basic_tools:hilbert_space
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basic_tools:hilbert_space [2017/10/22 18:27] jakobadmin [Why is it interesting?] |
basic_tools:hilbert_space [2018/05/03 13:22] 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 [[: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** |
| - | * [[:quantum_theory]] | + | |
| - | <tabbox Layman> | + | * 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 | ||
| + | |||
| + | ---- | ||
| + | |||
| + | <blockquote>The most fundamental property of a Hilbert space is that it is what is called | ||
| + | a vector space in fact, a complex vector space. This means that we are | ||
| + | allowed to add together any two elements of the space and obtain another | ||
| + | such element; and we are also allowed to perform these additions with | ||
| + | complex-number weightings. We must be able to do this because these are the | ||
| + | operations of quantum linear superposition that we have just been considering, namely the operations which give us $\Psi_t +\Psi_b$, $\Psi_t - \Psi_b$, $\Psi_t + i\Psi_b$ etc., for | ||
| + | the photon above. Essentially, all that we mean by the use of the phrase | ||
| + | 'complex vector space', then, is that we are allowed to form weighted sums of | ||
| + | this kind. | ||
| + | |||
| + | <cite>page 257 in "The Emperors new Mind" by R. Penrose</cite></blockquote> | ||
| - | <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> | + | <tabbox Abstract> |
| - | + | ||
| - | <note tip> | + | |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Researcher> | + | |
| <note tip> | <note tip> | ||
| Line 36: | Line 37: | ||
| </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_theory:quantum_mechanics theories:quantum_theory:quantum_field_theory}} | ||
basic_tools/hilbert_space.txt · Last modified: 2018/05/03 13:23 by jakobadmin
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