hw9-stability - < | x ] 2 > = R [ x ]...

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Physics 623, Spring 2010, O.W. Greenberg Problem set on stability of matter 1. Give a qualitative overview of the issues concerning the stability of matter. Include a. the stability of a single atom, b. the stability of bulk matter, and c. the existence of a thermodynamic limit for matter. For each case state the qualitative issue that is crucial for the demonstation of stability. 2. Construct a trial wave function that obeys the Heisenberg-Weyl inequality T ψ < | x ] 2 > ψ 9 / 4 , (1) where T ψ = R [ ψ ( x )] 2 d 3 x ,
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Unformatted text preview: &lt; | x ] 2 &gt; = R [ x ] 2 [ ( x )] 2 d 3 x , subject to R [ ( x )] 2 d 3 x = 1, and yet allows the energy of a single-electron atom to be unbounded below. 3. Use the Sobolev inequality T K s [ Z [ ( x )] 6 d 3 x ] 1 / 3 (2) to nd a nite lower bound for the energy of a single-electron atom. K s is a positive number whose value is not needed. The x s are three dimensional vectors. Reference: E.H. Lieb, Rev. Mod. Phys. 48, 553 (1976)....
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