Each particle keeps its own identity as it is always possible to distinguish between the two because an observation of them does not alter their behaviour. This is to be considered as one of the implications of the classical approach – that monitoring of the system has no real effect on the behaviour of the constituent components.

2 Atom Stability and Energy of the Electron

Cited from the book “The Stability of Matter: From Atoms to Stars: Selecta of Elliott H.Lieb”(2005).“There are methods available for calculating rather precise lower bounds for the energy of simple atoms or molecules. If complex atoms or molecules are to be investigated, these methods become inapplicable, or impracticable.” Quantum assists their process in coming to a way to find the lower energy levels of these bounds for atoms and molecules, a simple atom is referred as being the hydrogen atom as it is one of the main four elements that make up everything.

Highlighted by Randell L.Mills, the stability of a hydrogen atom is defined by what state it resides in. It is taught in some textbooks that hydrogen itself cannot go below the ground state. In figure 1 the different states of a hydrogen atom are shown:

Figure 1: Hydrogen atom with four lowest energy levels

Randell then goes on to state how atomic hydrogen has an experimental ground state of 13.6 eV that can only exist in a vacuum or in isolation, furthermore, how hydrogen cannot go below the ground state in isolation as well. Additionally, there is no defined composition of matter that contains hydrogen in the ground state of 13.6eV, it is considered a hypothetical that hydrogen has a theoretical ground state based on first principles.

In the past, there were many different ‘ways’ that solved the wave equation for hydrogen, one of these equations is the Schrodinger equation, this equation in particular gives the object being investigated spontaneously radiative states and the nonradiative energy level of atomic hydrogen.

$\widehat{H}\mathrm{\Psi }=\mathrm{E\; \Psi }$

This alone, it is warranted despite its inconsistency with physical laws as well as with numerous experiments. A hypothetical solution of where it would be suited to first principles and having first principles as the support of quantization was never found. Scattering results required the answer to be interpreted as probability waves that give foundation to the uncertainty principle, of which connected the basis of the wave particle duality. The correspondence principal speculates that quantum predictions must meet with classical predictions on a solid foundation in order to be deemed valid. In contrast to this, recent data has exhibited that the Heisenberg uncertainty principle, which serves as the foundation behind the wave particle duality and the correspondence principle taught are experimentally wrong.

In the year 2000, Randell proceeds to highlight how recent findings at the time describe how a reconsideration of the postulates of quantum mechanics has given a solution of a Schrodinger-like wave equation based on first principles. Hydrogen at hypothesised lower-energy levels has been recorded in the extreme ultraviolet emission spectrum from interstellar medium. Additionally, new compositions of matter that contain hydrogen at lower predicted energy levels were observed in experiments, which energy levels are achieved using the novel catalysts.

3 Infinite potential well example & Model of the electron

An electron in classical and quantum physics do not have the same meaning. In reference to INSERT REFERENCE HERE, classical physics states that objects such as atoms and electrons were to be treated as strictly particles and things like light and other forms of electromagnetic radiation to be treated as waveforms.

Quantum physics, there exists a limitation to how to determine the position and momentum of these properties both at once, as they are not measurable to any accuracy as the properties are independent to the process of observation. A particle can be described as a wave in quantum, of which encodes the probability of making a specific measurement. Possible observations are determined by these probabilities and hence are not fixed. There is no clear direction that can be defined by subsequent observations.

Additionally, from (Khatun, Joyner, Cosby, & Joe, 1998), they found that an electron experiences modulation whilst in conductance sees a reduction in the presence of the constriction in the structure. As the electron conductance has also been investigated through the stub-constriction structure by scanning the potential barrier, they found that the conductance of the surface is heavily modulated. The resonance and anti-resonance oscillatory structures are caused by quantum interference, of which is defined as scattering of electron waves between the incident wave by the boundaries and potential scatterer.

Hence, there exists many types of phenomenon unseen in classical mechanics that are adapted into the mathematic concepts of Quantum mechanics and for the probabilistic theory, it does give ‘hindsight’ into how things happen when and where. The issue that arises from it is how it affects most aspects of physics already defined. In the analogy given, in Quantum, the ball is no longer considered a ball, but instead an eigenvalue apart of a wave equation. And plays a part into how Quantum physics is truly a tricky physical concept, there are defined problems that can explain certain phenomenon, such as wave-particle duality and can be explained by using an infinite-potential well, but each have their limitations.

4 Quantum tunneling

From the textbook “Quantum theory of tunnelling” by Razhey Mohsen, states how Quantum tunnelling is a microscopic phenomenon, of where a particle can perforate and typically pass through a potential barrier. The maximum height of the barrier is larger than the kinetic energy of the particle, hence, this motion is not allowed by the law of classical mechanics. Scientists measured electrons escaping from atoms without having the necessary energy to do so. As outlined by Clara Moskowitz goes on to inform us how ”in the normal world around us, this would be like a child jumping into the air and somehow clearing a whole house.”

Quantum tunnelling itself is possible because of the wave-nature of matter. Within the quantum world, particles are referred to as acting like waves rather than solid objects clashing with other objects. Again drawing from Clara Moskowitz, an electron doesn’t exist in a singular place at a single point in time and with a certain energy, but rather a wave of probabilities. Physicist Dhor Shafir of Israel’s Weizmann Institute of Science elaborated in his essay of how he prompted electrons to tunnel out of atoms, and when they did he measured the time period for this process to happen was within 200 attoseconds (an attosecond is 10^-18 seconds).

Furthermore, stemming from Clara Moskowitz article, she defines that overall, an electron is given a certain probability of actually quantum tunnelling out of the atom. Let’s say for instance a non-zero value, five per cent chance per se, meaning that the electron when it tries to quantum tunnel out has a ninety-five per cent chance of not being able to do so. Now swapping out an electron for a human poses a greater problem, a person consists of 7^34of atoms according to Laurie L.Dove. Henceforth, let’s pose a similar probability as the electron, if a human were to approach a wall with the intention to pass through it, each individual atom would undergo this probability in order to determine if it will tunnel through the wall or reflect off of it. But in most instances, a person as a whole would bounce off of the wall, but, although we as the whole body of atoms may not have been able to tunnel through the wall. Some atoms on our hair, clothes, shoes or face did. This is the hypothetical idea anyway behind quantum tunnelling. Scientists have been able to observe and deduce that an electron as small and light as it may be, experiences quantum tunnelling that statistical mechanics cannot justify. It is, therefore, an interesting study into how particles behave in the quantum mechanics ‘universe’ in contrast to how particles behave in the classical mechanic’s observations.

Conclusion:

Overall, Quantum mechanics is an unfinished theory that from its origin by Schrödinger, Heisenberg and Dirac in 1925, alongside with Einstein’s and Bohr’s ideas is a solid theory with real experiments showing results. However, there exist some limitations to quantum mechanics, as everything isn’t fully discovered and experiments aren’t exactly producing the same results consecutively. However, with more experiments, theories and discoveries to investigate, a fully comprehensible explanation of quantum mechanics hopes to be achieved one day.

References:

• Khatun, M., Joyner, P. K., Cosby, R. M., & Joe, Y. S. (1998). Quantum interference in a stub-constriction structure containing an infinite strength potential barrier. Journal of Applied Physics, 84(6), 3409-3415. doi:10.1063/1.368500