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advanced_tools:quantization [2018/05/05 10:58] jakobadmin [Intuitive] |
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- | 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. |
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: | ||
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- | {{ :advanced_tools:quantization-box.png?nolink&400|}} | ||
- | 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. | + | |
+ | 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: | ||
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+ | {{ :advanced_tools:quantizationnot.png?nolink&400 |}} | ||
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+ | {{ :advanced_tools:quantization-box.png?nolink&400|}} | ||
The same thing now happens also in [[theories:quantum_mechanics|quantum mechanics]]. | 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 is quantized in quantum mechanics, as a result of the constraints imposed by the box. | + | 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. |
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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]]. |
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\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]]. |
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- | 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# | ||
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+ | <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> | ||
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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]]" | ||