Planck length and quantized position

[Lit]

After reading an article on Planck length, I began to wonder whether or not the theoretical limit implied that position could be quantized in whole integer multiples of Planck length?

To demonstrate what my question is asking mathematically I hope you will scrutinize the equations below:
If given two objects O₁ and O₂ (ignoring uncertainty for the time being) with positions (x₁,y₁,z₁) and (x₂,y₂,z₂), respectively, it seems the equations below would hold true:
$$L_p=\sqrt{\frac{\hbar G}{c^3}}$$
$$\frac{\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}}{L_p}=K$$ where K must be an integer.

If the above equation follows the integer condition, then change in position for an object moving from (x₁,y₁,z₁) to (x₂,y₂,z₂) should also follow the integer rule (One can treat object 1 as the object at its position before the change in position, and object 2 as the object after changing its position). Because of this, the wave function of a particle must take an argument which moves the particle an integer multiple of the Planck length away from its previous position. So:
$$\left\{\frac{\Psi \left(x_2,t_2\right)-\Psi \left(x_1,t_1\right)}{L_P}\right\}\subseteq \mathbb{Z}$$

Please let me know if I have made any mistakes in my understanding of Planck length/logic.
- Lit

 

[DennisN]

After reading an article on Planck length,

If you could post a link to that article you read, it would be very helpful.

 

[The_Duck]

After reading an article on Planck length, I began to wonder whether or not the theoretical limit implied that position could be quantized in whole integer multiples of Planck length?

No. The Planck length is simply the length scale at which we expect to need a quantum theory of gravity in order to describe physics properly.

 

[Lit]

If you could post a link to that article you read, it would be very helpful.

http://faculty.washington.edu/smcohen/320/GrainySpace.html

 

[Lit]

No. The Planck length is simply the length scale at which we expect to need a quantum theory of gravity in order to describe physics properly.

In the article it says "You have finally hit rock bottom: a span called the Planck length, the shortest anything can get. According to recent developments in the quest to devise a so-called "theory of everything," space is not an infinitely divisible continuum. It is not smooth but granular, and the Planck length gives the size of its smallest possible grains.", am I misinterpreting this section, or is there a controversy surrounding this claim?

 

[Vanadium 50]

The_Duck is right.

A New York Times article cited in a philosophy class is not a replacement for a physics text.

 

[bhobba]

In the article it says "You have finally hit rock bottom: a span called the Planck length, the shortest anything can get. According to recent developments in the quest to devise a so-called "theory of everything," space is not an infinitely divisible continuum. It is not smooth but granular, and the Planck length gives the size of its smallest possible grains.", am I misinterpreting this section, or is there a controversy surrounding this claim?

That is not what recent developments say.

What's going on at the plank scale is, at the moment, one big mystery. Of course research is ongoing, and hopefully it will eventually be resolved, but as of now we simply do not know.

As Vanadium 50 correctly says, The Duck is spot on.

Thanks
Bill