Relation between commutation and quantization

In summary: The momentumoperator can then be found from this equation, using the definition of the momentum operator in position space; -ip/hbar. Thus, the momentum operators for both x and p are oscillating functions of x and p respectively.Furthermore, the eigenvalue spectrum of these operators are quantized if the potential is bounded; like in a harmonic oscillator.
  • #1
osturk
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relation between "commutation" and "quantization"

Hi people;

Over the several texts I have read, I got the impression that position-momentum commutation relations is the cause of "quantization" of the system. Or, they are somehow fundamentally related.

The only relation I know of, is to derive the momentum operator in position space, [itex]-i\hbar\frac{d}{dx}[/itex], from the commutation relation [itex][x,p]=i\hbar[/itex], and then find the position and momentum eigenfunctions which turn out to be oscillating functions of [itex]x[/itex] and [itex]p[/itex]. Then, eigenvalue spectrum of these operators are then naturally "quantized", BUT only if the potential is bounding, like box, harmonic oscillator etc..

Now this demonstration of relation between commutation and quantization looks quite "indirect" to me, and also it is conditional (a bounding potential required to get quantized eigenvalues).

So my question is; is there a more fundamental demonstration of the relation between commutation relations and quantization of a system.

Thanks in advance for the answers.

Deniz
 
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  • #2


But, the commutation relation still holds between x and p when x and p are not quantized, for a free particle wave packet.
 
  • #3


Khashishi said:
But, the commutation relation still holds between x and p when x and p are not quantized, for a free particle wave packet.

Exactly.. I want to know if the notion that I've got, that commutation and quantization is fundamentally related to each other, is true or not.

If it is true; then, is there a mathematical way to show that in a more general way than the example that I gave above.
 
  • #4


Quantization means replacing classical functions on phase space, here x and p by, QM operators; in position space p becomes an operator -id/dx.

The commutation relation for the operators x and p can derived from this position space representation, i.e.

[x, -id/dx] f(x) = (id/dx x) f(x) = i f(x) for all f(x), so [x, -id/dx] = i
 

FAQ: Relation between commutation and quantization

1. What is commutation in relation to quantization?

Commutation refers to the process of converting a continuous signal into a discrete signal by dividing it into smaller segments. This is necessary for quantization, which involves representing a continuous signal with a finite number of discrete values.

2. How does commutation affect the accuracy of quantization?

The accuracy of quantization is directly affected by the size of the segments used during commutation. Smaller segments result in a more accurate representation of the continuous signal, while larger segments can lead to loss of information and reduced accuracy.

3. What is the relationship between commutation and sampling in quantization?

Commutation is a necessary step for quantization, as it divides the continuous signal into smaller segments that can then be sampled. Sampling involves measuring the amplitude of the signal at specific points within each segment, which ultimately determines the discrete values used for quantization.

4. How does the number of bits used in quantization affect the need for commutation?

The number of bits used in quantization refers to the number of discrete values that can be used to represent the continuous signal. As the number of bits increases, the need for commutation decreases because there are more available values to accurately represent the signal.

5. How does the type of signal being quantized impact the commutation process?

The type of signal being quantized, such as analog or digital, can impact the commutation process. Analog signals require a continuous conversion process, while digital signals can be quantized directly without the need for commutation. Additionally, the complexity and frequency of the signal can also affect the commutation process.

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