Energy Quantization
For Quantum States.
\(H\Psi_E = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\Psi_E + V(x)\Psi_E = E\Psi_E\).
\(\frac{d^2}{dx^2}\Psi_E = \frac{2m}{\hbar^2}[V(x)-E]\Psi_E(x)\)
N.B. Second derivative defines the concavity of the function
Finite Potential
Continuous first derivative. Thus the wave function is continuous.
Delta Function Potential
\(V(x) = C\delta(x-x_0)\). \(\frac{d}{dx}\Psi_E(x_0) = C\Psi_E(x_0)\).
First derivative at the delta function is discontinuous by some constant value.
Infinite Potential
Zero wave function in infinite region - discontinuous wavefunction derivative at boundary.
Example
Multileveled potential with a well with bottom of \(V_{min}\), one lip at \(V_-\) and the other lip at \(V_+\) with \(V_-\lt V_+\).
For \(E\lt V_{min}\), the wavefunction does not exist (it must be zero) - it is forbidden since the solutions intensify.
For \(E\lt V_-\), the wavefunction is bounded by the energy walls, with some decaying behaviour outside. Continuous function and discontinuity in derivative at both walls.
For \(E\lt V_+\), the wavefunction bounces off, scattering, the \(V_+\) wall. Continuous function and discontinuity in derivative at \(V_+\) wall.
For \(E\gt V_+\), the wavefuntion is a free particle. Continuous wave function and derivative.