# 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.