theories:quantum_mechanics
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theories:quantum_mechanics [2018/05/11 15:59] jakobadmin [Interpretations] |
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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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| <tabbox Interpretations> | <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. | + | 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 intereprtation of quantum mechanics are mostly a matter of taste. | + | 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 | Important notions regarding the interpretation of quantum mechanics are | ||
| * the EPR paradox, | * the EPR paradox, | ||
| - | * Bell's theorem, | + | * [[theorems:bells_theorem|Bell's theorem]], |
| * the no-clone theorem, | * the no-clone theorem, | ||
| * Schrödinger's cat, | * Schrödinger's cat, | ||
| Line 52: | Line 52: | ||
| ---- | ---- | ||
| - | The standard interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation. | + | 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. | 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. | 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. | ||
| ---- | ---- | ||
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| * A good book on the topic is Roland Omnes, Interpretation of Quantum Mechanics, Princeton U. Press, Princeton, 1994. | * 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 | * See also Elegance and Enigma - The Quantum Interviews by Schlosshauer | ||
| + | * Making Sense of Quantum Mechanics by Bricmont | ||
| - | <tabbox Equations> | ||
| - | The [[equations:schroedinger_equation|Schrödinger equation]] | + | ----- |
| - | $$i \hbar \partial_t \Psi(x,t) = H \Psi (x,t) $$ | + | <blockquote>“If you are not confused by quantum mechanics, then you haven’t really understood it.” <cite>Niels Bohr</cite></blockquote> |
| - | that describes the time-evolution of the wave function. The time-independent version for systems where the Hamiltonian is time-independent is given by | + | <blockquote>“I think I can safely say that nobody understands quantum mechanics.” <cite>Richard Feynman</cite></blockquote> |
| - | $$H \psi(x)= E\psi(x),$$ | ||
| - | 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 [[equations:canonical_commutation_relations|canonical commutation relations]] | ||
| - | |||
| - | $$ [\hat{p},\hat{x}] = -i \hbar .$$ | ||
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| - | __The Traditional Roadmap__ | + | -->The Traditional Roadmap# |
| **Basics** | **Basics** | ||
| Line 331: | Line 311: | ||
| </WRAP> | </WRAP> | ||
| + | |||
| + | </WRAP> | ||
| + | |||
| + | <-- | ||
| + | |||
| + | -->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. | 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. | ||
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| * the Born approximation, | * the Born approximation, | ||
| * Fermi's golden rule. | * Fermi's golden rule. | ||
| - | </WRAP> | + | <-- |
| - | + | ||
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