User Tools

Site Tools


advanced_tools:quantization

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
advanced_tools:quantization [2018/05/05 10:56]
jakobadmin [Intuitive]
advanced_tools:quantization [2022/06/12 06:02] (current)
192.154.255.53 [Intuitive]
Line 2: Line 2:
  
 <tabbox Intuitive> ​ <tabbox Intuitive> ​
-Quantization ​is a phenomenon where the constraints of the physical system have the effect that some physical quantity only appears in discrete jumps, while all values in between physically forbidden. ​+Sometimes when people talk about ``quantization",​ what they really mean is //​discretization//,​ i.e., a phenomenon where the constraints of the physical system have the effect that some physical quantity only appears in discrete jumps, while all values in between ​are physically forbidden. ​
  
-{{ :advanced_tools:​quantization-rope.png?​nolink&​500 |}}+The easiest example is a rope that is held under constant tension by two hands:
  
-The easiest example is a rope that is held under constant tension by two hands.+{{ :​advanced_tools:​quantization-rope.png?​nolink&​600 |}}
  
-The thing is, no matter how the two hands try to make the rope vibrate, the rope will only vibrate with a quantized set of modes. The two hands fix the rope at both ends. As a result of this constraint, the rope can only vibrate with fixed frequencies. The frequencies between this fixed set of frequencies are physically impossible. ​ 
  
-{{:​advanced_tools:​quantization-box.png?​nolink&​400|}} + 
-The same thing now happens also in quantum mechanics. Here we describe particles using waves. If we then consider, for example, a particle in a box we notice that only a specific discrete set of wave functions are physically possible. Physically this means that the energy levels within the box is quantized in quantum mechanics, as a result of the constraints imposed by the box.+ 
 +The thing is, no matter how the two hands try to make the rope vibrate, the rope will only vibrate with a quantized set of modes. The two hands fix the rope at both ends. As a result of this constraint, the rope can only vibrate with fixed frequencies. The frequencies between this fixed set of frequencies are physically impossible:​ 
 + 
 +{{ :​advanced_tools:​quantizationnot.png?​nolink&​400 |}} 
 + 
 +{{ :​advanced_tools:​quantization-box.png?​nolink&​400|}} 
 + 
 +The same thing now happens also in [[theories:​quantum_mechanics|quantum mechanics]] 
 + 
 +Here we describe particles using waves. If we then consider, for example, a particle in a box we notice that only a specific discrete set of wave functions are physically possible. Physically this means that the energy levels within the box are quantized in quantum mechanics, as a result of the constraints imposed by the box.
  
  
Line 83: Line 91:
 Now a crucial observation is that ordinary numbers and function commutate: $[f(x),​g(x)]=0$ or also $[3,​5]=3\cdot 5 - 5\cdot 3 =0$. Therefore, we learn here that the location $\hat{q}_i$ and the momentum $\hat{p}_j$ can no longer be mere numbers or functions, but must be __operators__. Operators are always denoted by a hat.  Now a crucial observation is that ordinary numbers and function commutate: $[f(x),​g(x)]=0$ or also $[3,​5]=3\cdot 5 - 5\cdot 3 =0$. Therefore, we learn here that the location $\hat{q}_i$ and the momentum $\hat{p}_j$ can no longer be mere numbers or functions, but must be __operators__. Operators are always denoted by a hat. 
  
-The operators are then interpreted as measurement operators. This means, for example, when we act with the momentum operator $\hat{p}_j$ on the wave function $\Psi$, which is the object that we use to describe, say a particle, we get as a result the momentum that the particle has in the $j$ direction: $\hat{p}_j \Psi = p_j \Psi$. (This only works so simple when the particle is in a momentum eigenstate, i.e. has a definite momentum. Otherwise the result is a superposition and more complicated.) For more on this, see the page about [[theories:​quantum_mechanics:​canonical_quantum_mechanics|quantum mechanics]]. ​+The operators are then interpreted as measurement operators. This means, for example, when we act with the momentum operator $\hat{p}_j$ on the wave function $\Psi$, which is the object that we use to describe, say a particle, we get as a result the momentum that the particle has in the $j$ direction: $\hat{p}_j \Psi = p_j \Psi$. (This only works so simple when the particle is in a momentum eigenstate, i.e. has a definite momentum. Otherwise the result is a superposition and more complicated.) For more on this, see the page about [[theories:​quantum_mechanics:​canonical|quantum mechanics]]. ​
  
 ----- -----
Line 178: Line 186:
 \end{eqnarray} \end{eqnarray}
  
-For more on this, see the page about [[theories:​quantum_field_theory|quantum field theory]].+For more on this, see the page about [[theories:​quantum_field_theory:canonical|quantum field theory]].
  
 ---- ----
Line 210: Line 218:
  
  
-A nice summary can be found in [[https://​arxiv.org/​abs/​math-ph/​0405065|Quantization Methods: A Guide for Physicists and Analysts]] by S. Twareque Ali, Miroslav Engliš. See also [[https://​arxiv.org/​abs/​math-ph/​9809011|Obstructions to Quantization]] by Mark J. Gotay and Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. Springer-Verlag,​ New York, 1998.+  * A nice summary can be found in [[https://​arxiv.org/​abs/​math-ph/​0405065|Quantization Methods: A Guide for Physicists and Analysts]] by S. Twareque Ali, Miroslav Engliš. 
 +  *  ​See also [[https://​arxiv.org/​abs/​math-ph/​9809011|Obstructions to Quantization]] by Mark J. Gotay and Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. Springer-Verlag,​ New York, 1998. 
 +  * and the book Quantization of Gauge Systems by Henneaux and Teitelboim
  
 -->​Canonical Quantization#​ -->​Canonical Quantization#​
Line 400: Line 410:
  
 ---- ----
 +
 +<​blockquote>​Quantization is an art form which, when applied to classical physical theories, yields predictions of subatomic behavior which are in spectacular agreement with experiments.
 +<​cite>​[[https://​arxiv.org/​abs/​alg-geom/​9705010|Ron Y. Donagi]]</​cite>​
 +</​blockquote>​
 +
  
 <​blockquote>​ <​blockquote>​
Line 439: Line 454:
 just the condition for constructive interference of the phases of waves just the condition for constructive interference of the phases of waves
 differing slightly in the parameter E. The procedure based on Hamilton-Jacobi differing slightly in the parameter E. The procedure based on Hamilton-Jacobi
-theory works in [[theories:​classical_mechanics:​newtonian_mechanics|classical mechanics]] because it is supported by the+theory works in [[theories:​classical_mechanics:​newtonian|classical mechanics]] because it is supported by the
 [[equations:​schroedinger_equation|Schrodinger equation]]"​ [[equations:​schroedinger_equation|Schrodinger equation]]"​
  
advanced_tools/quantization.1525510601.txt.gz · Last modified: 2018/05/05 08:56 (external edit)