- #1
Lit
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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:
[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 (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:
[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
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:
[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 (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:
[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