A one-dimensional potential well is given in the form of a delta function at x=0,
V(x)=Cd(x), C<0.
(a) A non relativistic particle of mass m and energy E is incident from one
side of the well. Derive an expression for the coefficient of transmission T(E).
(b) Since a bound state can exist with the attractive potential, find the binding energy
of the ground state of the system.
 | Solution: This is a "square potential" problem. (a) E>0.
f is continuous at x=0. f1(0)=f2(0). A1+A1’=A2.
Let us evaluate this equation at x=e and at x=-e and write down a difference equation.
If V does not remain finite at the step, then
We eliminate A1’.
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f is finite at infinity. A1’=A2=0. f is continuous at x=0. f1(0)=f2(0). A1=A2’.
Only one bound state exists.
Consider the non relativistic motion in one dimension of a particle outside an infinite barrier at x£0 with an additional delta function potential at x=a,
where F is a positive constant. Derive an analytical expression for the phase shift d(k) for a particle approaching the origin from x=+¥ with momentum hk.
| Solution: The most general solution of the" time-independent" Schroedinger equation in region 1 is f1(x)=Asinkx, because f1(0)=0 due to the boundary condition at x=0. The most general solution in region 2 is f2(x)=Bsin(kx+d(k)). The boundary conditions at x=a are
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has a finite discontinuity at the step.
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