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ps3d - Notice this is a positive quantity as expected for a...

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Unformatted text preview: Notice this is a positive quantity as expected for a kinetic energy. Without an explicit lorrn for the potential we can‘t calculate the exact expectation value, but we can find a bound. If Vtrj) is attractive then we can find a V’(ac:) such that V’ is at finite square well (with depth V0 and width 2a) and satisfies V(:17:\ g V’(a:) for all so (see figure below). Then /i7 "L . ,9 -) ~ 72,1717" (Lilli/No : “I’ll/”MW : ~V :I3VO dr 6: {30) 7 - ,0 E ) 2 g ~V/e3l/b2aeq‘8'a’ (31) 7r Therefore ,, v , IZU Q i 2 ‘ <q/rI\\1/> g ,3 (I ' — /4/52ae*13 02> (32) 2771, i W This last expression can be made less than zero by taking fl sufficiently small To see this notice that the first term goes to zero as ‘3 goes to zero, but the second term approaches a finite value Since the ground states energy is always less than the expectation value of the Hamiltonian for any trial wave function; we conclude that an attractive potential always has a bound state. ...
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