theories:quantum_mechanics:canonical
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theories:quantum_mechanics:canonical [2018/05/04 16:33] jakobadmin [Concrete] |
theories:quantum_mechanics:canonical [2020/04/02 14:39] (current) 62.178.252.198 [Intuitive] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | {{ :theories:quantum_mechanics:wavesqm.png?nolink&400|}} | ||
| - | In quantum mechanics, we no longer describe the trajectories of individual particles but only talk about probabilities that certain events can happen. | + | In quantum mechanics, we no longer describe the trajectories of individual particles but only talk about probabilities that certain events can happen. In the canonical description of quantum mechanics, we calculate these probabilities using a wave description for the particles. |
| - | So instead of describing the path between some points $A$ and $B$, we ask instead: "What's the probability that a particle which started at $A$ ends up at $B$?". | + | So instead of describing the path between some points $A$ and $B$, we ask instead: "What's the probability that a particle which started at $A$ ends up at $B$?". |
| + | Note that because particles are described by a wave, the points $A$ and $B$ will now correspond to points with highest probability of finding a particle there (being the highest point of the wave), as opposed to an exact point in space and time. | ||
| - | This idea is rooted in the observation that the only things that are actually important are those that we observe. Whatever happens between two measurements is not important, since we do not measure it. So when we do not measure the position of the particle between $A$ and $B$ it could have taken any path. | + | This idea of describing particles and their trajectories by waves is rooted in the observation that the only things that are actually important are those that we observe, the observation being any form of measurement. Whatever happens between two measurements is not important, since we do not measure it. So when we do not measure the position of the particle between $A$ and $B$ it could have taken any path. |
| While this sounds strange for everyday objects, this is the natural point of view for much tinier particles. We measure the position of a ball whenever we look at it. Such a measurement has no significant effect on the ball. | While this sounds strange for everyday objects, this is the natural point of view for much tinier particles. We measure the position of a ball whenever we look at it. Such a measurement has no significant effect on the ball. | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | [{{ :theories:wavefunction.png?nolink&400|[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | + | {{ :theories:quantum_mechanics:wavefunction.png?nolink&400|}} |
| The state of a system in quantum mechanics is represented by a wave function $\Psi(\vec x,t)$. This wave function contains all the information about the system we are considering. The wave function is a complex number at each point in space and time and it is useful to write [[basic_tools:complex_analysis|complex numbers]] in polar form | The state of a system in quantum mechanics is represented by a wave function $\Psi(\vec x,t)$. This wave function contains all the information about the system we are considering. The wave function is a complex number at each point in space and time and it is useful to write [[basic_tools:complex_analysis|complex numbers]] in polar form | ||
| Line 78: | Line 80: | ||
| $$ \int_R dV |\Psi(\vec x,t)|^2 .$$ | $$ \int_R dV |\Psi(\vec x,t)|^2 .$$ | ||
| + | We get the wave function that describes the system in question by solving the [[equations:schroedinger_equation|Schrödinger equation]]. The object in the Schrödinger equation that describes the system in question is the [[formalisms:hamiltonian_formalism|Hamiltonian]] and the [[basic_notions:boundary_conditions|boundary conditions]]. | ||
| - | + | -->A short introduction# | |
| - | **A short introduction** | + | |
| At the heart of quantum mechanics is the idea that we know nothing about an elementary particle until we measure its properties like position or momentum. (The same is true of course for a big object, like a rolling ball, but the statement is on the macroscopic scale trivial, because when we talk about the position and momentum of the ball it‘s clear that we mean the position and momentum that we measure with a camera etc.) | At the heart of quantum mechanics is the idea that we know nothing about an elementary particle until we measure its properties like position or momentum. (The same is true of course for a big object, like a rolling ball, but the statement is on the macroscopic scale trivial, because when we talk about the position and momentum of the ball it‘s clear that we mean the position and momentum that we measure with a camera etc.) | ||
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| There are still a lot of things of missing to transform our rough framework into an actual physical theory. For example, we must know how our vectors change in time. This is what the [[equations:schroedinger_equation|Schrödinger equation]] tells you and you will spend a lot of time-solving it for different situations. | There are still a lot of things of missing to transform our rough framework into an actual physical theory. For example, we must know how our vectors change in time. This is what the [[equations:schroedinger_equation|Schrödinger equation]] tells you and you will spend a lot of time-solving it for different situations. | ||
| + | <-- | ||
| + | ---- | ||
| **Examples** | **Examples** | ||
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| * Modern Quantum Mechanics by Jun John Sakurai | * Modern Quantum Mechanics by Jun John Sakurai | ||
| * The Principles of Quantum Mechanics by Paul Dirac | * The Principles of Quantum Mechanics by Paul Dirac | ||
| - | * Foundations of Quantum Mechanics by Gregory Naber for students who prefer a more mathematical treatment. | ||
| * Nice free lecture notes can be found [[https://www.colorado.edu/physics/phys7270/phys7270_fa16/lecnotes.html|here]]. | * Nice free lecture notes can be found [[https://www.colorado.edu/physics/phys7270/phys7270_fa16/lecnotes.html|here]]. | ||
| + | * See also [[https://www.isaacbooks.org/files/Ch1_qm2b.pdf|A Cavendish Quantum Mechanics Primer]] by M. Warner, FRS & A. C. H. Cheung which is a very gentle introduction aimed at highschool students. | ||
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| <note tip> | <note tip> | ||
| - | Usually, quantum mechanics is formulated using the [[formalisms:hamiltonian_formalism|Hamiltonian framework]]. This formulation is known as the canonical formulation. Another possibility is to use the [[formalisms:lagrangian_formalism|Lagrangian framework]]. The formulation of quantum mechanics using the Lagrangian formalism is known as [[theories:quantum_mechanics:path_integral|path integral]] formulation. In addition, quantum mechanics can be formulated in the Newtonian framework. This formulation of quantum mechanics is known as [[theories:quantum_mechanics:bohmian_mechanics|Bohmian mechanics]]. </note> | + | Usually, quantum mechanics is formulated using the [[formalisms:hamiltonian_formalism|Hamiltonian framework]]. This formulation is known as the canonical formulation. Another possibility is to use the [[formalisms:lagrangian_formalism|Lagrangian framework]]. The formulation of quantum mechanics using the Lagrangian formalism is known as [[theories:quantum_mechanics:path_integral|path integral]] formulation. In addition, quantum mechanics can be formulated in the Newtonian framework. This formulation of quantum mechanics is known as [[theories:quantum_mechanics:bohmian|Bohmian mechanics]]. </note> |
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| <tabbox Abstract> | <tabbox Abstract> | ||
| - | Mathematically canonical quantum mechanics is all about unitary operators acting on a [[basic_tools:hilbert_space|Hilbert space]]. | + | Mathematically canonical quantum mechanics is analysis of unitary (self-adjoint) operators acting on [[basic_tools:hilbert_space|Hilbert spaces]]. |
| Line 332: | Line 335: | ||
| **The best abstract quantum mechanics textbooks are** | **The best abstract quantum mechanics textbooks are** | ||
| + | * Foundations of Quantum Mechanics by Gregory Naber for students who prefer a more mathematical treatment. | ||
| * Quantum Theory: Concepts and Methods by Asher Peres | * Quantum Theory: Concepts and Methods by Asher Peres | ||
| * Quantum Mechanics by L. E. Ballentine | * Quantum Mechanics by L. E. Ballentine | ||
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| * [[https://arxiv.org/pdf/gr-qc/9706069.pdf|Geometrical Formulation of Quantum Mechanics]] by Abhay Ashtekar and Troy A. Schilling | * [[https://arxiv.org/pdf/gr-qc/9706069.pdf|Geometrical Formulation of Quantum Mechanics]] by Abhay Ashtekar and Troy A. Schilling | ||
| * [[http://lanl.arxiv.org/pdf/0810.1019v2|Classical and quantum mechanics via Lie algebras]] by Arnold Neumaier Dennis Westra | * [[http://lanl.arxiv.org/pdf/0810.1019v2|Classical and quantum mechanics via Lie algebras]] by Arnold Neumaier Dennis Westra | ||
| + | * Quantum Mechanics and Quantum Field Theory: A Mathematical Primer by J. Dimock | ||
| + | |||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Quantum mechanics is a better approximate description of nature than [[theories:classical_mechanics:newtonian_mechanics|classical mechanics]]. | + | Quantum mechanics is a better approximate description of nature than [[theories:classical_mechanics:newtonian|classical mechanics]]. |
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| + | <tabbox Equations> | ||
| + | |||
| + | The [[equations:schroedinger_equation|Schrödinger equation]] | ||
| + | |||
| + | $$i \hbar \partial_t \Psi(x,t) = H \Psi (x,t) $$ | ||
| + | |||
| + | that describes the time-evolution of the wave function. The time-independent version for systems where the Hamiltonian is time-independent is given by | ||
| + | |||
| + | $$H \psi(x)= E\psi(x),$$ | ||
| + | where the complete wave function $\Psi$ is then given by | ||
| + | |||
| + | $${\Psi(x,t) = \phi(t) \psi(x) = e^{-Et/\hbar} \psi(x)}$$ | ||
| + | ---- | ||
| + | |||
| + | The standard Hamiltonian is | ||
| + | |||
| + | $$ H = - \frac{\hbar^2}{2m} \Delta^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} +\hat{V}. $$ | ||
| + | ---- | ||
| + | |||
| + | The [[equations:heisenberg_equation|Heisenberg equation]] | ||
| + | |||
| + | $$\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t},$$ | ||
| + | |||
| + | which described the time-evolution of operators and the related time dependence of an expectation value | ||
| + | |||
| + | $$ \langle \frac{\mathrm{d}\hat F}{\mathrm{d}t} \rangle = \langle -\frac{i}{\hbar}[\hat F,\hat H] \rangle + \langle \frac{\partial \hat F}{\partial t} \rangle .$$ | ||
| + | |||
| + | ---- | ||
| + | |||
| + | The [[advanced_notions:uncertainty_principle|uncertainty principle]] | ||
| + | |||
| + | $$ \sigma_x \sigma_p \geq \hbar/2,$$ | ||
| + | |||
| + | and the generalized version | ||
| + | |||
| + | $$ \sigma_A \sigma_B \geq \big | \frac{1}{2i} \langle [A,B] \rangle \big|^2 .$$ | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | The [[formulas:canonical_commutation_relations|canonical commutation relations]] | ||
| + | $$ [\hat{p},\hat{x}] = -i \hbar .$$ | ||
| | | ||
theories/quantum_mechanics/canonical.1525444385.txt.gz · Last modified: 2018/05/04 14:33 (external edit)
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