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?(adsbygoogle = window.adsbygoogle || []).push({});

To demonstrate what my question is asking mathematically I hope you will scrutinize the equations below:

If given two objects O_{1}and O_{2}(ignoring uncertainty for the time being) with positions (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}), respectively, it seems the equations below would hold true:

[tex]L_p=\sqrt{\frac{\hbar G}{c^3}}[/tex]

[tex]\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[/tex] where K must be an integer.

If the above equation follows the integer condition, then change in position for an object moving from (x_{1},y_{1},z_{1}) to (x_{2},y_{2},z_{2}) 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:

[tex]\left\{\frac{\Psi \left(x_2,t_2\right)-\Psi \left(x_1,t_1\right)}{L_P}\right\}\subseteq \mathbb{Z}[/tex]

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

- Lit

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# Planck length and quantized position

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