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basic_tools:hilbert_space [2018/03/12 15:13] jakobadmin |
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====== Hilbert Space ====== | ====== Hilbert Space ====== | ||
- | <tabbox Why is it interesting?> | + | //see also [[basic_tools:configuration_space|Configuration Space]] and [[basic_tools:phase_space|Phase space]]// |
- | <blockquote>Recall that in Chapter 5 the concept of [[basic_tools:phase_space|phase space]] was introduced for the | + | <tabbox Intuitive> |
- | description of a classical system. A single point of phase space would be used | + | |
- | to represent the (classical) state of an entire physical system. In the [[theories:quantum_theory|quantum | + | |
- | theory]], the appropriate analogous concept is that of a Hilbert space. A single | + | |
- | point of Hilbert space now represents the quantum state of an entire system. | + | |
- | <cite>page 257 in "The Emperors new Mind" by R. Penrose</cite></blockquote> | ||
- | **Important Related Concepts:** | + | A Hilbert space is the natural arena of quantum mechanics. Each point in it represents one specific configuration a given system can be in. |
- | * [[basic_tools:phase_space]] | + | <tabbox Concrete> |
- | * [[basic_tools:configuration_space]] | + | **Recommended Resources** |
- | * [[theories:quantum_theory]] | + | |
- | <tabbox Layman> | ||
- | |||
- | <note tip> | ||
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
- | </note> | ||
- | | ||
- | <tabbox Student> | ||
* The best introduction can be found in [[https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwiR2N7ajcXXAhVMPxoKHQlcAHMQFggtMAE&url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9783319587318-c2.pdf%3FSGWID%3D0-0-45-1610032-p180855298&usg=AOvVaw3e2e_1g_oXSm4c0aqbzyOV|chapter 2 of Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers]] by G. Fano and S. M. Blinder | * The best introduction can be found in [[https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwiR2N7ajcXXAhVMPxoKHQlcAHMQFggtMAE&url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9783319587318-c2.pdf%3FSGWID%3D0-0-45-1610032-p180855298&usg=AOvVaw3e2e_1g_oXSm4c0aqbzyOV|chapter 2 of Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers]] by G. Fano and S. M. Blinder | ||
* See the nice explanations at page 257ff in "The Emperors new Mind" by R. Penrose | * See the nice explanations at page 257ff in "The Emperors new Mind" by R. Penrose | ||
+ | ---- | ||
<blockquote>The most fundamental property of a Hilbert space is that it is what is called | <blockquote>The most fundamental property of a Hilbert space is that it is what is called | ||
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| | ||
- | <tabbox Researcher> | + | <tabbox Abstract> |
<note tip> | <note tip> | ||
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</note> | </note> | ||
- | --> Common Question 1# | + | <tabbox Why is it interesting?> |
- | + | <blockquote>Recall that in Chapter 5 the concept of [[basic_tools:phase_space|phase space]] was introduced for the | |
- | <-- | + | description of a classical system. A single point of phase space would be used |
+ | to represent the (classical) state of an entire physical system. In the quantum | ||
+ | theory, the appropriate analogous concept is that of a Hilbert space. A single | ||
+ | point of Hilbert space now represents the quantum state of an entire system. | ||
- | --> Common Question 2# | + | <cite>page 257 in "The Emperors new Mind" by R. Penrose</cite></blockquote> |
- | |||
- | <-- | ||
- | | ||
- | <tabbox Examples> | ||
- | --> Example1# | ||
- | |||
- | |||
- | <-- | ||
- | |||
- | --> Example2:# | ||
- | |||
- | |||
- | <-- | ||
- | | ||
- | <tabbox History> | ||
</tabbox> | </tabbox> | ||
- | {{tag>theories:quantum_theories:quantum_mechanics theories:quantum_theories:quantum_field_theory}} | + | {{tag>theories:quantum_theory:quantum_mechanics theories:quantum_theory:quantum_field_theory}} |