1. A model of interest for quantum mechanics, due to its simplicity, corresponds to a delta wall. A delta wall is you can think of as a rectangular wall with width L and height s taking the limit when L - 0. This in mathematical terms translates into the potential V (x) = 6 (x) where 6 (x) is the delta of dirac. This system meets the boundary condition dy lo+ dy 2mS y(0) lo- - dx dx a) Solve the Schrodinger equation on both sides of the wall (x <0 and x> 0) for the case where the incident particles have energy 0

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1. A model of interest for quantum mechanics, due to its simplicity, corresponds to a delta wall. A delta wall is
you can think of as a rectangular wall with width L and height s
taking the limit when L - 0. This in mathematical terms translates into the potential V (x) = ô (x) where 6 (x) is
the delta of dirac. This system meets the boundary condition
dw
lo+
dự
lo-
2mS
v(0)
dx
dx
a) Solve the Schrodinger equation on both sides of the wall (x <0 and x> 0) for the case where the incident
particles have energy 0 <E.
b) Apply the continuity of W and the boundary condition at x = 0. Solve the resulting equations and obtain the
transmission coefficient T as a function of the energy of the particle E. Sketch T (E) for E > 0.
c) If we allow E to be negative, we find that it diverges by a particular energy E0. Find the E0 value
d) What fraction of the particles incident to the well with energy E = | E0 | is transmitted and what fraction is
reflected?
Transcribed Image Text:1. A model of interest for quantum mechanics, due to its simplicity, corresponds to a delta wall. A delta wall is you can think of as a rectangular wall with width L and height s taking the limit when L - 0. This in mathematical terms translates into the potential V (x) = ô (x) where 6 (x) is the delta of dirac. This system meets the boundary condition dw lo+ dự lo- 2mS v(0) dx dx a) Solve the Schrodinger equation on both sides of the wall (x <0 and x> 0) for the case where the incident particles have energy 0 <E. b) Apply the continuity of W and the boundary condition at x = 0. Solve the resulting equations and obtain the transmission coefficient T as a function of the energy of the particle E. Sketch T (E) for E > 0. c) If we allow E to be negative, we find that it diverges by a particular energy E0. Find the E0 value d) What fraction of the particles incident to the well with energy E = | E0 | is transmitted and what fraction is reflected?
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