(2) Please show that a discontinuity arises in the first derivative of the wave function of...

(2) Please show that a discontinuity arises in the first derivative of the wave function of the infinite potential well (0 to L).

Expert Solution

We can use the Schrödinger Equation to show that the first derivative of the wave function should be continuous, unless the potential is infinite at the boundary.

Integrate both sides from just below a boundary (assumed to be at ) to just above.

Let go to zero and the right hand side must go to zero for finite potentials.

Infinite potentials are unphysical but often handy. The delta function potential is very handy, so we will derive a special continuity equation for it. Assume . Integrating the Schrödinger Equation, we get

As before, finite terms in the right hand integral go to zero as , but now the delta function gives a fixed contribution to the integral.

So from here we conclude that in the presence of infinite potential there is always a discontinuity in the first derivative of potential.

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genius_generous answered 2 years ago

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(2) Please show that a discontinuity arises in the first derivative of the wave function of...

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