39-3 An Electron in a Finite Well

39-3 An Electron in a Finite Well#

Prompts

  • Sketch a finite potential well: \(U = 0\) for \(0 < x < L\), \(U = U_0 > 0\) outside. How does it differ from an infinite well? Why can’t we assume nodes at the walls?

  • For a finite well, the wave function penetrates the walls. What happens to \(|\psi|^2\) inside the walls? How does the leakage depend on quantum number \(n\)?

  • Compare a finite well and an infinite well of the same width \(L\): which has (a) more bound states, (b) longer de Broglie wavelengths, (c) lower energies for a given \(n\)?

  • What is the nonquantized region? When is an electron “free” rather than trapped? How much energy must a ground-state electron absorb to barely escape?

  • A finite well has \(U_0 = 450\) eV and \(L = 100\) pm; \(E_1 \approx 27\) eV. What photon wavelength is needed to barely free the electron? If \(\lambda = 2.00\) nm, what happens?

Lecture Notes#

Overview#

  • A finite potential well has \(U = 0\) inside (\(0 < x < L\)) and \(U = U_0 > 0\) outside—a more realistic model than the infinite well.

  • The wave function penetrates the walls: \(|\psi|^2\) decays exponentially in the classically forbidden region. Leakage is greater for higher \(n\).

  • Compared to an infinite well of the same width: finite well has fewer bound states, longer de Broglie wavelengths, and lower energies for each \(n\).

  • Electrons with \(E > U_0\) are not trapped—their energy is nonquantized. To escape, a trapped electron must absorb at least \(U_0 - E_n\).

Finite well: setup and wave penetration#

A finite potential well has depth \(U_0\) and width \(L\): \(U = 0\) for \(0 < x < L\), and \(U = U_0\) for \(x < 0\) or \(x > L\). Unlike the infinite well, we cannot assume the wave function vanishes at the walls—there are no strict nodes at \(x = 0\) and \(x = L\).

Solving Schrödinger’s equation shows that the wave function extends into the walls, where it decays exponentially. This is analogous to tunneling (section 38-9): the matter wave “leaks” into the classically forbidden region (\(E < U_0\)). The leakage is greater for higher quantum numbers \(n\)—higher \(n\) states have more energy and thus penetrate farther.

Physical picture

Inside the barrier, the wave function decays like \(e^{-\kappa x}\) where \(\kappa\) depends on \(U_0 - E\). The wave is not exactly zero at the boundary—it smoothly extends into the wall and decays. The electron has a finite probability of being found in the “classically forbidden” region.

Finite vs infinite well#

Property

Infinite well

Finite well (same \(L\))

Bound states

Infinitely many

Finite number (only those with \(E < U_0\))

de Broglie \(\lambda\)

Shorter

Longer (wave extends into walls)

Energy \(E_n\)

Higher

Lower (same \(n\))

Wave at walls

Nodes (\(\psi = 0\))

No nodes; penetrates

Because the wave leaks into the walls, the effective wavelength inside the well is longer than in an infinite well of the same \(L\). From \(\lambda = h/\sqrt{2mE}\), a longer \(\lambda\) means lower \(E\)—so each \(E_n\) is lower in the finite well.

Energy-level diagram: Start with the infinite-well levels for the same \(L\). Remove levels above \(U_0\). Shift the remaining levels down, with the highest \(n\) shifted the most (greatest leakage).

Nonquantized region and escape#

An electron with \(E > U_0\) has enough energy to be free—it is not confined. Its energy is nonquantized (any value \(\geq U_0\) is allowed).

Minimum energy to escape: For an electron in state \(n\) with energy \(E_n\), the minimum energy it must absorb to become free is

(393)#\[ \Delta E_{\text{escape}} = U_0 - E_n \]

If a photon provides exactly this energy, the electron barely escapes with zero kinetic energy. If the photon provides more (\(hf > U_0 - E_n\)), the electron escapes with kinetic energy \(K = hf - (U_0 - E_n)\).

For photon absorption: \(hf = hc/\lambda = \Delta E\). The threshold wavelength for escape from the ground state is \(\lambda = hc/(U_0 - E_1)\).

Example

Finite well: \(U_0 = 450\) eV, \(L = 100\) pm, \(E_1 \approx 27\) eV. To barely escape: \(\Delta E = 450 - 27 = 423\) eV. Threshold wavelength \(\lambda = hc/(423\,\text{eV}) \approx 2.94\) nm. Light with \(\lambda = 2.00\) nm has photon energy \(hf \approx 622\) eV \(> 423\) eV—the electron escapes with \(K \approx 199\) eV.

Summary#

  • Finite well: \(U = 0\) inside, \(U = U_0\) outside; wave penetrates walls (exponential decay).

  • Comparison: Finite well has fewer bound states, longer \(\lambda\), lower \(E_n\) than infinite well of same \(L\).

  • Nonquantized: \(E > U_0\) \(\Rightarrow\) electron free; energy not quantized.

  • Escape: Minimum absorption \(U_0 - E_n\); photon with \(hf = hc/\lambda = \Delta E\).