====== Hilbert Space ======

//see also [[basic_tools:configuration_space|Configuration Space]] and [[basic_tools:phase_space|Phase space]]//

<tabbox Intuitive> 


A Hilbert space is the natural arena of quantum mechanics. Each point in it represents one specific configuration a given system can be in. 


<tabbox Concrete> 
**Recommended Resources**

  * 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>

  
<tabbox Abstract> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<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.

<cite>page 257 in "The Emperors new Mind" by R. Penrose</cite></blockquote>


</tabbox>