- #1

jigsaw21

- 20

- 0

- Homework Statement
- I'm trying to figure out how to derive the appropriate de Broglie Relation: k = p / h-bar

- Relevant Equations
- E = mc^2

E = hf

lambda = h / p

h-bar = h / 2*pi

Kinetic Energy = 1/2m*v^2

I began by taking E = mc^2 and E = hf , where h is Planck's constant, and then rewrote E as 1/2mv^2.

I rewrote f as c / λ, which made hf become h*c / λ. I then set this expression equal to the Kinetic Energy equation 1/2mv^2, which gave me:

1/2mv^2 = h*c / λ

I then replaced c on the right side with v, because although that equation initially represented a photon (from E = hf), it can apply to the energy of any particle, which means we can use v for that speed (is that correct??)

So then one of the v's on the left will cancel out with the v term on the right and simplify to:

1/2 mv = h / λ

I then solved for λ and got λ = 2h / mv. I then replaced mv with p (momentum) and got λ = 2h / p

From here, I may have done some redundant steps that were unnecessary, but I was trying to do this from scratch. So I multiplied both sides by (h/2π), and got: hλ / 2π = 2h^2 /p*2π

On the right side, the 2's canceled out. On the left, I recalled from a prior lesson that a value of k was k = 2π / λ. So since I had its reciprocal on the left side next to the, I rewrote the left side as just h / k, which would be equal to h^2 / p*π

From here, I was a bit lost, and decided to multiply both sides by k. That would give me h = h^2 k / p*π

I then multiplied both sides by pπ and got h*p*π = h^2*k

I then canceled out the h from the left with one of the h's on the right, and got pπ = hk

I then divided both sides by h which gave pπ/h = k. At this point I thought I was close, but not sure. I decided to multiply both sides by 2 since I knew that ħ = h / 2π. So after that step, I got p*(2π / h) = 2k.

I then replaced (2π/h) with 1/ħ , since that's the reciprocal, and that gave me p / ħ = 2k, which was really close to the answer I should've gotten which should be k = p / ħ. I have 2k instead of k, and I"m not sure how I got that, or even if this would still be correct since k is a constant. I'm not sure.

Can someone please just check and verify that my steps are correct, and let me know if there's another equation I may be missing, or if I made any mistakes with my math.

Thanks for any help, and apologize for the lengthiness of this.

I rewrote f as c / λ, which made hf become h*c / λ. I then set this expression equal to the Kinetic Energy equation 1/2mv^2, which gave me:

1/2mv^2 = h*c / λ

I then replaced c on the right side with v, because although that equation initially represented a photon (from E = hf), it can apply to the energy of any particle, which means we can use v for that speed (is that correct??)

So then one of the v's on the left will cancel out with the v term on the right and simplify to:

1/2 mv = h / λ

I then solved for λ and got λ = 2h / mv. I then replaced mv with p (momentum) and got λ = 2h / p

From here, I may have done some redundant steps that were unnecessary, but I was trying to do this from scratch. So I multiplied both sides by (h/2π), and got: hλ / 2π = 2h^2 /p*2π

On the right side, the 2's canceled out. On the left, I recalled from a prior lesson that a value of k was k = 2π / λ. So since I had its reciprocal on the left side next to the, I rewrote the left side as just h / k, which would be equal to h^2 / p*π

From here, I was a bit lost, and decided to multiply both sides by k. That would give me h = h^2 k / p*π

I then multiplied both sides by pπ and got h*p*π = h^2*k

I then canceled out the h from the left with one of the h's on the right, and got pπ = hk

I then divided both sides by h which gave pπ/h = k. At this point I thought I was close, but not sure. I decided to multiply both sides by 2 since I knew that ħ = h / 2π. So after that step, I got p*(2π / h) = 2k.

I then replaced (2π/h) with 1/ħ , since that's the reciprocal, and that gave me p / ħ = 2k, which was really close to the answer I should've gotten which should be k = p / ħ. I have 2k instead of k, and I"m not sure how I got that, or even if this would still be correct since k is a constant. I'm not sure.

Can someone please just check and verify that my steps are correct, and let me know if there's another equation I may be missing, or if I made any mistakes with my math.

Thanks for any help, and apologize for the lengthiness of this.