Overview#

• Confinement principle: Confining a wave to a finite region leads to quantization—only discrete wavelengths and energies are allowed. This applies to string waves, light in a cavity, and matter waves.

• An electron trapped in a one-dimensional infinite potential well of length \(L\) can have only the energies \(E_n = h^2 n^2/(8mL^2)\) for \(n = 1, 2, 3, \ldots\).

• The ground state (\(n=1\)) has the lowest energy; the electron cannot have \(E = 0\). Quantum jumps between levels require \(\Delta E = E_{\text{high}} - E_{\text{low}}\).

• Photon absorption (upward jump) and photon emission (downward jump) obey \(hf = hc/\lambda = \Delta E\).

Confinement and quantization#

A stretched string of infinite length can support traveling waves of any frequency. A string of finite length \(L\) with fixed ends supports only standing waves—and only those for which \(L = n\lambda/2\) (\(n = 1, 2, 3, \ldots\)). The ends are nodes; the wavelength is quantized.

This idea applies to all waves, including matter waves. Confining an electron to a finite region forces its matter wave to fit that region—only certain wavelengths (and thus certain energies) are allowed.

Confinement principle: Confinement of a wave leads to quantization—discrete states with discrete energies. States with intermediate energies are not allowed.

Infinite potential well#

An idealized trap: an electron confined to \(0 < x < L\) with \(U = 0\) inside and \(U \to \infty\) outside. The walls are impenetrable—the electron cannot escape. This is a one-dimensional infinite potential well.

The matter wave must have nodes at \(x = 0\) and \(x = L\) (the wave function must vanish where \(U \to \infty\)). So the well length must equal an integer number of half-wavelengths:

The de Broglie wavelength is \(\lambda = h/p = h/\sqrt{2mK}\). Inside the well, \(U = 0\), so \(E = K\) (kinetic energy). Thus \(\lambda = h/\sqrt{2mE}\). Substituting into Eq. (384) and solving for \(E\):

The quantum number \(n\) labels the allowed states. The electron cannot have an energy between, say, \(E_1\) and \(E_2\)—only the discrete values Eq. (385) are possible.

Energy-level diagram#

We represent the allowed energies as energy levels on a vertical ladder. The lowest level is the ground state (\(n=1\)); higher levels are excited states (\(n=2\): first excited, \(n=3\): second excited, etc.).

The electron tends to be in the ground state. It can occupy an excited state only if it receives the required energy from an external source.

Quantum jumps and photons#

A quantum jump (or transition) is a change from one allowed state to another. The energy change is

Upward jump (electron gains energy): requires absorption of a photon (or other energy transfer). The photon energy must equal \(\Delta E\):

Downward jump (electron loses energy): the electron emits a photon with energy \(\Delta E\).

The electron cannot exist “between” levels—it jumps directly from one allowed state to another. It may skip intermediate levels (e.g., \(n=1 \to n=4\)).

Emission and absorption spectra#

Absorption spectrum: Shine light on the trapped electron. Photons with energies matching \(\Delta E\) for some upward jump are absorbed; the rest pass through. The spectrum shows dark lines at the absorbed wavelengths.

Emission spectrum: An excited electron drops to a lower level and emits a photon. The emitted light has discrete wavelengths corresponding to the possible \(\Delta E\) values. The spectrum shows bright lines.

Summary#

• Confinement \(\Rightarrow\) quantization of wavelength and energy.

• Infinite well: \(U=0\) for \(0 < x < L\); \(U \to \infty\) elsewhere; nodes at boundaries.

• Allowed energies \(E_n = h^2 n^2/(8mL^2)\); ground state \(n=1\) has \(E_1 > 0\) (zero-point energy).

• Quantum jump \(\Delta E = E_{\text{high}} - E_{\text{low}}\); photon absorption (up) or emission (down) with \(hf = hc/\lambda = \Delta E\).