Derive Planck Length in terms of c, G, and h_bar

[MCS5280]

Homework Statement

a. The first part of this problem was to derive the escape speed (Vesc) for a star of mass M and radius R.

b. The second part is where I am having trouble. It says to equate the Vesc calculated above and derive a formula for Planck length in terms of c, G, and h_bar.

Homework Equations

This is from the Heisenberg uncertainty section for position and momentum so:

$$\Delta$$X * $$\Delta$$P = h_bar/2

E = mc²

The Attempt at a Solution

The escape velocity I calculated from part a was Vesc = $$\sqrt{2GM/R}$$

Equating Vesc to C gives:

C = $$\sqrt{2GM/R}$$

Am I supposed to assume that $$\Delta$$p = Mc and solve for $$\Delta$$x using the Heisenberg uncertainty principle?

I have tried a couple of different methods and I can't get this to simplify down to the form I need. I am pretty sure this is an easy problem, I just am not seeing the trick.

 

[Delphi51]

According to Wikipedia the Planck length is defined as sqrt(hG/c^3)
I think it is just a way to get units of meters out of those fundamental constants.
I don't think it has a physical meaning, except in quantum mechanics which doesn't seem to apply to the problem you have.

 

[MCS5280]

Ok I think I am getting close to this one:

c=$$\sqrt{\frac{2GM}{R}}$$

M=$$\frac{c^2 R}{2G}$$

Using the Heisenberg Principal:
$$\Delta$$X $$\Delta$$P = $$\frac{h_bar}{2}$$

$$\Delta$$P = c M

Solving this for $$\Delta$$X
$$\Delta$$X = $$\frac{G h_bar}{c^3 R}$$

How do I get the square root and get rid of the R?

 

[pavi_elex]

Gravity level
c =under root of (2GM/R)
where R=L (Planck's Length) on gravity level
so c =under root of (2GM/L)

Uncertainty principle
uncertainty in momentum x uncertainty in position = reduced Planck constant/2
mv x L = reduced Planck constant/2
here on quantum level, Uncertainty in position is Planck length and velocity is c (velocity of light)
mcL= reduced h/2
m=reduced h/(2cL)

put the value of m in value of c in gravity level
c square = 2 G reduced h/(2cL*R)
c square = 2 G reduced h/(2cL*L)
where R=L (Planck length on gravity level)
Derive L (Planck length) from above equation.

 

[rwisz]

To derive the Planck length in terms of c, G, and h_bar, we can start by equating the escape velocity (Vesc) to the speed of light (c):

Vesc = c

Next, we can substitute the expression for the escape velocity from part a:

\sqrt{2GM/R} = c

Squaring both sides and rearranging, we get:

2GM = c^2R

Next, we can use the definition of gravitational constant (G) and the mass-energy equivalence equation (E=mc^2) to rewrite this expression:

2G(Mc^2) = c^2R

Using the Heisenberg uncertainty principle, we can also write:

\Deltax * \Deltap = h_bar/2

Substituting \Deltap = Mc, we get:

\Deltax * Mc = h_bar/2

Solving for \Deltax, we get:

\Deltax = h_bar/2Mc

Substituting this value for \Deltax into our previous equation, we get:

2G(Mc^2) = c^2R

h_bar/2Mc * Mc = c^2R

Simplifying, we get:

h_bar/2 = c^2R

Finally, we can solve for the Planck length (l_p) by rearranging this equation:

l_p = \sqrt{h_bar*c^2/2}

Substituting the value of c (speed of light in a vacuum) and h_bar (reduced Planck constant) in terms of their numerical values and units, we get:

l_p = \sqrt{(1.0545718*10^-34 m^2 kg/s)*(2.99792458*10^8 m/s)^2/2}

Simplifying, we get the final expression for Planck length in terms of c, G, and h_bar:

l_p = \sqrt{1.616199*10^-35 m^2 kg}