Quantum Mechanics for Beginners

Introduction to Quantum Mechanics

RNfinity | 04-03-2023

Introduction to Quantum Mechanics

The majority of quantum mechanics development occurred in the first half of the twentieth century. It is focused on the smallest particles with their characteristics, the microphysical objects and quantities. The smallest quantum mechanical particles are invisible to the human eye. Hence, it is impossible to experience a quantum system with our senses. Particularly, quantum mechanics defies the accepted theory that nature is always being generated and quantifiable.

Quantum mechanics (QM) is study of physics on extremely short length scales, though it also directly applies to some macroscopic systems. The word "quantum" refers to the notion that certain quantities can only have discrete values in quantum physics, as opposed to classical mechanics. Nonetheless, some quantities continue to have continuous values [1].

A Brief History of QM

Particles in quantum physics exhibit wavelike characteristics, and the Schrodinger equation, a specific wave equation, determines how these waves behave. It is a mathematical representation of the probability of finding a particle in a particular location. This is contrary to the classical physics interpretation of reality, where matter is made of particles, which are discrete objects that behave in a predictable way and have specific, determinable locations.

Let's briefly review the development of quantum mechanics throughout time.

Max Planck:

Max Planck hypothesized in 1900 that energy-rich quantized lumps of light with frequency v are released in integral multiples of the quantity.

E = hν

Albert Einstein:

According to Albert Einstein in 1905, the quantization is a property of light, and the lumps can be thought of as particles that we now refer to as "photons." His research on the photoelectric effect, which deals with how light is absorbed by a material and how it emits electricity, led to this suggestion.

E = ω

Niels Bohr:

According to Niels Bohr in 1913, atoms' electrons exhibit wave-like characteristics. A few aspects of hydrogen were successfully explained by this, including the known quantized energy levels.

Louis de Broglie:

According to Louis de Broglie in 1924, all particles are connected to waves, with the frequency and wavenumber of the wave being determined by the same relationships that we discovered for photons above, namely E = ω, p= k. The so-called wave particle duality results from the fact that every particle has a wave attached to it.

Werner Heisenberg:

Matrix mechanics was incorporated into Werner Heisenberg's formulation of quantum mechanics in 1925. Instead of discussing this matrix formulation, we will examine the wave formulation that follows that was proposed by Schrodinger.

Erwin Schrodinger:

A wave-based interpretation of quantum mechanics was developed by Erwin Schrodinger in 1926. He recorded a wave equation (known as the Schrodinger equation) that describes how waves change through time and space.

Max Born:

Max Born accurately identified the probability amplitude in Schrodinger's wave. To get the desired probability, the wave must be squared, which is what we mean by "amplitude." More specifically, we must square the absolute value of the wave because it is inherently complicated. This results in the probability of discovering a particle at a specific location.

Paul Dirac:

Paul Dirac demonstrated that the interpretations of quantum mechanics proposed by Heisenberg and Schrodinger were similar in the sense that they might both be obtained from a more general interpretation of the theory.

John Bell:

Bell's Inequality (1964): In 1964, John Bell proposed a way of testing whether particles were truly "entangled," or linked, such that the state of one particle is immediately affected by the state of another, regardless of the distance between them, or whether their properties were inherent at their moment of generation and behaved in accordance with Einstein, Podolsky, Rosen hidden variables explanation.

Due to the fundamental differences between the physical principles of quantum mechanics and those of classical mechanics, it is commonly considered as new branch of physics [2]. Superposition, entanglement, and quantum teleportation all work differently than we are accustomed to thinking about them in our Newtonian world. Even Einstein had trouble describing this fact, implying to entanglement as "spooky activity at a distance" [1], and it is impossible to envisage explanations like Schrödinger's cat [3]. Richard Feynman, a scientist, pioneered the idea of quantum computation by arguing that computers may apply the fundamental ideas of quantum mechanics to solve quantum mechanical issues.

Quantum computers have also been possible due to the shrinking size of computer parts. It is expected that components the size of a single atom will exist. Here, quantum mechanics is required since the laws of classical physics are no longer relevant [4].

The amount of information amassed regarding the field of quantum mechanics has greatly increased in recent years. Nonetheless, many people find it difficult to begin studying quantum mechanics. The principles and their implications are challenging to understand. As quantum computers become more prevalent, more people may be required to study quantum computer programming, which calls for deep knowledge of quantum physics. The quantum circuits which are core of many quantum computers, have been the subject of countless attempts in the past. Since games engender an engaging approach, several of them employ game theory and contests to establish an easy and interesting entrance point to quantum computing [5]. It is becoming more and more important, to study quantum computing because of the great progress and potential of this technology [6].

According to Nita et al. [6], quantum literacy has to be made more approachable for a variety of students. According to Stadermann et al. [6], who investigated the secondary education quantum physics curricula of 15 different countries, quantum physics and quantum mechanics are being introduced earlier and earlier. They also see the opportunity to have "unusual" discussions about this difficult subject. Perhaps children who play quantum games as they grow up may develop an instinctively grasp of quantum phenomena that our young people lack.

Quantum computers cannot employ the boolean algebra logic of conventional computers because of quantum mechanics. A conventional computer uses bits, and binary state with two possible values, to perform calculations. These variables could be categorised as active or inactive, true or false. Yet, the most common ways to express them are as 1 or 0. A quantum computer, in contrast, runs on concept of "qubits," or "quantum bits." Instead of adopting a single state of 1 or 0, a qubit, unlike a normal bit, adopts a true combination of these states. "Superposition" is another name for this orderly collection of states.

In some varieties of quantum computers, quantum gates are applied to these qubits as fundamental operations. Quantum gates are not devices, in contrast to digital gates. These could be considered changes to the state of a qubit (spin). This process alters a qubit's ground state.

A quantum circuit is made up of numerous quantum gates used in succession. It could be viewed as a particular type of computer programs that describes several manipulations of the fundamental states of various qubits.

Rules to understand quantum mechanics:

When viewed across the lens of Newtonian universe, quantum mechanical phenomena are challenging to understand. The three following rules are of the vital importance:

Superposition:

A capacity of quantum particle to coexist in multiple states at once with differing probabilities. For example, a particle can exist in two different locations at the same time, or may have two different energy levels simultaneously.

This idea was famously illustrated by Schrödinger's cat experiment, in which a hypothetical cat is placed in a box with a vial of poison that will be released if a radioactive particle decays. According to quantum mechanics, until the box is opened, and the observer looks inside, the cat exists in a superposition of both being alive and dead.

Entanglement:

It is a process in which far elements of a quantum system show mysterious relationships. In this case, status of one component affects the state of another component. This means that if the properties of one particle are measured, then the properties of the other particle can be determined immediately, even if the particles are separated by vast distances.

This phenomenon was famously described by Einstein as "spooky action at a distance," and it has been demonstrated in many experiments.

Quantum Uncertainty

One of the core principles of quantum mechanics is the uncertainty principle. The uncertainty principle states that the position and momentum of a particle cannot both be known with absolute certainty. The more precisely one is known, the less precisely the other is known.

Collapse:

The method through which quantum states of system are transformed into classical states. Collapses happen when we evaluate quantum circuit and so make system smaller. It is crucial to understand that measurement alone can determine the state of qubits.

A base state of | 0> or | 1> results from state’s collapse at this point. This procedure destroys the qubit's state, which can only be restored by retrying quantum gates in original way [7].

Applications of Quantum Mechanics

While the principles of quantum mechanics may seem abstract and theoretical, they have many useful technological applications. One of the most well-known applications is the laser, which relies on the principles of quantum mechanics to produce a focused beam of light. Other applications include quantum cryptography, which uses the principles of entanglement to create secure communication channels, and quantum computing, which uses the principles of superposition and entanglement to perform complex calculations much faster than classical computers.

Quantum information Processing

Quantum information processing makes extensive use of key aspect of quantum mechanics. Superposition states:

Compared to their classical counterparts, quantum systems have far deeper and more intriguing reality. A single bit, the most fundamental component of any traditional information processor, can only exist in one of two potential states: 0 or 1. A single quantum bit, or qubit, has access to an infinite number of such superposition states. In a similar fashion to how a musical note has several harmonic frequencies, nature permits it to simultaneously have a component that corresponds to 0 and a part that corresponds to 1.

Information technology and quantum mechanics

There are two ways that information technology (IT) can benefit from quantum physics, which might be loosely categorised as evolutionary and revolutionary. Both of which are very active research fields with huge technological potential. When quantum physics is used as a tool in evolutionary research, a significant portion of its impact can be understood and appreciated without having a firm grasp of the theory itself. Paradoxically, quantum mechanics plays an enormous role in revolutionary work.

Quantum physics is fundamentally working in evolutionary IT work to better comprehend and thereby enhance existing technologies. For example, understanding a quantum behavior of electrons in a materials is beneficial for the creation of lesser and quicker silicon or other semiconducting devices. The switch to superconducting Josephson junction devices from silicon transistors would be a little more radical. Superconductors may be naturally quantum in nature, but this would not create a fundamentally new technology. Here, faster digital switching and less energy use would be advantages of superconducting.

Nonetheless, the logical operations carried out and the physical bits that are manipulated in such devices are identical to those of other devices. These well-known logical processes continue to follow the same classical physics laws as they have always done. If quantum physics has an additional, very distinct effect on information technology, then a truly radical revolution occurs. Consider machines that genuinely process information and conduct logical operations in accordance with quantum physics rather than improved copies of what we already have. Such devices, which would be a component of the emerging quantum information technology (QIT), are significantly different from their classical counterparts [8].

Fundamental particles like electrons, in contrast to billiard balls, can display interference phenomena that resemble waves, and two of them can become entangled. Similarly, machines that store, analyse, and transfer information in a quantum mechanical manner are able to perform tasks with that information that would seem completely uncharacteristic or even impossible for classical machines. However, it is not that simple; if it were, QIT would have been in existence for considerable amount of time by now. The issue is that detecting electrons usually causes them to move around, destroying entanglement and interference.

In mathematics, the state of quantum particle is often indicated by |Ψ>) is a vector in an abstract Hilbert space of potential states for system. A basis made up of the two potential classical states, |0> and |1>, spans a space for single qubit. With the right selection of the complex coefficients a and b, each state of a qubit could be reduced into superposition.

|Ψ> =  a|0> + b|1>

In this case, |Ψ> is represented by vector, which is an orthogonal 2D unit vector used in a common representation of the basis  ( 01) and (10)  [9].

Quantum mechanics of today

The development of quantum mechanics led to the Nobel Prize in Physics being awarded to Planck, Einstein, Bohr, and many others. But if they were still living today, they wouldn't recognize the world today, which has changed as a result of there discoveries. Quantum mechanics runs computers due to transistors. Transistors are little electronic components found within computers that serve as data storage. A transistor conducts electricity when it is turned on, and the computer interprets this as a "1." A transistor stops conducting electricity when it is turned off, and the computer interprets this as a "0." Ones and zeros are the language of computers. To encrypt data, they turn transistors on or off. A semiconductor substance is used to create transistors. According to quantum mechanics, an electron can only occupy a limited range of energy levels. These levels are "bands," or ranges of permitted energy values, when examining a big group of electrons, such as those present in semiconductors [10].

The semiconductor conducts electricity when it is linked to a voltage that is within the energy band. It does not conduct electricity when attached to a voltage that is outside of the permitted energy band. As an insulator, it works. Transistors turn on or off in this manner, which the computer interprets as a 1 or 0. Digital refers to anything that operates using a binary code of 1s and 0s. Semiconducting transistors are used in almost every digital device you can imagine. even your television, as well as your computer and mobile device! Think of a world without these items or the Internet. This would be world without quantum mechanics.

The Future of Quantum Mechanics

As technology continues to advance, it is likely that our understanding of quantum mechanics will also continue to develop. New experiments may shed light on some of the remaining mysteries of quantum mechanics, and lead to new applications and technologies that we can't even imagine yet.as an example, scientists are exploring the possibility of using quantum entanglement for faster-than-light communication, or even for teleportation.

Conclusion

Quantum mechanics has fundamentally changed our understanding of the universe. While it may seem daunting at first, with patience, anyone can begin to grasp the basic concepts of quantum mechanics. With further progress, many new technologies and insights that could revolutionize the world as we know it, may be around the corner.

Reference

[3] C. Monroe, D. M.Meekhof, B. E. King, and D. J. Wineland, "A “Schrödinger cat”superposition state of an atom," Science,vol. 272, no. 5265, pp. 1131-1136, 1996.

[4] S. Gudder,"Spooky Action at a Distance," arXivpreprint arXiv:2005.11870, 2020.

[6] A. Kultima, L.Piispanen, and M. Junnila, "Quantum Game Jam–Making Games with QuantumPhysicists," in Proceedings of the24th International Academic Mindtrek Conference, 2021, pp. 134-144.

[9] H.-K. Lo, T.Spiller, and S. Popescu, Introduction toquantum computation and information. World Scientific, 1998.