So, with Assignment 1, with the question of the particle in a 3D box, the energy eigenvalues to be shown seem a bit odd.
If we work with ℏ in our derivation then we will have the 2 in the denominator as in the question, but there is ℏ2 missing in the numerator. If we do our derivation with h, then we don't have the π as is absent in the question, but we end up with a denominator of 8, not 2.
I think Andy found this too. Anyone else?
Apologies I didn't ask this in tute.
Digressing slightly now, we discussed this today, but it seems interesting that we would bother to define a whole new symbol, say ℏ, for an existing constant h, just divided by 2π.
This also led to a rather lengthy discussion involving the upcoming Pi Day, should we use π or τ (so 2π).
So, if we redefined π to be τ, then our current ℏ would just be h divided by τ. However, the new ℏ, would we still keep it as h divided by 2 π (so then it would be h divided by 2τ, or 4π) or would we still keep the ℏ definition (so h divided by τ, or 2π)?
I acknowledge that it is just constants which are (arbitrarily) assigned, but just out of curiosity...
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I don't follow your point.
ReplyDeleteThere is only one correct answer. It is independent of whether one works with hbar or h.
Ann, I think you got a different answer cause you used different boundary conditions. I got confused as well in the beginning. I suggest you read the section on chapter 2 of Ashcroft that talks about periodic boundary conditions for a particle in a cubic box. If you used infinite potentials at the boundary you will get a different answer as your wavefunction would be standing waves in a box so it can't get out. If you use periodic boundary conditions that allow running waves so the electron can get out of the box, you get the answer that Ross got.
ReplyDeleteSeems to look better now.
ReplyDeleteThanks!