A particle in 1D potential well

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SUMMARY

A particle with mass "m" in a one-dimensional potential well described by the potential function V(x) = -αδ(x) for |x| < a and ∞ for |x| ≥ a experiences a unique potential landscape. The potential is zero for all points within the well except at x = 0, where it is defined by the delta function, resulting in an infinite potential at that specific point. The boundary conditions require that the wavefunction must vanish at the boundaries x = ±a. This setup is crucial for understanding quantum mechanics in confined systems.

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Students and professionals in physics, particularly those focusing on quantum mechanics, as well as researchers interested in potential wells and their applications in particle physics.

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Hello,

What does it means when a particle having mass "m" in a one dimensional potential well has the potential given by:
V(x)=
\stackrel{-\alpha δ(x) for |x|&lt;a}{∞ for |x|≥a}

where δ(x) is the delta function and \alpha is a constant.

I understand that the well boundries have infinite potential but what about the well? Does it have -\alpha potential only for x=0? What is the potential for |X|<a and x≠0? And how do I write the boundary conditions here?

10x!
Y.
 
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the potential in for X>a or X<-a is infinite
the potential on point X=0 is -infinite due to the delta function (but its integral for example is not).
the rest |X|<a and not 0, the potential is zero.

The boundary conditions on +/-a are the same (your wavefunction must vanish).
http://en.wikipedia.org/wiki/Delta_potential
 

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