theories:quantum_mechanics
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| - | Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as **[[theories:quantum_mechanics:canonical|canonical quantum mechanics]]**. The relevant mathematical stage for both formulations is [[basic_tools:hilbert_space|Hilbert space]]. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the [[equations:canonical_commutation_relations|canonical commutation relations]] | + | Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as **[[theories:quantum_mechanics:canonical|canonical quantum mechanics]]**. The relevant mathematical stage for both formulations is [[basic_tools:hilbert_space|Hilbert space]]. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the [[formulas:canonical_commutation_relations|canonical commutation relations]] |
| $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ | $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ | ||
| We usually call these two formulations the "**Heisenberg picture**" and the "**Schrödinger picture**", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"((https://en.wikipedia.org/wiki/Heisenberg_picture)). | We usually call these two formulations the "**Heisenberg picture**" and the "**Schrödinger picture**", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"((https://en.wikipedia.org/wiki/Heisenberg_picture)). | ||
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| * For an interesting alternative introduction to quantum mechanics, see http://www.scottaaronson.com/democritus/lec9.html. | * For an interesting alternative introduction to quantum mechanics, see http://www.scottaaronson.com/democritus/lec9.html. | ||
| * see also https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics | * see also https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics | ||
| + | |||
| + | <tabbox Interpretations> | ||
| + | |||
| + | There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations((At least in the standard, Hilbert space formulation)) in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them. | ||
| + | |||
| + | The thing is that experimentally outcomes stay the same no matter which interpretation we believe in((This is similar to the statement that it doesn't matter which formulation we use. But here it makes at least some difference since some scenarios can be calculated more easily in a specific formulation.)). In this sense, discussions about the interpretation of quantum mechanics are mostly a matter of taste. | ||
| + | |||
| + | Important notions regarding the interpretation of quantum mechanics are | ||
| + | |||
| + | * the EPR paradox, | ||
| + | * [[theorems:bells_theorem|Bell's theorem]], | ||
| + | * the no-clone theorem, | ||
| + | * Schrödinger's cat, | ||
| + | * the quantum Zeno paradox. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | The standard (orthodox) interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation. | ||
| + | |||
| + | According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement. | ||
| + | |||
| + | The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again. | ||
| + | |||
| + | Regarding the question, whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that | ||
| + | |||
| + | >"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan. | ||
| + | |||
| + | In contrast, hidden variable interpretations which are also called realist interpretations, state that | ||
| + | |||
| + | >“the position of the particle was never indeterminate, but was merely unknown to the experimenter.” - Bernard d'Espagnat. | ||
| + | |||
| + | A third popular interpretation is called the agnostic interpretation states that it makes no sense to ask such a question since how can we discuss anything that we can never measure. By definition, a property like momentum is undetermined until we measure it and a discussion about its value before measurement makes no sense: | ||
| + | |||
| + | >"One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about the ancient question of how many angels are able to sit on the point of a needle." - Wolfgang Pauli | ||
| + | |||
| + | An amazing discussion of the Copenhagen interpreation and how it came about can be found in Quantum Dialogue by Mara Beller. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | There are dozens of other interpretations of what quantum mechanics really means: | ||
| + | |||
| + | * Hidden-Variable Interpretations | ||
| + | * [[theories:quantum_mechanics:bohmian|Bohmian Mechanics]] | ||
| + | * Stochastic Interpretation | ||
| + | * "//It is well known that under certain restrictions, quantum mechanics can be interpreted as a Markov process for classical coordinates. Various authors have derived Schrodinger's equation within the context of classical diffusion theory. These works interpret quantum statistics or quantum indeterminacy as originating from the interaction of nonrelativistic point particles with random forces. These forces must be present even in the vacuum. This view does not conflict with present understanding of nature, since vacuum fluctuations in quantum field theory are known to exist and are well verified experimentally. The random forces experienced by particles in the stochastic interpretation may be viewed as coming about due to random fluctuations of the various fields in the vacuum.//" ((https://aip.scitation.org/doi/abs/10.1063/1.523893)) | ||
| + | * [[http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012011/meta|Review of stochastic mechanics]] by Edward Nelson | ||
| + | * Dynamical Collapse Interpretations | ||
| + | * https://plato.stanford.edu/entries/qm-collapse/ | ||
| + | * Quantum Bayesianism | ||
| + | * [[http://blogs.discovermagazine.com/cosmicvariance/files/2011/11/banks-qmblog.pdf|The Interpretation of Quantum Mechanics]] by Tom Banks | ||
| + | * Everett (many-worlds) Interpretation | ||
| + | * "//Everett's thesis on the many worlds interpretation of quantum mechanics, which is an attempt to reformulate quantum mechanics using Shannon information tools.//"((https://physics.stackexchange.com/questions/43175/uncertainty-principle-for-information)) | ||
| + | * [[http://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/|Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct]] by Sean Carroll | ||
| + | * Relational Quantum Mechanics | ||
| + | * [[https://arxiv.org/abs/quant-ph/9801057|What is quantum mechanics trying to tell us?]] by N. David Mermin | ||
| + | * [[https://arxiv.org/abs/quant-ph/9609002|Relational Quantum Mechanics]] by Carlo Rovelli | ||
| + | * Quantum Mechanics Without State Vectors | ||
| + | * [[http://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/|The Trouble with Quantum Mechanics]] by Steven Weinberg | ||
| + | * [[https://arxiv.org/abs/1405.3483|Quantum Mechanics Without State Vectors]] by Steven Weinberg | ||
| + | * The Cellular Automaton Interpretation of Quantum Mechanics | ||
| + | * [[https://arxiv.org/abs/1405.1548|The Cellular Automaton Interpretation of Quantum Mechanics]] by Gerard 't Hooft | ||
| + | * Consistent histories interpretation | ||
| + | * [[http://quantum.phys.cmu.edu/CQT/|Consistent Quantum Theory]] by Robert B. Griffiths | ||
| + | * Shut up and calculate | ||
| + | * This is basically the agnostic approach where we say that since the interpretation doesn't matter we shouldn't spent time discussing about it. | ||
| + | * [[https://arxiv.org/abs/0709.4024|Shut up and calculate]] by Max Tegmark | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | **Recommended Resources:** | ||
| + | |||
| + | * For a quick overview, see http://www.preposterousuniverse.com/blog/2014/05/29/quantum-mechanics-smackdown/ | ||
| + | * For an overview see https://arxiv.org/pdf/1703.08341.pdf | ||
| + | * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994. | ||
| + | * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer | ||
| + | * Making Sense of Quantum Mechanics by Bricmont | ||
| + | |||
| + | |||
| + | ----- | ||
| + | |||
| + | <blockquote>“If you are not confused by quantum mechanics, then you haven’t really understood it.” <cite>Niels Bohr</cite></blockquote> | ||
| + | |||
| + | <blockquote>“I think I can safely say that nobody understands quantum mechanics.” <cite>Richard Feynman</cite></blockquote> | ||
| + | |||
| + | |||
| + | |||
| <tabbox Roadmaps> | <tabbox Roadmaps> | ||
| + | |||
| + | |||
| + | |||
| -->Recommended Background# | -->Recommended Background# | ||
| Line 75: | Line 166: | ||
| <-- | <-- | ||
| - | -->Traditional Roadmap# | + | |
| + | |||
| + | -->The Traditional Roadmap# | ||
| **Basics** | **Basics** | ||
| Line 216: | Line 310: | ||
| </WRAP> | </WRAP> | ||
| + | |||
| + | |||
| </WRAP> | </WRAP> | ||
| <-- | <-- | ||
| - | <tabbox Interpretations> | + | -->Applications# |
| + | Quantum mechanics is technically difficult. Only a few extremely artificial textbook examples can be solved exactly. For everything else, we need to use approximation techniques to tackle realistic systems. | ||
| - | The standard interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation. | + | The most important approximation schemes in quantum mechanics are |
| - | According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement. | + | * time-independent perturbation theory, |
| + | * WKB approximation | ||
| + | * time-dependent perturbation theory, | ||
| + | * adiabatic approximation, | ||
| + | * semi-classical approximation. | ||
| - | The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again. | + | To describe scattering processes in quantum mechanics additional tools are needed, especially |
| - | ---- | + | * partial wave analysis and |
| + | * the Born approximation, | ||
| + | * Fermi's golden rule. | ||
| + | <-- | ||
| - | There are dozens of other interpretations of what quantum mechanics really means: | ||
| - | * Hidden-Variable Interpretations | ||
| - | * [[theories:quantum_mechanics:bohmian|Bohmian Mechanics]] | ||
| - | * Stochastic Interpretation | ||
| - | * "//It is well known that under certain restrictions, quantum mechanics can be interpreted as a Markov process for classical coordinates. Various authors have derived Schrodinger's equation within the context of classical diffusion theory. These works interpret quantum statistics or quantum indeterminacy as originating from the interaction of nonrelativistic point particles with random forces. These forces must be present even in the vacuum. This view does not conflict with present understanding of nature, since vacuum fluctuations in quantum field theory are known to exist and are well verified experimentally. The random forces experienced by particles in the stochastic interpretation may be viewed as coming about due to random fluctuations of the various fields in the vacuum.//" ((https://aip.scitation.org/doi/abs/10.1063/1.523893)) | ||
| - | * [[http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012011/meta|Review of stochastic mechanics]] by Edward Nelson | ||
| - | * Dynamical Collapse Interpretations | ||
| - | * https://plato.stanford.edu/entries/qm-collapse/ | ||
| - | * Quantum Bayesianism | ||
| - | * [[http://blogs.discovermagazine.com/cosmicvariance/files/2011/11/banks-qmblog.pdf|The Interpretation of Quantum Mechanics]] by Tom Banks | ||
| - | * Everett (many-worlds) Interpretation | ||
| - | * "//Everett's thesis on the many worlds interpretation of quantum mechanics, which is an attempt to reformulate quantum mechanics using Shannon information tools.//"((https://physics.stackexchange.com/questions/43175/uncertainty-principle-for-information)) | ||
| - | * [[http://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/|Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct]] by Sean Carroll | ||
| - | * Relational Quantum Mechanics | ||
| - | * [[https://arxiv.org/abs/quant-ph/9801057|What is quantum mechanics trying to tell us?]] by N. David Mermin | ||
| - | * [[https://arxiv.org/abs/quant-ph/9609002|Relational Quantum Mechanics]] by Carlo Rovelli | ||
| - | * Quantum Mechanics Without State Vectors | ||
| - | * [[http://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/|The Trouble with Quantum Mechanics]] by Steven Weinberg | ||
| - | * [[https://arxiv.org/abs/1405.3483|Quantum Mechanics Without State Vectors]] by Steven Weinberg | ||
| - | * The Cellular Automaton Interpretation of Quantum Mechanics | ||
| - | * [[https://arxiv.org/abs/1405.1548|The Cellular Automaton Interpretation of Quantum Mechanics]] by Gerard 't Hooft | ||
| - | * Consistent histories interpretation | ||
| - | * [[http://quantum.phys.cmu.edu/CQT/|Consistent Quantum Theory]] by Robert B. Griffiths | ||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | **Recommended Resources:** | ||
| - | |||
| - | * For a quick overview, see http://www.preposterousuniverse.com/blog/2014/05/29/quantum-mechanics-smackdown/ | ||
| - | * For an overview see https://arxiv.org/pdf/1703.08341.pdf | ||
| - | * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994. | ||
| - | * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer | ||
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| see https://physics.stackexchange.com/questions/177800/what-is-the-relation-between-phase-space-formulation-with-wigner-quasi-probabili | see https://physics.stackexchange.com/questions/177800/what-is-the-relation-between-phase-space-formulation-with-wigner-quasi-probabili | ||
| <-- | <-- | ||
| - | |||
| - | <tabbox Equations> | ||
| - | |||
| - | __Schrödinger equation__ | ||
| - | |||
| - | $$i \hbar \partial_t \Psi(x,t) &= H \Psi (x,t) $$ | ||
| - | |||
| - | and the time-independent version for systems where the Hamiltonian is time-independent | ||
| - | |||
| - | $$H \psi(x)= E\psi(x),$$ | ||
| - | where the complete wave function $\Psi$ is then given by | ||
| - | |||
| - | $${\Psi(x,t) = \phi(t) \psi(x) = e^{-Et/\hbar} \psi(x)}$$ | ||
| - | ---- | ||
| - | |||
| - | __Standard Hamiltonian__ | ||
| - | |||
| - | $$ H = - \frac{\hbar^2}{2m} \Delta^2 + V \hat{=} \frac{\hat{p}^2}{2m}. $$ | ||
| - | ---- | ||
| - | |||
| - | __Time dependence of an expectation value__ | ||
| - | |||
| - | ---- | ||
| - | |||
| - | __Uncertainty Principle__ | ||
| - | |||
| - | ---- | ||
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