Comments on: Why Everything is Hard http://cornellmath.wordpress.com/2007/08/11/why-everything-is-hard/ Geometry, Topology, Categories, Groups, Physics, . . . Everything Sat, 19 Jul 2008 10:32:00 +0000 http://wordpress.org/?v=MU By: jpivarski http://cornellmath.wordpress.com/2007/08/11/why-everything-is-hard/#comment-253 jpivarski Thu, 16 Aug 2007 22:27:05 +0000 http://cornellmath.wordpress.com/2007/08/11/why-everything-is-hard/#comment-253 The uncertainty principle actually comes from two inequalities. The Cauchy-Schwartz inequality is the first one; the second is the observation that |A|^2 >= Im{A}^2. Wikipedia has an okay derivation (step-by-step, but with little explanation), but the best derivation I've seen is in Shankar's red textbook. After posting this, I wished I had written the reduced-space argument more carefully. The Figures were fun to make, but they don't add much. Here's what I'm thinking: when two wells of length L are not overlapping, the energy of electrons in those wells is independent of separation, in agreement with our observation of zero contact force between objects which are not in contact. When the wells just barely touch each other, the length is 2L. As they overlap more, the total length shrinks to less than 2L, reaching a lower limit of 1L when they completely overlap. The force I derived is the force needed to shrink the well, starting at 2L with all electrons filling the entire combined well. I never handled the issue of what happens when separated blocks *first* come into contact. For infinite square wells, there's a net reduction in potential energy: energy of separated metal blocks > energy of combined block 2 Sum_n^N (n/L)^2 > Sum_n^2N (n/2L)^2 but this difference is tiny in the limit of large N. This means there's a tiny attractive force between metal blocks until they touch and conduct electrons: an attractive contact force! It would be hard to measure such a force in a macroscopic block because there are a lot of other small attractive forces (polarization, the Casmir effect, residual charge, and so on). (For two blocks with 1e23 valence electrons, the fractional energy loss is 0.75e-23. Multiply this by the number of electrons, and we've only lost a quarter of an average electron's energy, for the whole block. It might be noticible in nanofabricated systems.) The uncertainty principle actually comes from two inequalities. The Cauchy-Schwartz inequality is the first one; the second is the observation that |A|^2 >= Im{A}^2. Wikipedia has an okay derivation (step-by-step, but with little explanation), but the best derivation I’ve seen is in Shankar’s red textbook.

After posting this, I wished I had written the reduced-space argument more carefully. The Figures were fun to make, but they don’t add much. Here’s what I’m thinking: when two wells of length L are not overlapping, the energy of electrons in those wells is independent of separation, in agreement with our observation of zero contact force between objects which are not in contact. When the wells just barely touch each other, the length is 2L. As they overlap more, the total length shrinks to less than 2L, reaching a lower limit of 1L when they completely overlap. The force I derived is the force needed to shrink the well, starting at 2L with all electrons filling the entire combined well.

I never handled the issue of what happens when separated blocks *first* come into contact. For infinite square wells, there’s a net reduction in potential energy:

energy of separated metal blocks > energy of combined block

2 Sum_n^N (n/L)^2 > Sum_n^2N (n/2L)^2

but this difference is tiny in the limit of large N. This means there’s a tiny attractive force between metal blocks until they touch and conduct electrons: an attractive contact force! It would be hard to measure such a force in a macroscopic block because there are a lot of other small attractive forces (polarization, the Casmir effect, residual charge, and so on).

(For two blocks with 1e23 valence electrons, the fractional energy loss is 0.75e-23. Multiply this by the number of electrons, and we’ve only lost a quarter of an average electron’s energy, for the whole block. It might be noticible in nanofabricated systems.)

]]> By: Aaron F. http://cornellmath.wordpress.com/2007/08/11/why-everything-is-hard/#comment-249 Aaron F. Thu, 16 Aug 2007 02:54:28 +0000 http://cornellmath.wordpress.com/2007/08/11/why-everything-is-hard/#comment-249 Oooooooh, cool! It's great to finally hear a solid explanation for something that everyone traditionally glosses over. Incidentally, another such frequently-glossed concept is the uncertainty principle itself... I've never seen a decent derivation of it, and now that you've mentioned the Cauchy-Schwarz inequality, I'm tempted to have a go at it myself. :) p.s. I'm still having trouble with the idea that the electrons in your finger and the book would have less available living space if the two objects interpenetrated. Consider two electrons in two potential wells of width L. If the wells are separate, each electron has a position uncertainty of ~L. If the wells overlap, both electrons still have a position uncertainty of ~L! To me, therefore, it makes more sense to say "It's hard to make the wells overlap because one electron must gain energy due to the Pauli exclusion principle" than to say "It's hard to make the wells overlap because both electrons gain energy when their living spaces are reduced." Of course, I'm not the world's brightest quantum mechanic... :P Oooooooh, cool! It’s great to finally hear a solid explanation for something that everyone traditionally glosses over. Incidentally, another such frequently-glossed concept is the uncertainty principle itself… I’ve never seen a decent derivation of it, and now that you’ve mentioned the Cauchy-Schwarz inequality, I’m tempted to have a go at it myself. :)

p.s. I’m still having trouble with the idea that the electrons in your finger and the book would have less available living space if the two objects interpenetrated. Consider two electrons in two potential wells of width L. If the wells are separate, each electron has a position uncertainty of ~L. If the wells overlap, both electrons still have a position uncertainty of ~L! To me, therefore, it makes more sense to say “It’s hard to make the wells overlap because one electron must gain energy due to the Pauli exclusion principle” than to say “It’s hard to make the wells overlap because both electrons gain energy when their living spaces are reduced.” Of course, I’m not the world’s brightest quantum mechanic… :P

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