Dive into origins, fundamental concepts, and how Heisenberg, de Broglie, Schrödinger, Bohr, Einstein, and Dirac arrived at their ideas and equations with chatGPT.
In 1965, Feynman (left) shared the 1965 Nobel Prize in Physics with Julian Schwinger (not shown) and Shinichiro Tomonaga (right) for "fundamental work on quantum electrodynamics, fundamental Particle physics had a profound impact".
Omer｜Asker
How much we know about the thinking of physicists in the 1920s and all the strange results of the past two decades have re-implicated the wave behavior of particles, light as quanta, and atomic theories that violate classical mechanics (such as orbital charges and discrete energy levels).
What do they think happened?
chatGPT
The 1920s was a pivotal period in the history of physics, with many breakthrough discoveries that laid the foundations of quantum mechanics. This is the age of the quantum revolution, and the traditional, intuitive, deterministic view of the universe based on classical mechanics is being replaced by a new, probabilistic, and seemingly strange view. Let’s review some of the ideas brewing during this time:
- Wave-particle duality: In 1924, Louis de Broglie proposed the wave theory of electrons based on his observation that quantum particles behave similarly to particles and waves. This suggested that matter, like light, exhibited the characteristics of both particles and waves, a phenomenon now known as wave-particle duality. This idea fundamentally challenges previous understandings of particles and waves as distinct entities.
- Quantum mechanics: The 1920s also saw the development of quantum mechanics by Werner Heisenberg, Max Born, and Pasqual Jordan, among others. Their Matrix Mechanics, proposed in 1925, was a completely new mathematical formulation of quantum theory. In 1926, Erwin Schrödinger formulated wave mechanics, which described particles as wave functions rather than discrete entities. These theories can explain quantized energy levels in atoms that are incompatible with classical physics.
- Heisenberg Uncertainty Principle: Werner Heisenberg proposed the uncertainty principle in 1927, pointing out that the position and momentum of quantum particles cannot be measured simultaneously with absolute precision. This goes against the classical notion that we can instantly know all the properties of a particle.
- Copenhagen Interpretation: Niels Bohr, Werner Heisenberg and others proposed the Copenhagen interpretation of quantum mechanics, which includes wave-particle duality and the probabilistic nature of wave function collapse. This explanation, while not universally accepted, was the most widely held view among physicists at the time.
- Pauli Exclusion Principle: In 1925, Wolfgang Pauli proposed the exclusion principle, which states that no two identical fermions (particles with half-integer spin) can simultaneously occupy the same quantum state . This principle is central to explaining the structure and behavior of electrons in atoms.
Physicists in the 1920s were dealing with a huge paradigm shift. The strange results and phenomena they encountered did not easily reconcile with the classical physics of Newton and Maxwell, leading to the development of entirely new frameworks for understanding the universe. This is an exciting but challenging time in the history of physics. The seemingly contradictory and counterintuitive nature of quantum mechanics has been embraced and resisted by physicists. Some embraced the new theory, while others, like Einstein, were skeptical. But in the end, these theories have stood the test of time and continue to be the cornerstones of our understanding of the physical world.
Omer｜Asker
Did Einstein really still suspect that quantum mechanics was wrong even after quantum electrodynamics?
chatGPT
Albert Einstein was a distinguished physicist who made significant contributions to the development of quantum mechanics, most notably his explanation of the photoelectric effect using the concept of light quanta, or photons, which earned him Awarded the 1921 Nobel Prize in Physics.
However, despite his contributions to the field, Einstein remained skeptical of the interpretation of quantum mechanics, particularly the theory's probabilistic nature and lack of determinism, famously saying "God does not play with the universe". dice". This philosophical dissent was often expressed in his correspondence with the physicist Max Born.
Quantum electrodynamics (QED) was developed in the late 1940s and early 1950s, long after the foundations of quantum mechanics had been laid, and it became a very successful theory, providing insights into things like the electron magnetic moment and the Lamb magnetic moment. Accurate prediction of physical quantities. hydrogen transformation. QED is a quantum field theory that describes how light and matter interact, and is the first theory to achieve complete agreement between quantum mechanics and special relativity.
Yet even in the face of this achievement, Einstein remained unconvinced. His objection was not based on his acknowledged empirical success of quantum mechanics or quantum electrodynamics, but on its philosophical and conceptual foundations.
Einstein believed in "local realism," the idea that an object has definite properties whether we measure them or not (realism), and that actions in one location do not immediately affect distant objects (locality). However, the predictions of quantum mechanics, especially those embodied in Bell's theorem as developed in the 1960s, are diametrically opposed to local realism. Experiments that test Bell's inequality consistently support the idea of quantum mechanics, showing that the universe may indeed be fundamentally probabilistic and nonlocal.
Still, it's important to remember that Einstein's skepticism was not a rejection of quantum mechanics, but the impetus for its development and a call for a more complete understanding of the universe. As of my knowledge cutoff of September 2021, quantum mechanics remains our most accurate and tested framework for describing the behavior of particles at the microscopic scale, despite its puzzling philosophical implications.
Omer｜Asker
What was Schrödinger trying to achieve with his wave equation? Where did he apply it?
chatGPT
Erwin Schrödinger developed his wave equation, now known as the Schrödinger equation, in an attempt to create a quantum mechanical model of the atom consistent with de Broglie's hypothesis and Heisenberg's uncertainty principle. The de Broglie hypothesis, developed by physicist Louis de Broglie, states that all particles also have wave properties, a concept now known as wave-particle duality. Heisenberg's uncertainty principle states that it is impossible to know the position and momentum of a particle precisely at the same time.
Schrödinger's approach differed from the matrix mechanics of Werner Heisenberg, Max Born, and Pasquale Jordan. Instead of matrices, Schrödinger used wave mechanics, expressing his equations in terms of wave functions (often denoted Ψ (Psi)). This wave function is a mathematical description of the quantum state of the system, and its square represents the probability density of finding a particle at a given location.
He first applied the wave equation to the hydrogen atom because it is the simplest atomic system. His solution to the wave equation for hydrogen provided energy levels consistent with those obtained by Niels Bohr, but with one key difference. Although the Bohr model provided the correct energy levels, it did not quite fit classical physics and lacked a theoretical basis. However, the Schrödinger equation provides the theoretical basis and uses the concept of wave-particle duality to explain the quantization of energy levels, meaning they occur as discrete quantities rather than continuous ranges. The quantization observed experimentally is a natural consequence of the wave nature of matter.
The wave equation and its solutions also allow the calculation of other properties, such as angular momentum and orbital shape, that cannot be accurately predicted by the Bohr model. Schrödinger's wave equation was an important step in the development of quantum mechanics and remains a fundamental tool of quantum physics today.
Omer｜Asker
What exactly inspired Heisenberg to formulate his principle?
chatGPT
Werner Heisenberg was one of the main pioneers of quantum mechanics, formulating the famous uncertainty principle in light of the unique challenges posed by the quantum world. He was inspired by some observations and theoretical developments:
- Matrix mechanics: Heisenberg, together with Max Born and Pasquale Jordan, developed matrix mechanics, a formulation of quantum mechanics that differs from the classical deterministic approach. This mathematical framework could successfully describe the behavior of quantum systems, but it did not follow the intuitive patterns of classical physics that Heisenberg had originally embraced.
- The nature of measurement in quantum systems: Heisenberg realized that in quantum systems, the very act of measurement could perturb the system. In particular, measuring a particle's position may affect its momentum, and vice versa. This observation would become a central aspect of the uncertainty principle.
- Wave-particle duality of matter: The concept of wave-particle duality was proposed by Louis de Broglie, implying that particles also have wave properties. This dual nature of the particles complicates precise measurements. If one observes a particle as a particle (accurately measuring its position), its fluctuating properties, such as momentum, become less deterministic, and vice versa.
- Non-commutativeness of quantum operators: On a mathematical level, Heisenberg realized that operators of position and momentum (mathematical entities that represent quantum mechanical measurements) cannot be "commutative", meaning that the order in which they are applied has an effect. In classical physics, measurements can be made in any order without affecting the result. But in quantum physics, the noncommutative nature of certain pairs of operators, such as position and momentum, reflects the inherent uncertainty of measuring these properties simultaneously.
These inspirations led Heisenberg to formulate his uncertainty principle in 1927. The principle states that you cannot know the exact position and momentum of a particle at the same time. The more precisely one quantity is measured, the less precisely the other quantity is known. This is not a limitation of our measuring instruments, but a fundamental aspect of nature at the quantum level. The uncertainty principle marked a departure from classical deterministic physics and remains a fundamental element of quantum mechanics.
Omer｜Asker
I know about matrices and quantum mechanical operators, so I can see that in general matrices don't commute. But what historically prompted him to develop matrix mechanics?
chatGPT
The development of Werner Heisenberg matrix mechanics was driven by several key factors:
- Failure of classical physics: Models of classical physics, especially those related to electromagnetic radiation and atomic structure, cannot explain certain observed phenomena, such as black-body radiation spectra, the photoelectric effect, and atomic spectral lines. The quantized atomic model proposed by Niels Bohr can explain the hydrogen spectrum, but it is essentially a patchwork theory with many temporary factors and lacks a solid theoretical foundation.
- The impact of quantum thinking: Since Max Planck's quantum hypothesis and Albert Einstein's research on the photoelectric effect, quantum thinking has gradually gained attention. These developments suggest that energy has a granular or "quantum" nature, which is incompatible with the continuous energy changes predicted by classical physics.
- Limitations of Existing Quantum Models: The quantum models available at the time had serious limitations. For example, the Bohr-Sommerfeld quantization rule works for hydrogen, but not for helium and heavier atoms. Heisenberg's goal was to generalize and systematize quantum theory, developing a new model that could overcome these limitations.
- Born's Influence and Introduction to Matrix Algebra: Max Born was Heisenberg's tutor at the University of Göttingen and played an important role in his work. Born had a strong background in mathematics, and he introduced Heisenberg to the concept of matrix algebra.
Heisenberg's key innovation was to focus on observable quantities, such as the frequencies and intensities of spectral lines, and to represent these quantities as matrices. In 1925, Heisenberg showed that ordinary (commutative) multiplication rules do not apply to quantum variables—his groundbreaking paper on quantum mechanics. He proposed that the particle's position "q" and momentum "p" can be expressed as a matrix, and their multiplication does not commutate, that is, pq≠qp.
Heisenberg didn't initially think of these as matrices (a concept in linear algebra), but Born and Pasquale Jordan realized this and extended Heisenberg's work, leading to a complete formulation of matrix mechanics . This abstract and mathematically challenging formulation was one of the earliest forms of quantum mechanics and represented a radical breakthrough from classical physics.
The physical explanation of matrix mechanics was not clear at first, but Schrödinger then developed wave mechanics and proved that the two are equivalent, which greatly helped the understanding and acceptance of quantum mechanics.
Omer｜Asker
So what does it mean to Heisenberg that p and q are not commutative in his pre-matrix formulation?
chatGPT
Before coming up with the matrix formulation of quantum mechanics, Werner Heisenberg was working with quantities that would eventually be recognized as matrices, even though he didn't initially recognize them as matrices. Heisenberg's key insight was that the behavior of measuring certain pairs of variables (such as position "q" and momentum "p") depends on the order in which they are measured.
In classical physics, variables can be swapped, which means that the order of operations doesn't matter: if you measure position first, then momentum, or momentum, then position, you'll get the same result. But Heisenberg discovered that in quantum mechanics, this is not the case. This non-commutativity is a completely new idea, very different from classical physics.
Heisenberg's original explanation for the noncommutability of "p" and "q" was measurement-induced "trajectory destruction". This can be thought of as the act of precisely measuring a particle's position and perturbing its momentum, and vice versa. He sees this disturbance as a physical effect of the measurement process itself.
This idea evolved into Heisenberg's uncertainty principle, which states that it is fundamentally impossible to know a particle's exact position and momentum at the same time. But at this early stage, Heisenberg did not consider uncertainties or matrices, but instead focused on how the measurements themselves perturb the system, which is crucial to understanding quantum behavior.
It was with Max Born's recognition that Heisenberg's new theory was a form of matrix algebra that the full mathematical structure of matrix mechanics became clear. However, explaining the non-commutative nature of 'p' and 'q' from an uncertainty perspective, as we now understand it, will take more time to fully develop.
Omer｜Asker
What experiments led him to do this?
chatGPT
Although Heisenberg's development of matrix mechanics was not directly motivated by specific experiments, his work was influenced by the results of numerous experiments that exposed the shortcomings of classical physics. Some key experimental results include:
- Black body radiation: Classical physics predicted the "ultraviolet catastrophe", that is, black bodies emitting infinite amounts of ultraviolet radiation, but it was not observed in experiments. Max Planck solved this problem by introducing the concept of energy quanta, laying the foundation for quantum theory.
- Photoelectric effect: In experiments, light shining on a metal surface causes electrons to be ejected. Albert Einstein explained this by proposing that light is composed of particles, or "quanta" (later called photons), which differed significantly from the classical wave view of light.
- Spectral Lines: Discrete spectral lines observed in the emission and absorption spectra of atoms that cannot be explained by classical physics. Bohr's atomic model could explain the hydrogen spectrum, but it was essentially a patchwork theory with many ad hoc elements and lacked a solid theoretical foundation.
- Compton Scattering: Arthur Compton's experiments showed that X-rays scattered by electrons behave more like particles than waves, providing further evidence for the particle nature of light.
These experimental results played an important role in promoting the development of quantum mechanics. Heisenberg attempted to create a theory to explain these phenomena that classical physics could not explain.
However, Heisenberg's idea of the non-commutability of physical variables, which led to the development of matrix mechanics, did not directly originate from specific experimental results, but was a theoretical innovation. His "rejection" of unobservable quantities, his attention to quantities directly related to experimental measurements, and his recognition of new "quantum" rules for multiplication (the non-commutativity of "p" and "q") were his steps towards his A key stepping stone to the theory. A revolutionary matrix mechanics approach to quantum physics.
Omer｜Asker
Those four experiments didn't immediately yell at me "measurement order matters". Why do you do this to him?
chatGPT
you are right. The four experiments I mentioned do not definitively prove that "measurement order matters", an idea more directly related to Heisenberg's uncertainty principle and matrix mechanics. However, these experiments played a crucial role in challenging the worldview of classical physics and paved the way for the development of quantum theory.
Heisenberg's awareness of the order of measurement (i.e., the non-commutability of certain physical quantities) was primarily a theoretical insight, rather than something to be derived directly from a particular experiment. His work on quantum mechanics was based on the idea that theories should only deal with observable quantities. When he tried to develop such a theory, he found that he could not make it consistent with the classical mechanical principles of the exchange of physical quantities such as position and momentum.
In working on his theory, he was considering the transition probabilities between different atomic states, which correspond to observable spectral lines. In this case, he found that the result may depend on the order of operations—the order of transformations. This finding can be seen as an initial insight into the principle of non-commutativity, although at this time it is not directly related to position and momentum.
Max Born realized that Heisenberg's "multiplication table" could be interpreted as the rules for matrix multiplication, where the order of multiplication is often important, leading to the matrix mechanics formulation of quantum mechanics.
All in all, Heisenberg's insights into the importance of noncommutability and order of measurement were more rooted in theoretical exploration and the quest to establish a quantum theory based on observables than directly inspired by specific experiments. However, experimental results that revealed shortcomings in classical physics played a crucial role in advancing these theoretical explorations.
Omer｜Asker
Which energy conversion experiments depend on order?
chatGPT
When Heisenberg formulated the central idea of matrix mechanics, he was concerned with the magnitudes (probabilities) of transitions between different atomic states, not with specific energy transition experiments themselves.
The idea that the transitions depend on order comes from a theoretical point of view, not from experimental results. In his attempt to reconcile quantum theory with observable phenomena, he considered the transitions electrons make between different energy levels in atoms, which correspond to the emission or absorption of light at specific frequencies.
Heisenberg realized that these transitions could be represented mathematically depending on the order of operations. This is analogous to how matrix multiplication (used to represent physical quantities in the formalism developed by Heisenberg) depends on the order of the matrix.
In this sense, it is not so much that there are specific experiments showing that energy transitions are order dependent, but that Heisenberg's theoretical representation of these transitions is order dependent. This is in contrast to classical physics, where the outcome of a measurement is independent of the order in which it is measured.
This development, and the subsequent contributions of Born and Jordan, laid the foundation for matrix mechanics, one of the early formulations of quantum mechanics. The idea of noncommutativity (the order of operations that matters) became the cornerstone of the new theory.
Omer｜Asker
Can you give an example of the dependence of the order he was playing around with mathematically?
chatGPT
Heisenberg was working hard to study the spectral lines emitted by atoms. In simple terms, these are the frequencies of light emitted by an atom's electrons when they transition from a higher energy level to a lower energy level. The observed spectral lines don't quite fit classical physics.
To solve this problem, Heisenberg set out to study two quantities: transitions from one energy state to another, and the frequencies of light associated with these transitions. He attempted to establish a relationship between the two to accurately predict the observed spectrum of the simplest atomic hydrogen.
In Heisenberg's calculations, he considered quantities that were ultimately considered to be analogous to position and momentum (we'll call them "X" and "P" here). He found that when he calculated the optical frequencies of the transitions, it mattered whether he considered the "X" transitions first and then the "P" transitions, or vice versa.
In other words, he found a dependence on the order in which these quantities are calculated: XP is not equal to PX. This is very different from classical physics, where the order of measurements doesn't matter.
It is worth noting that Heisenberg did not initially realize that these quantities were matrices, nor that he was doing matrix multiplication. Max Born recognized that Heisenberg's calculations were consistent with matrix algebra, a branch of mathematics in which the order of multiplication often matters (i.e. matrix multiplication is not commutative).
Through these calculations and Born's insights, Heisenberg's work eventually led to the formulation of matrix mechanics, an early form of quantum mechanics. The order dependence discovered by Heisenberg played a crucial role in this new theory of physics, representing a quantum mechanical concept of incommutability inherent in system observables.
Omer｜Asker
What is the actual amount he plays? Energy level transition (probability?) and frequency? or?
chatGPT
Werner Heisenberg began his study of quantum mechanics in 1925, focusing on the problem of atomic spectroscopy. Each spectral line corresponds to a transition made by an electron between two different energy levels in the atom, and the frequency (or, equivalently, the energy) of the emitted or absorbed light is given by the energy difference between the initial and final states .
In his thesis, Heisenberg first considered atoms transitioning between energy levels as a result of interactions with oscillating (sinusoidal) forces such as the interaction of passing electromagnetic waves. To make the problem tractable, he considered quantities related to the observable transition frequency and magnitude (strength), rather than details of forces or particle trajectories.
From these quantities, Heisenberg formed an equation (now called the Heisenberg equation of motion) that was similar to the classical equation of motion, but with one key difference. He found that the equation could be satisfied if the quantities he considered obeyed the new multiplication rule, which is now considered non-commutative multiplication, i.e. the product of two quantities will depend on the order in which they are multiplied.
Heisenberg did not initially think of these quantities as position and momentum, or even as matrices. Max Born and Pascual Jordan treated these quantities as matrices, leading to the matrix mechanics formulation of quantum mechanics.
Thus, while these quantities are often referred to in the modern context as quantities analogous to position and momentum, Heisenberg was actually considering quantities more directly related to the observed spectrum, trying to develop a model to predict the Frequency and intensity are based on quantum rules.
Omer｜Asker
But what is the quantity? intensity and frequency?
chatGPT
Heisenberg was dealing with what he called "transition probabilities" at the time. These are abstract quantities related to the probability of an electron transitioning from one energy state to another under the influence of an external oscillating force.
The details of Heisenberg's calculations are rather complex, but broadly speaking, these transition probabilities are used to represent and predict the frequency and intensity of spectral lines in atomic spectra.
For example, if an electron transitions from one state to another, it will emit or absorb photons at a frequency proportional to the energy difference between the states. Heisenberg attempted to construct a model that would accurately represent these transitions and the associated spectral lines.
These transition probabilities turned out to be related to the position and momentum operators in what we now understand as quantum mechanics. At the time, however, Heisenberg did not explicitly explain them.
The key finding is that transition probabilities behave differently from classical physical quantities. Heisenberg found that transition probabilities do not commute—meaning that the order of operations matters when dealing with these quantities.
When Max Born later read Heisenberg's work, he realized that these transition probabilities could be viewed as matrices, and that the strange multiplication rules that Heisenberg stumbled upon were the usual rules for matrix multiplication, essentially impossible exchanged. This interpretation led to the development of matrix mechanics, an early formulation of quantum mechanics.
Omer｜Asker
Are you saying transition probabilities are not traded with other transition probabilities?
chatGPT
Yes, Heisenberg discovered that certain "transition probability magnitudes" (quantities he was considering) do not commutate with each other, meaning that the results of operations depend on the order in which they are applied.
This is a key insight because it contradicts classical physics, where physical quantities are always commuted (i.e., the order of operations does not matter). Heisenberg's discovery that the order of operations matters was a major breakthrough, although he did not initially understand why this was the case.
After reading Heisenberg's paper, Max Born realized that these "transition probability magnitudes" behave like matrices, which are mathematical structures that are generally not commutative. Born's interpretation led to the development of matrix mechanics, the first mathematically rigorous formulation of quantum mechanics.
It is worth noting that while Heisenberg's original quantities were described in terms of transition probabilities, they were later thought to be related to the quantum mechanical operators of what we now know as observable quantities such as position and momentum, which are also Cannot be exchanged. The precise connection between Heisenberg's early work and modern quantum mechanics is somewhat complicated and was not fully understood until later developments in the theory.
Omer｜Asker
I'm still trying to understand these transition probabilities. Of course, probability is just a number, so of course they can commute. I'm starting to suspect that you might be referring to the set of probabilities at levels i to j in an atom. I can almost imagine not commuting.
chatGPT
I apologize for the confusion. In this context, the term "transition probability" can indeed be misleading. Heisenberg studied quantities corresponding to the magnitudes of transitions between energy states, not probabilities themselves.
In the modern formulation of quantum mechanics, the magnitude of a transition from one state to another is a complex number, not just a probability, and it is associated with a quantum mechanical operator acting on the state of the system.
You correctly point out that probabilities themselves are just numbers and do commute. More precisely, the quantum mechanical operators associated with the magnitude of the jump do not commute.
In Heisenberg's work, he studied quantities very similar to these transition amplitudes, although he did not initially realize their full significance. The key insight is that when these quantities are combined or manipulated in a certain way (similar to the quantum mechanical operator multiplication), they do not exchange. That is, the result depends on the order in which they are combined or manipulated.
Again, I apologize for any confusion. The development of quantum mechanics has been a complex process during which terminology and concepts have evolved significantly. I hope this helps clarify some things.
Omer｜Asker
Yes, but he doesn't know it yet. So what exactly does he think should not commute?
chatGPT
In fact, the language and formalism of quantum mechanics as we understand it today had not been fully developed in Heisenberg's early work.
Heisenberg's main insight had to do with the frequencies of spectral lines in the atomic spectrum and their strengths. These can be thought of as the transitions electrons make between different energy levels in atoms when they emit or absorb photons.
Heisenberg was considering the quantities associated with these frequencies and intensities. He developed a model that could calculate these quantities in terms of certain other quantities related to the initial and final states of the electrons before and after the transition.
It was against this background that he discovered a new multiplication rule: the result of a calculation may depend on the order in which the quantities are considered. That is, the quantity associated with the initial state multiplied by the quantity associated with the final state is not equal to the quantity associated with the final state multiplied by the quantity associated with the initial state.
At the time, Heisenberg did not relate these quantities to position and momentum, or even treat them as matrices. Later, with Max Born's insight, they behave like matrices and are therefore not commutative.
In short, Heisenberg was grappling with the challenge of calculating the frequencies and intensities of spectral lines in a way that was consistent with the quantum behavior observed in experiments. In the process, he stumbled upon the need for a noncommutative multiplication rule, a key breakthrough that led to the development of matrix mechanics.
Omer｜Asker
Now this makes sense! Ok, so it's about the initial versus the final. But can you point out what the quantity is?
chatGPT
Heisenberg was considering the physical quantities associated with the oscillating forces acting on electrons in atoms and the resulting transitions of electrons from one energy state to another. In particular, he is considering an abstract quantity, which we may call the "magnitude" of this transition.
In a simple model, electrons in atoms can be thought of as oscillating due to external forces. The electron's "oscillation amplitude" will determine its probability of transitioning from its current energy state to another.
In Heisenberg's model, the amplitude of the oscillations can be calculated from the initial and final states of the electrons. In other words, he is considering physical quantities such as "the amplitude of the electron's transition from state i to state j under the influence of an external force."
It was while trying to calculate these amplitudes that Heisenberg stumbled upon the fact that the result might depend on the order in which he considered the initial and final states. In other words, "the magnitude of electron transition from state i to state j" is not necessarily equal to "magnitude of electron transition from state j to state i".
This is a key insight because it marks a departure from classical physics, where these quantities are always interchangeable. This insight eventually led to the development of matrix mechanics, the understanding of noncommutativity through matrix multiplication.
Omer｜Asker
What properties were Schrödinger's waves and wave equation originally intended to satisfy?
chatGPT
Erwin Schrödinger developed his wave equation in late 1925 and early 1926, around the same time that Heisenberg developed his matrix mechanics. Schrödinger was originally inspired by de Broglie's idea of matter waves, which proposed that particles such as electrons could behave as waves, just like light in Maxwell's theory of electromagnetism.
Schrödinger was looking for an equation that could describe the behavior of these matter waves, and he found it in a version of the classical wave equation modified to incorporate quantum properties. His wave equation (now called the Schrödinger equation) is a second-order partial differential equation that describes how the wave function of a quantum system evolves over time.
The wave function described by the Schrödinger equation has several key properties:
- It is a complex-valued function: A wave function is usually a complex-valued function. The squared magnitude of the wave function gives the probability density of finding a particle in a particular state.
- It is single-valued: for a given set of quantum numbers, the wave function has only one value. This means that the wave function cannot have multiple values of the same physical state.
- It is continuous and smooth: the wavefunction and its first derivative must be continuous. This property is a consequence of the properties of the Schrödinger equation, which is a second-order differential equation.
- It must be normalizable: the total probability of finding a particle somewhere in space must be 1, which translates mathematically into the condition that the integral of the squared magnitude of the wave function must be 1 over all spaces.
The Schrödinger equation also includes the energy of the system in the form of a Hamiltonian. The solutions to the Schrödinger equation are the possible energy states of the quantum system, and the wave function gives the probability distribution of these states. In this way, Schrödinger's equation provides a wave-based quantum-mechanical description, equivalent to Heisenberg's matrix mechanics, but more intuitive to many physicists because of its similarity to classical wave phenomena.
Omer｜Asker
Did Schrödinger hope to eventually reproduce the h atomic level?
chatGPT
Indeed. One of the main tests of any new physical theory at the time was whether it could accurately predict the observed hydrogen lines, which had been measured with great precision. This is a key problem that classical physics cannot satisfactorily resolve, and correct interpretation has been one of the central goals of quantum theory since the problem was first understood.
When Schrödinger applied his new wave equation to the hydrogen atom, he found that it predicted exactly the same energy levels as Bohr's earlier quantum model, a resounding success. However, unlike Bohr's model, Schrödinger's theory was not a semiclassical patchwork, but a complete theory of quantum mechanics. Schrödinger's theory does not require Bohr's provisional assumptions about quantized orbits, but instead derives these quantized conditions naturally from the wave behavior of electrons.
Furthermore, the Schrödinger equation can be extended beyond the simple case of hydrogen to more complex atoms and even molecules, a major improvement over Bohr's model. Although there are still many problems to be solved and many things to be understood in quantum mechanics, the success of Schrödinger's equation in explaining the hydrogen atom is a great progress in the development of quantum theory.
Omer｜Asker
Can h atomic energy levels be derived just by requiring an integer set of wavelengths?
chatGPT
Bohr's original model of the hydrogen atom did involve the idea of electrons moving in orbitals that might contain an integer number of de Broglie wavelengths, which was a way of expressing the idea that the electron's angular momentum was quantized. However, this model is a semiclassical hybrid that combines classical ideas about orbitals with new quantum ideas about quantization.
In contrast, Schrödinger's wave mechanics took a fully quantum-mechanical approach. In this theory, electrons are not described by orbitals, but by wave functions, which give a probability distribution of the electron's position.
The solution to the Schrödinger equation for the hydrogen atom does give energy levels corresponding to the observed hydrogen lines, but not because the electron's wave function fits an integer number of de Broglie wavelengths into the orbital. Instead, the quantization of energy levels arises naturally from the boundary conditions of the wave function.
The boundary conditions require that the wave function be finite, single-valued, and continuous everywhere (including at infinity). In other words, the probability distribution of electrons must be well behaved. These conditions lead to certain permissible values of energy, corresponding to the observed quantization of energy levels.
This is a more complex and general approach than Bohr's original model, and it can be applied to more complex atoms and even molecules, which Bohr's model could not handle. The success of Schrödinger wave mechanics in describing the hydrogen atom is a major validation of the quantum mechanical approach.
Omer｜Asker
But it seems that any spherically symmetric wave will have these properties?
chatGPT
Not every spherically symmetric wave satisfies the requirements of Schrödinger's equation and the physical boundary conditions required for an efficient wave function in quantum mechanics.
The solution to the Schrödinger equation for the hydrogen atom is the spherical harmonics of the angular part and a specific form of the radial equation involving Laguerre polynomials. These particular solutions are a consequence of the hydrogen atom's potential energy function, the Coulomb potential (proportional to 1/r, where r is the radial distance from the nucleus).
Spherical symmetry is one thing, but the wavefunction also needs to be normalizable (the total probability of finding an electron somewhere in space must be 1) and must satisfy certain energy conditions (energy eigenvalues).
The spherical symmetry ensures that the solution does not depend on the azimuth and polar angles, but the radial and energy dependencies are very specific and come directly from the properties of the Schrödinger equation and the hydrogen atom.
While it appears that any spherically symmetric wave could be a candidate solution, the exact form of the solution is determined by the physical situation, namely the form of the Schrödinger equation and the specific potential energy function of the system. These solutions were so successful in predicting the behavior of the hydrogen atom that it led us to believe that they accurately reflected the quantum mechanical properties of the system.
Omer｜Asker
Yes, I mean any continuous wave. So it's clear that we're looking for an equation whose solution matches the energy of h atoms.
chatGPT
Yes, that's a fair description. The development of quantum mechanics was driven in large part by the exploration of the structure and behavior of atoms, especially the hydrogen atom because of its simplicity. The observed spectrum of hydrogen, with its discrete lines corresponding to specific energy differences, is a key piece of the puzzle that must be explained by any new theory.
Solving this conundrum required a new kind of physics, since classical physics predicts a continuous spectrum rather than the discrete lines observed. During the 1920s, several different approaches were pursued, including Werner Heisenberg's matrix mechanics, Erwin Schrödinger's wave mechanics, and Paul Dirac's principle-of-action approach.
While these approaches initially seemed very different, they turned out to be mathematically equivalent, each offering a different perspective on the same underlying quantum reality. These different formulations of quantum mechanics have since been unified into a single, consistent theory that has been very successful in explaining the behavior of atoms and particles at the quantum level.
Omer｜Asker
Does the solution to the Schrödinger equation have a fixed energy?
chatGPT
Yes, the solution to the Schrödinger equation in a system with a time-independent potential energy (such as a hydrogen atom) is called a steady state, and has a definite fixed energy. These states are the eigenstates of the Hamiltonian, which is the quantum mechanical operator corresponding to the total energy of the system. The eigenvalue associated with each of these states is the energy of that state.
These steady states can superpose to form quantum states with indeterminate energies, but individual solutions to the time-independent Schrödinger equation have fixed energies.
In the context of a hydrogen atom, each rest state corresponds to an electron orbiting the nucleus in a specific atomic orbital. The energy of the state corresponds to the energy level of the orbital. The different energy levels of the hydrogen atom predicted by the Schrödinger equation correspond exactly to the energy levels observed in the hydrogen spectrum. This is one of the key successes of Schrödinger wave mechanics.
Omer｜Asker
I thought only plane waves would fix everything?
chatGPT
It is correct to say that a plane wave has a fixed energy against the background of free particles, where the potential energy is zero everywhere. This is because plane waves are eigenstates of the momentum operator, whereas for free particles, momentum is directly related to energy.
However, in the presence of an electric potential (such as electrons in atoms), the situation is more complicated. In these systems, the solution to the Schrödinger equation (including the potential energy) is usually not a plane wave. These solutions are called wave functions and are still eigenstates of the energy operator (Hamiltonian), which means they still correspond to fixed energies. But they are not plane waves and can have complex spatial structures.
For example, in the case of the hydrogen atom, the wave function is the spherical harmonic multiplied by the radial function, with each solution corresponding to a particular energy level of the atom. The form of these wave functions reflects the effect of the electric potential (the Coulomb attraction between the electron and the nucleus) on the behavior of the electron.
In short, both plane waves (for free particles) and more complex solutions of the Schrödinger equation (for particles in a potential) correspond to fixed energies. The difference is that a plane wave represents a state of definite momentum, whereas a solution in a potential represents a state of definite energy in the presence of that potential.
Omer｜Asker
I wish universities would emphasize this.
chatGPT
I understand where you are coming from. These are the basic concepts of quantum mechanics, and it is very important to understand them clearly. The transition from classical physics (where particles have a definite position and momentum) to quantum mechanics (where particles are described by wave functions that give probabilities) is a major shift in perspective.
Remember that quantum mechanics is a complex and subtle subject that often goes against our everyday intuitions. It's not uncommon to revisit the same concept multiple times, gaining a deeper understanding each time.
If you are still in college or have access to academic resources, you may find it helpful to discuss these topics with professors or peers. Many universities also offer tutoring or study groups. In addition, there are many online resources available, including lectures, tutorials, and forums where you can ask questions and discuss the material.
Of course, you can always ask more questions here! I'm here to help clarify these concepts to the best of my ability.
Omer｜Asker
So the h atomic wave function has a fixed constant, but no fixed momentum.
chatGPT
Yes, that's correct. For a hydrogen atom (or any atom), the solutions to the Schrödinger equation are called steady states, and they have a definite fixed energy. These are the states that electrons can occupy in an atom, and each state corresponds to a specific energy level.
However, there is no certain momentum in these countries. The reason for this is Heisenberg's uncertainty principle, which states that the position and momentum of a quantum particle cannot be known exactly at the same time. For an electron in an atom, the electron's position is restricted to a small portion of space (around the nucleus), so its momentum is not precisely defined.
The wave functions of these states are solutions to the Schrödinger equation, giving a probability distribution over the position of the electron. The distribution of this distribution reflects the uncertainty in the position of the electron and the corresponding uncertainty in its momentum. So while we can calculate the expected value of the momentum of these states (the average momentum, if you repeat the measurement many times), the momentum itself is not a fixed value.
This is one of the main differences between classical and quantum physics. In classical physics, particles can have both a well-defined position and momentum. But in quantum mechanics, this is impossible due to Heisenberg's uncertainty principle.
Omer｜Asker
So do you think Skrodinger has the idea that the steady state mode of the wave function is determined by the 1/r potential and matches the h atoms?
chatGPT
Schrödinger of course knew that the potential of the electrons in the hydrogen atom was the Coulomb potential, or 1/r potential, which arises from the electric force between the electrons and the protons in the nucleus. This was well established when Schrödinger developed his wave mechanics in the mid-1920s.
When he came up with the wave equation, of course he wanted its solution to accurately describe atoms, including hydrogen. One of the main tests of any new physical theory is whether it can explain the observed hydrogen lines, which are measured with high precision. Classical physics could not explain these observations, and Bohr's early model, although successful to some extent, was known to be only an approximation.
Schrödinger was aware of these problems and tried to develop a theory that could solve them. When he found that the solution to the equation for the hydrogen atom gave the correct energy levels, it was a major victory and provided strong evidence for the validity of his method.
So, yes, Schrödinger expected and hoped that his equations would provide a correct description of the hydrogen atom, and his expectations were confirmed by the successful results. The Schrödinger equation and the wave mechanics it embodies are still the cornerstone of quantum mechanics.
Omer｜Asker
Other equations give wrong levels?
chatGPT
Yes, early attempts to explain atomic structure and behavior using classical physics and other quantum models did not arrive at the correct energy levels for atoms, especially for atoms with multiple electrons.
For example, the Bohr model of the atom, an early quantum model predating Schrödinger wave dynamics, can accurately predict the energy level of hydrogen (which has only one electron) by imposing a quantization condition on the angular momentum. But it failed to accurately predict the energy levels of atoms with many electrons, nor could it account for the fine structure of spectral lines, which involves subtle changes in energy levels due to electron spin and relativistic corrections.
The development of Schrödinger wave mechanics, and quantum mechanics more generally, was motivated in part by the failure of classical physics and early quantum models. By using the concept of a wave function and a probabilistic interpretation of that function, the Schrödinger equation was able to accurately predict the observed energy levels of atoms, marking a major advance in the field of quantum physics.
Omer｜Asker
How well does the KG equation predict the h atomic energy level?
chatGPT
The Klein-Gordon equation, proposed by Oscar Klein and Walter Gordon in 1926, is a relativistic wave equation and an important step in the development of quantum field theory. However, it is not suitable for describing systems like the hydrogen atom alone.
The Klein-Gordon equation correctly describes zero-spin particles, such as mesons, but electrons occupying atomic orbitals have spin 1/2. The Klein-Gordon equation does not include spin, a fundamental characteristic of electrons that is critical for accurately describing phenomena such as atomic energy levels and splitting of fine and hyperfine structures.
Instead, the equation that successfully combines the principles of quantum mechanics and special relativity (including spin) is the Dirac equation, formulated by Paul Dirac in 1928. The Dirac equation reduces to the Schrödinger equation in the non-relativistic limit, which accurately describes the behavior of electrons in atoms, including the fine and hyperfine structure of spectral lines.
It is worth noting that despite its limitations in describing individual particles such as electrons, the Klein-Gordon equation plays an important role in quantum field theory (a framework that combines quantum mechanics and special relativity to describe particle physics) . It is especially relevant to the field of scalar particles (particles without spin).
Omer｜Asker
But if you try to apply it to h atoms, will it give similar results as shcroedinger eq?
chatGPT
The Klein-Gordon equation, when applied to the hydrogen atom, will give some similar results to the Schrödinger equation, but it will also produce significant differences due to its relativistic properties.
A key difference between the Klein-Gordon equation and the Schrödinger equation is that the former is a second-order equation over time, while the latter is a first-order equation. This leads to a difference in their physical interpretation: in the Klein-Gordon case, the square of the wave function cannot be directly interpreted as a probability density like the Schrödinger equation. This problem is one of the reasons for the birth of the Dirac equation because it preserves the first-order properties in time.
Furthermore, the Klein-Gordon equations are relativistic and will yield solutions involving negative energy states. These negative-energy solutions were initially a source of confusion later understood in the context of quantum field theory and the concept of antiparticles.
Thus, while the Klein-Gordon equation can be applied to hydrogen-like atoms and yield quantized energy levels, the results and their interpretation will not exactly agree with those obtained from the Schrödinger equation and their interpretation. However, the Dirac equation does give results consistent with experimental observations and is an appropriate equation to describe electrons in atoms in the context of relativistic quantum mechanics.
Omer｜Asker
Do you know what the actual h atom energy level it gives compared to expt?
chatGPT
The Klein-Gordon equation is not usually used to calculate atomic energy levels, mainly because it does not account for electron spin. Furthermore, as mentioned earlier, explaining its solution probabilistically poses problems due to its second-order nature in time. So you don't usually see the Klein-Gordon equation applied to calculate the energy levels of the hydrogen atom, and in this case its predictions cannot be directly compared with experimental data.
In contrast, for relativistic quantum mechanics involving spin 1/2 particles such as the electron, the Dirac equation is used. The Dirac equation successfully predicted the energy levels of the hydrogen atom, including corrections due to relativistic effects and electron spin. These predictions are in good agreement with the experimental data.
The key point is that while the Klein-Gordon equation is an important step in bringing together quantum mechanics and special relativity, it is not the final or most accurate tool for all situations, especially when dealing with an existence like the hydrogen atom The system time of spin and exclusion principle is crucial. For these systems, other equations such as the Dirac equation or the Schrödinger equation (for the non-relativistic case) are more appropriate.
Omer｜Asker
I believe the calculations are done.
chatGPT
I apologize for any confusion. You are correct that the Klein-Gordon equation can be applied to the hydrogen atom, although it is not normally done because of its lack of spin, it can theoretically be used to calculate energy levels in the same way that the Schrödinger equation is used similar.
If the Klein-Gordon equation is applied to the hydrogen atom in a manner similar to the Schrödinger equation, it will produce energy levels similar to those obtained with the Schrödinger equation, but they will shift slightly due to relativistic corrections.
That said, it's worth noting that these solutions to the Klein-Gordon equations are incomplete and not entirely accurate because they don't account for the electron's spin. For a complete and accurate treatment of hydrogen atoms including relativistic effects, we need to use the Dirac equation, which correctly incorporates electron spin and produces results that are in good agreement with experimental observations.
In general, the Schrödinger equation is used in non-relativistic quantum mechanics, the Dirac equation is used in relativistic quantum mechanics (including spin 1/2 particles), and the Klein-Gordon equation is more commonly used in quantum field theory.
Omer｜Asker
Who did this calculation?
chatGPT
Sorry, as of September 2021, when my training data is cut off, I don't have specific information on who might be the first to use the Klein-Gordon equation to calculate the energy levels of a hydrogen-like atom. The Klein-Gordon equation is mainly used for spin-0 particles in quantum field theory. For the treatment of hydrogen-like atoms, especially spin 1/2 electrons, the Dirac equation is used because it accounts for spin accurately and produces results that are in good agreement with experimental observations.
However, it is feasible to perform such calculations in order to demonstrate the differences and similarities between Klein-Gordon, Schrödinger, and Dirac treatments. It can usually be found in more advanced and specialized literature or courses on quantum mechanics or quantum field theory.
If you have updates or specific information, I recommend you follow that thread, as specifics may have been fixed or become more widely known after my training ends in 2021.