import numpy as npimport plotly.graph_objects as goimport plotly.io as piopio.renderers.default ="plotly_mimetype+notebook_connected"# Create figurefig = go.Figure()# Define the Bloch sphereu = np.linspace(0, 2* np.pi, 100)v = np.linspace(0, np.pi, 100)x = np.outer(np.cos(u), np.sin(v))y = np.outer(np.sin(u), np.sin(v))z = np.outer(np.ones(np.size(u)), np.cos(v))uc = np.outer(u, np.ones(np.size(v)))vc = np.outer(np.ones(np.size(u)), v)fig.add_trace(go.Surface(x=x, y=y, z=z, opacity=0.5, colorscale='Blues', showscale=False, customdata=customdata, hovertemplate=hovertemplate))# Define the main quantum statesstates = {'|0⟩': (0, 0, 1),'|1⟩': (0, 0, -1),'|+⟩': (1, 0, 0),'|−⟩': (-1, 0, 0),'|R⟩': (0, 1, 0),'|L⟩': (0, -1, 0)}# Add markers for the main quantum statesfor state, coords in states.items(): fig.add_trace(go.Scatter3d( x=[coords[0]], y=[coords[1]], z=[coords[2]], mode='markers+text', marker=dict(size=5, color='red'), text=[state], textposition='top center' ))fig.show()
1 Section 1.1: Historical Introduction
1.1 1.1 Historical Introduction
At the dawn of the 20th century, the field of physics was considered nearly complete. This era was marked by significant advancements in various branches of physics, including Rational Mechanics, Thermodynamics, Electromagnetism, Fluid Mechanics, and Optics. These disciplines collectively provided a comprehensive understanding of the physical phenomena observed in nature. The prevailing belief was thus that all fundamental principles governing the natural world had been discovered.
However, despite the remarkable progress, there were two unresolved issues that cast a shadow over the otherwise clear understanding of classical physics. Lord Kelvin famously referred to these issues as “two clouds on the horizon.” The first cloud was the Michelson-Morley experiment, which failed to detect the expected motion of the Earth as the speed of light was found to be constant in all directions. This lead to the development of the Theory of Relativity by Albert Einstein. The second cloud was the problem of blackbody radiation, which classical physics could not adequately explain. This problem eventually led to the birth of Quantum Physics, with significant contributions from Max Planck, Albert Einstein, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Paul Dirac, Wolfgang Pauli, and Richard Feynman, among others.
The resolution of these two issues marked the beginning of Modern Physics, characterized by the development of the Theory of Relativity and Quantum Physics. These two pillars of modern physics revolutionized the understanding of our universe, providing new insights into the nature of space, time, and matter. The Theory of Relativity, with its concepts of spacetime and the equivalence of mass and energy, explained gravity and the large-scale structure of the universe. Quantum Physics, on the other hand, revealed the discrete and probabilistic nature of the microscopic world, fundamentally altering our conception of particles and waves.
Richard Feynman, one of the most influential physicists of the 20th century, famously remarked, “I think I can safely say that nobody understands quantum mechanics.” This statement underscores the counterintuitive and often perplexing nature of quantum phenomena, which continue to challenge our classical intuitions. Despite its complexity, over the last century, quantum mechanics has proven to be an extraordinarily successful theory, providing accurate predictions and leading to numerous technological advancements that include semiconductors, lasers, and quantum computing.
1.2 1.2 Black Body Radiation Problem
A black body is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of its frequency or angle of incidence. This concept is crucial in the study of thermal radiation and quantum mechanics. When a black body is in thermal equilibrium, it emits radiation with a characteristic spectrum that depends solely on its temperature, a phenomenon described by Wien’s Law. This law states that the wavelength at which the emission of a black body spectrum is maximized is inversely proportional to the temperature of the black body.
However, classical physics faced a significant challenge with the black body radiation problem, often referred to as the “Ultraviolet Catastrophe.” According to classical theories, such as Rayleigh-Jeans law, the energy radiated at ultraviolet frequencies would be infinite, which clearly contradicted experimental observations. This discrepancy highlighted the limitations of classical physics in explaining phenomena at atomic and subatomic levels.
The resolution to this problem came from the German scientist Max Planck. He proposed that electromagnetic waves could be considered as quantized stationary waves. Planck introduced the idea that the energy of these waves is quantized and can be expressed as \(E = h\nu\), where $E $ is the energy, \(h\) is Planck’s constant (\(6.626 \times 10^{-34}\) J·s), and \(\nu\) is the frequency of the radiation. This revolutionary concept laid the foundation for quantum theory, fundamentally altering our understanding of energy and matter interactions at microscopic scales. Planck’s radiation formula successfully explained the observed black body spectrum, resolving the ultraviolet catastrophe and marking the birth of quantum mechanics.
Another experimental observation that supported the quantized nature of light was the photoelectric effect. This is a phenomenon in which electrons are emitted from a material when it is exposed to light, and several results did not agree with a classical interpretation of this photoelectric effect:
Immediate Emission of Electrons: When light shines on a material, electrons are emitted almost instantaneously. This observation contradicts the classical wave theory of light, which would predict a time delay as the electrons accumulate enough energy to escape the material.
Intensity and Electron Count: Increasing the intensity of the incident light increases the number of emitted electrons but does not affect their energy. This result is inconsistent with the classical wave theory, which would suggest that higher intensity (more energy) should increase the energy of the emitted electrons.
Threshold Frequency: Light below a certain frequency, regardless of its intensity, does not eject electrons from the material. For example, red light, even at high intensities, fails to emit electrons, while weak violet light, which has a higher frequency, can eject electrons with significant energy. This threshold frequency is specific to the material and indicates that the energy of the incident photons must be above a certain value to overcome the work function of the material.
Energy of Emitted Electrons: The energy of the emitted electrons depends on the frequency of the incident light, not its intensity. Weak violet light can eject fewer electrons, but those electrons have higher energy compared to those emitted by more intense red light.
Albert Einstein provided a theoretical explanation for these observations by proposing that light consists of discrete packets of energy called photons. According to Einstein, each photon has an energy \(E = h\nu\), where \(h\) is Planck’s constant and \(\nu\) is the frequency of the light. When a photon strikes an electron in the material, it transfers its energy to the electron. If the photon’s energy exceeds the work function \(\phi\) of the material, the electron is emitted with a kinetic energy given by the photoelectric equation:
\(KE_{max} = h\nu - \phi\)
Here, $ KE_{max} $ is the maximum kinetic energy of the emitted electrons, $ h $ is Planck’s constant, \(\nu\) is the frequency of the incident light, and \(\phi\) is the work function of the material. This equation elegantly explains the experimental results and underscores the quantum nature of light, marking a significant departure from classical physics.
3 Section 1.4: Wave-Particle Duality
3.1 1.4 Wave-Particle Duality
Wave-particle duality is a fundamental concept in quantum mechanics that describes how light and matter exhibit both wave-like and particle-like properties. This duality is one of the cornerstones of quantum theory and has profound implications for our understanding of the nature of reality.
One of the most famous experiments demonstrating wave-particle duality is Young’s double-slit experiment. In this experiment, a beam of light is directed at a barrier with two closely spaced slits. When the light passes through the slits, it creates an interference pattern on a screen behind the barrier, which is characteristic of wave behavior. The pattern consists of alternating bright and dark fringes, indicating constructive and destructive interference of the light waves. However, when the experiment is conducted with individual photons or particles, the interference pattern still emerges, but only when the particles are not observed. When detectors are placed at the slits to determine through which slit each particle passes, the interference pattern disappears, and the particles behave as if they are classical particles, hitting the screen in two distinct bands. This experiment highlights the dual nature of light and matter, behaving as waves in the absence of measurement and as particles when observed.
The wave-particle duality of matter was further extended by Louis de Broglie, who proposed that particles such as electrons also exhibit wave-like properties. According to de Broglie’s hypothesis, the wavelength $ $ of a particle is inversely proportional to its momentum $ p $, given by the equation $ = $, where $ h $ is Planck’s constant. This idea was experimentally confirmed by the observation of electron diffraction patterns, similar to those produced by light waves, when electrons were passed through a crystal lattice.
Wave-particle duality is closely related to Heisenberg’s Uncertainty Principle, which states that certain pairs of physical properties, such as position $ x $ and momentum $ p $, cannot be simultaneously measured with arbitrary precision. The uncertainty principle is mathematically expressed as $ x p $, where $ x $ and $ p $ are the uncertainties in position and momentum, respectively, and $ $ is the reduced Planck’s constant. This principle implies that the more precisely one property is measured, the less precisely the other can be known. The uncertainty principle is a direct consequence of the wave-like nature of particles, as the wave properties impose fundamental limits on the precision of simultaneous measurements of conjugate variables.
Wave-particle duality and the uncertainty principle are central to the quantum mechanical description of nature, challenging classical intuitions and providing a deeper understanding of the behavior of light and matter at microscopic scales. These concepts have led to the development of various quantum technologies, including electron microscopy, quantum cryptography, and quantum computing, which exploit the unique properties of quantum systems.
4 Section 1.5: The Uncertainty Principle
In 1927, Werner Heisenberg introduced a groundbreaking concept in quantum mechanics known as the Uncertainty Principle. This idea fundamentally challenges the classical notion that certain pairs of physical properties (such as position and momentum) can be measured simultaneously with arbitrary precision. According to Heisenberg’s Uncertainty Principle, there is an inherent limit to the precision with which these properties can be known. Mathematically, this relationship is expressed as $ x p $, where $ x $ represents the uncertainty in position, $ p $ represents the uncertainty in momentum, and $ $ is the reduced Planck’s constant.
The Uncertainty Principle implies that the act of measurement itself affects the quantum system being observed. When we measure the position of a particle with high precision, its momentum becomes highly uncertain, and vice versa. This is a stark departure from classical mechanics, where it is assumed that all properties of a system can be measured without influencing the system. In quantum mechanics, however, the observer plays a crucial role, and the process of measurement disturbs the system in a way that introduces uncertainty.
This principle has profound implications for our understanding of the quantum world. It suggests that at a fundamental level, nature is probabilistic rather than deterministic. The Uncertainty Principle also highlights the limitations of classical mechanics in describing phenomena at atomic and subatomic scales. It underscores the need for a new framework—quantum mechanics—to accurately describe the behavior of particles at these scales. The principle has been experimentally verified and is a cornerstone of modern quantum theory, influencing various fields such as quantum computing, quantum cryptography, and the study of quantum systems.
5 Section 1.6: The Wave Function
In quantum mechanics, the wave function $ (x,t) $ is a fundamental concept that encapsulates the state of a quantum system. The wave function is a complex-valued function of space and time, and it contains all the information about the system’s physical properties. The Schrödinger equation, a key equation in quantum mechanics, governs the evolution of the wave function over time. This equation is analogous to Newton’s laws in classical mechanics, providing a deterministic description of how quantum states evolve.
The wave function’s magnitude squared, $ |(x,t)|^2 $, represents the probability density of finding a particle at a given position $ x $ and time $ t $. This probabilistic interpretation is a departure from classical mechanics, where particles have definite positions and velocities. The normalization condition, $ |(x,t)|^2 dx = 1 $, ensures that the total probability of finding the particle somewhere in space is equal to one. This condition is crucial for the consistency of the probabilistic interpretation of the wave function.
Upon measurement, the wave function undergoes a process known as wave function collapse. Before measurement, the wave function can describe a superposition of multiple states, representing various possible outcomes. However, when a measurement is made, the wave function collapses to a single eigenstate corresponding to the observed outcome. This collapse is instantaneous and non-deterministic, reflecting the inherent uncertainty and probabilistic nature of quantum mechanics.
One of the most famous thought experiments illustrating the peculiarities of the wave function is Schrödinger’s cat paradox. In this scenario, a cat is placed in a sealed box with a radioactive atom, a Geiger counter, and a vial of poison. If the atom decays, the Geiger counter triggers the release of the poison, killing the cat. According to quantum mechanics, until the box is opened and an observation is made, the cat is in a superposition of being both alive and dead. This paradox highlights the counterintuitive nature of quantum superposition and the role of the observer in determining the outcome of a quantum system.
6 Section 1.7: The Bohr’s Hydrogen Atom Model
In 1913, Niels Bohr proposed a revolutionary model for the hydrogen atom that addressed the limitations of classical physics in explaining atomic structure and spectral lines. Bohr’s model introduced the concept of quantized energy levels, where electrons orbit the nucleus in discrete, stable orbits without radiating energy. This was a significant departure from classical electrodynamics, which predicted that electrons in orbit should continuously emit radiation and spiral into the nucleus.
According to Bohr’s model, electrons can only occupy certain allowed orbits, each corresponding to a specific energy level. The energy levels are quantized and are given by the formula $ E_n = - $, where $ E_n $ is the energy of the nth level, and $ n $ is a positive integer known as the principal quantum number. The negative sign indicates that the energy is lower (more negative) for orbits closer to the nucleus, reflecting the bound state of the electron.
Electrons can transition between these energy levels by absorbing or emitting photons with energy equal to the difference between the initial and final energy levels. This process explains the discrete spectral lines observed in the emission and absorption spectra of hydrogen. When an electron absorbs a photon, it gains energy and moves to a higher energy level (excitation). Conversely, when an electron falls to a lower energy level, it emits a photon with energy corresponding to the difference between the two levels (de-excitation).
Bohr’s model successfully explained the Rydberg formula for the spectral lines of hydrogen and provided a foundation for the development of quantum mechanics. Despite its success, the model had limitations, such as its inability to accurately describe atoms with more than one electron and its reliance on classical concepts. Nevertheless, Bohr’s hydrogen atom model was a key step in the evolution of atomic theory and remains a fundamental concept in the study of quantum mechanics.
7 Section 1.8: Quantum Numbers
In quantum mechanics, the state of an electron in an atom is described by a set of four quantum numbers. These quantum numbers arise from the solutions to the Schrödinger equation for the hydrogen atom and are essential for understanding the arrangement of electrons in atoms and the resulting chemical properties.
Principal Quantum Number ($ n $): The principal quantum number $ n $ determines the energy level of an electron in an atom. It is a positive integer (1, 2, 3, …) and indicates the relative size and energy of atomic orbitals. Higher values of $ n $ correspond to orbitals that are larger and have higher energy. The principal quantum number is crucial in defining the electron shell in which an electron resides.
Orbital Quantum Number ($ l $): The orbital quantum number $ l $, also known as the azimuthal or angular momentum quantum number, defines the shape of the electron’s orbital. It can take on integer values from 0 to $ n-1 $. Each value of $ l $ corresponds to a specific type of orbital: $ l = 0 $ for s-orbitals, $ l = 1 $ for p-orbitals, $ l = 2 $ for d-orbitals, and $ l = 3 $ for f-orbitals. The shape of the orbital influences the electron’s distribution around the nucleus.
Magnetic Quantum Number ($ m $): The magnetic quantum number $ m $ describes the orientation of the orbital in space relative to an external magnetic field. It can take on integer values ranging from \(-l\) to \(+l\), including zero. For example, if $ l = 1 $, $ m $ can be -1, 0, or +1, corresponding to the three possible orientations of a p-orbital. The magnetic quantum number is essential for understanding the spatial orientation of orbitals and their behavior in magnetic fields.
Spin Quantum Number ($ s $): The spin quantum number $ s $ specifies the intrinsic angular momentum (or spin) of an electron. It can take on one of two possible values: $ + $ or $ - $. The spin quantum number is fundamental to the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of all four quantum numbers. This principle explains the unique arrangement of electrons in atomic orbitals and the structure of the periodic table.
The combination of these four quantum numbers uniquely identifies the quantum state of an electron in an atom. The Pauli Exclusion Principle, formulated by Wolfgang Pauli in 1925, is a key principle in quantum mechanics that has profound implications for the structure of atoms and the behavior of electrons in solids. It ensures that electrons occupy distinct quantum states, leading to the complex electron configurations observed in multi-electron atoms and the resulting chemical properties of elements.
7.1 1.9 Quantum Tunneling
Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential barrier that it classically shouldn’t be able to surmount. This effect arises due to the wave-like nature of particles in quantum mechanics, as described by the Schrödinger equation. When a particle encounters a potential barrier, there is a non-zero probability that it will tunnel through the barrier, even if its energy is less than the height of the barrier. This phenomenon cannot be explained by classical physics, which dictates that a particle must have sufficient energy to overcome a barrier. Quantum tunneling is a direct consequence of the probabilistic nature of quantum mechanics and the principle of superposition, which allows particles to exist in multiple states simultaneously.
One of the most well-known examples of quantum tunneling is electron tunneling in semiconductors. In semiconductor devices such as tunnel diodes and transistors, electrons can tunnel through thin insulating barriers, allowing for the flow of current even when classical physics would predict no conduction. This tunneling effect is crucial for the operation of many modern electronic devices, enabling the miniaturization and increased performance of integrated circuits. The ability of electrons to tunnel through barriers also plays a significant role in the behavior of superconductors, where Cooper pairs of electrons can tunnel through insulating layers between superconducting materials, leading to phenomena such as the Josephson effect.
Quantum tunneling is also the basis for the operation of scanning tunneling microscopes (STMs), which are powerful tools for imaging surfaces at the atomic level. In an STM, a sharp metal tip is brought very close to the surface of a conductive sample. When a voltage is applied between the tip and the sample, electrons can tunnel through the vacuum gap, creating a measurable tunneling current. By scanning the tip across the surface and measuring the tunneling current, researchers can obtain detailed images of the surface’s atomic structure. This technique has provided a unique access to the nanoscopic world, allowing for the direct observation and manipulation of individual atoms and molecules.
8 Section 2: Quantum Entanglement
8.1 2. Quantum Entanglement
Two particles can become linked regardless of distance.
Measurement on one affects the other instantly.
Basis for quantum communication and cryptography.
9 Section 3: Quantum Computing
9.1 3. Quantum Computing
Qubits instead of classical bits.
Superposition and entanglement enable parallel computation.
Potential applications in cryptography, optimization, and simulation.
Note
This introduction has been adapted from Borja’s bibliography: references.bib PhD thesis [@requena2024].
10 What is machine learning?
With the progress of technology, we encounter increasingly more complex and abstract problems that are hard to formalize mathematically. For instance, the problem of face recognition in images cannot be easily presented in a formal mathematical way, or tasks such as detecting new phases of matter may not even have a known mathematical formulation. Hence, these types of problems cannot be effectively addressed using standard hard-coded algorithms.
The field of machine learning (ML) emerges as a new paradigm to develop algorithms that are not explicitly programmed, but learned from experience instead, typically in the form of data. Figure 1 provides a visual comparison between traditional programming and ML. In traditional programming, an input task and a program to solve it yield the desired result after some computation. In ML, an input task and related data instances result in a program to solve the task. For example, the data can be pairs of problem examples and their solutions.
This process heavily relies on applied statistics, emphasizing the use of computers to approximate complex functions. Deep learning [@goodfellow:2016], a subfield of ML, is the maximum exponent of this trend. It employs parametrized hierarchical models to extract intricate patterns from data, achieving outstanding results across various tasks.
Figure 1: Traditional programming vs machine learning
Code
import plotly.graph_objects as goimport numpy as npimport plotly.io as piopio.renderers.default ="plotly_mimetype+notebook_connected"# Create figurefig = go.Figure()# Add traces, one for each slider stepfor step in np.arange(0, 5, 0.1): fig.add_trace( go.Scatter( visible=False, line=dict(color="#00CED1", width=6), name="𝜈 = "+str(step), x=np.arange(0, 10, 0.01), y=np.sin(step * np.arange(0, 10, 0.01))))# Make 10th trace visiblefig.data[10].visible =True# Create and add slidersteps = []for i inrange(len(fig.data)): step =dict( method="update", args=[{"visible": [False] *len(fig.data)}, {"title": "Slider switched to step: "+str(i)}], # layout attribute ) step["args"][0]["visible"][i] =True# Toggle i'th trace to "visible" steps.append(step)sliders = [dict( active=10, currentvalue={"prefix": "Frequency: "}, pad={"t": 50}, steps=steps)]fig.update_layout( sliders=sliders)fig.show()
This circuit performs the following operations: 1. A CNOT gate is applied with the first qubit as the control and the second qubit as the target. 2. A Hadamard gate is applied to the first qubit. 3. Both qubits are measured in the computational basis.
This circuit performs the following operations: 1. A CNOT gate is applied with the first qubit as the control and the second qubit as the target. 2. A Hadamard gate is applied to the first qubit. 3. Both qubits are measured in the computational basis.
Hola \(\sqrt{2}\) y \(\text{\LaTex}\)
Just say hello!
This could be a good example or inlined :
%Here ends the furst plot %Here begins the 3d plot
