In a finite potential well, what is the qualitative form of a bound-state wave function inside and outside the well?

For a finite well with outside potential \(V_0\) and bound-state energy \(E<V_0\), write the outside decay constant.

If the outside decay constant is \(\kappa=2.0\,\mathrm{nm^{-1}}\), find the penetration depth.

For a finite well, what happens to the outside tail as \(E\) approaches \(V_0\) from below?

State the matching conditions at the edge of a finite square well.

Why are the bound-state energies of a finite well discrete?

A finite well is made wider while its depth stays fixed. Predict the change in the number of bound states.

A finite well is made deeper while its width stays fixed. Predict the change in bound-state energies relative to the bottom of the well.

Why are finite-well energies lower than infinite-well energies for the same central width?

For a symmetric finite well, why can bound states be chosen to have definite parity?

For a symmetric finite well with interior even solution \(A\cos(kx)\) and exterior tail \(Be^{-\kappa x}\) at \(x=a\), derive the even-state matching condition.

For a symmetric finite well with interior odd solution \(A\sin(kx)\) and exterior tail \(Be^{-\kappa x}\) at \(x=a\), derive the odd-state matching condition.

Why must the growing exponential outside a finite well be rejected for a bound state?

How does increasing particle mass affect the leakage of a finite-well bound state, all else fixed?

A finite well has ground and first excited bound states. Which has a node, and what does that imply about energy ordering?

A finite well has \(V_0-E\) increased by a factor of \(4\). How does \(\kappa\) change if \(m\) is unchanged?

Use the dimensionless parameter \(z_0=a\sqrt{2mV_0}/\hbar\) to explain qualitatively why a wider or deeper symmetric well supports more states.

Explain why a finite-well bound state can have nonzero probability in a classically forbidden region without having negative kinetic energy as a measurement result.

A finite well approaches the infinite-well limit as \(V_0\to\infty\). What happens to the penetration depth and the boundary condition?

Prove that a finite well bound-state wave function cannot simply become zero at one finite edge and stay nonzero inside if the potential is finite.