What is Earth's de Broglie Wavelength and Quantum Number?

[physecks]

Homework Statement

Earth has the mass 6 · 10^24 kg and the orbit radius r = 1.5 · 10^11 m.
(a) Compute its de Broglie wavelength. (b) Apply de Broglie quantization condition
as in the case of the Hydrogen atom and compute the quantum number n for the
orbit of Earth. (c) What is the difference between the radius of nth and n + 1th
orbit?

Homework Equations

lamba=h/p=h/mv

The Attempt at a Solution

I am using the above equation to solve this but I am stuck at getting the velocity of earth. I am completely stumped.

 

[nickjer]

Well you know lambda will need to be periodic about the circumference of a circle. So what would you get for lambda if the radius of the circle is 'r'?

 

[physecks]

I'm not sure I follow what you are asking.

 

[nickjer]

If you drew a wave around the circumference of a circle, you will need to connect the wave back to itself. Otherwise you will have a discontinuity when try reattaching the wave back on itself. So the wave will need to be periodic around the circle, so it is continuous the whole way around.

 

[physecks]

Yes, but I don't understand the relationship between the wavelength and the radius.

 

[nickjer]

Well if you straightened out the circumference of the circle into a box of length L. Then you specify that the wave at both ends of the box need to be periodic, you are left with the only possibilities of:

$$L = n\lambda$$

where n=1,2,3... Now going back to the circle, ask yourself what is L for a circle?

 

[physecks]

Yea L is the circumference of the circle. I understand I use the equation n*lambda=2*pi*r for b. I am stuck at a) because I am not given the velocity for Earth and asked to figure out the de Broglie.

 

[nickjer]

Well you will have to compute the velocity the old fashioned way. You know the period of the orbit of the earth. It is just 1 year. Using that and the radius, get a speed.

 

[physecks]

oh wow. Thank you so much!