Consider a particle of mass m held in the potential: V(x) =-V 0 [(x)+(x-L)] wher

Question

Consider a particle of mass m held in the potential: V(x) =-V0[(x)+(x-L)] where L is a constant. Findthe bound states of the particles. Show that the energies are givenby the relation: e-L = ±(1-2/)where E = -h22/2m and = 2mV0/h2.

Explanation / Answer

This problem has a symmetry with respect to the midpoint betweenthe 2 delta functions, i.e., at x = L/2. We can use this to our advantage by shifting our origin in x-spaceto x = L/2= b (say), solve the problem and then shift back tooriginal coordinate. Note that if you are only interested inenergies, they are not affected by the shift; only thewavefunctions are. So we start with the potential V(x) = -V0((x-b)+(x+b)) >>> L =2b so it looks like 2 downward infinite spikes at x = b and -b. (i tried to include a picture but the picture drawing tool crashedand so i gave up.) So we have to solve the Schroedinger equation in the 3 regions x> i have used E =-hbar2p2 /2m which is positivesince we are looking for a bound state E < 0. Generally, the solutions are of the form = Aepx + Be-px >>>where A and B are constants. In the three regions, we get the 3 sets of solutions x < -b: = Aepx x > b: = Be-px -b

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