Index
Chapter 4 · Item 4.6
Finite potential well: bound-state setup
Regions I, II, and III for \(-V_0<E<0\)
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Guided reading

The finite well is divided into three regions. The bound-state range \(-V_0<E<0\) gives exponential wave functions outside the well and oscillatory wave functions inside it. The boundary conditions at \(x=0\) and \(x=a\) connect these three solutions.

Finite well and its regions
Book figure of a finite potential well with regions I, II and III, depth V0, width a, and a bound-state energy.
Figure 4.3: finite potential well with regions I, II, and III. Bound states have \(-V_0<E<0\).
Potential and energy range
\[V(x)=\left\{\begin{array}{ll}0,&x<0\ \text{and}\ x>a,\\-V_0,&0\le x\le a,\end{array}\right.\qquad -V_0<E<0.\]

The energy range makes the particle bound to the well while allowing nonzero wave-function tails in regions I and III.

Regions I and III

Outside the well, the Schrödinger equation becomes

\[\frac{d^2}{dx^2}\psi(x)-k^2\psi(x)=0,\qquad k^2=\frac{2m}{\hbar^2}|E|>0.\]

Convergence far from the potential selects

\[\psi_{\mathrm I}(x)=Ae^{kx},\qquad \psi_{\mathrm{III}}(x)=Ge^{-kx}.\]
Region II

Inside the well, the equation is oscillatory:

\[\frac{d^2}{dx^2}\psi(x)+\tilde{k}^{2}\psi(x)=0,\qquad \tilde{k}^{2}=k_0^2-k^2>0,\qquad k_0^2=\frac{2m|V_0|}{\hbar^2}.\]
\[\psi_{\mathrm{II}}(x)=Ce^{i\tilde{k}x}+De^{-i\tilde{k}x}.\]
Continuity of the wave function
\[\psi_{\mathrm I}(0)=\psi_{\mathrm{II}}(0),\qquad \psi_{\mathrm{II}}(a)=\psi_{\mathrm{III}}(a).\]
\[A=C+D,\qquad Ce^{i\tilde{k}a}+De^{-i\tilde{k}a}=Ge^{-ka}.\]
Continuity of the derivative
\[\psi'_{\mathrm I}(0)=\psi'_{\mathrm{II}}(0),\qquad \psi'_{\mathrm{II}}(a)=\psi'_{\mathrm{III}}(a).\]
\[kA=i\tilde{k}(C-D),\qquad i\tilde{k}\left(Ce^{i\tilde{k}a}-De^{-i\tilde{k}a}\right)=-Gke^{-ka}.\]
What the boundary conditions determine

The four continuity conditions relate \(A,C,D,G\) and leave a condition on \(\tilde{k}\). Since \(\tilde{k}^{2}=k_0^2-k^2\), determining \(\tilde{k}\) determines the allowed bound-state energies.

Exercise-ready boundary

This page supports exercises on the finite-well potential, the three regional wave functions, and the four boundary conditions.

  • Use from this page: the potential, energy range, regional equations, and continuity conditions.
  • Keep in the book: the complete algebra leading from the four conditions to the transcendental equations.
  • Good exercise balance: identify the correct region before applying a boundary condition.
Practice anchors

Use these anchors to design compact exercises from the finite-well setup.

  • Focus: regional wave functions and matching at \(x=0\) and \(x=a\).
  • Conceptual check: explain why the external solutions must decay.
  • Equation: \[\psi_{\mathrm I}(x)=Ae^{kx},\qquad \psi_{\mathrm{III}}(x)=Ge^{-kx}\]
  • Equation: \[\psi_{\mathrm{II}}(x)=Ce^{i\tilde{k}x}+De^{-i\tilde{k}x}\]
  • Boundary: use the regional equations for setup; use the book for the complete elimination of amplitudes.
  • Typical task: write and match the wave functions in the three regions.
Original book and previews:
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Source note: Original auxiliary summary based on Chapter 4 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Consult the original book for complete derivations and exercises.