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Bound-State Spectra for Two Delta Function Potentials

​
strength of the δ function potential λ
1
This Demonstration shows the bound-state spectra
E
b
(d)
of a particle of mass
m
in the presence of two attractive
δ
potentials separated by a distance
2d
,
V(x)=-λ[δ(x+d)+δ(x-d)]
. Since the Fourier transform of this potential is factorizable,
V(k-k')=-2λcos((k-k')a)=-2λ[cos(ka)cos(k'a)+sin(ka)sin(k'a)]
, the bound-state spectra are easily obtained using the momentum-space Schrödinger equation. The energies are normalized to the magnitude of the symmetric-state energy at
d=0
. Note that the second (antisymmetric) bound state appears only when the distance between the
δ
functions exceeds the critical value
D
c
=2
2
ℏ
/λm
.
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