Energy Quantization

For Quantum States.

$H Ψ_{E} = - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} Ψ_{E} + V (x) Ψ_{E} = E Ψ_{E}$ .

$\frac{d^{2}}{d x^{2}} Ψ_{E} = \frac{2 m}{ℏ^{2}} [V (x) - E] Ψ_{E} (x)$

N.B. Second derivative defines the concavity of the function

Finite Potential

Continuous first derivative. Thus the wave function is continuous.

Delta Function Potential

$V (x) = C δ (x - x_{0})$ . $\frac{d}{d x} Ψ_{E} (x_{0}) = C Ψ_{E} (x_{0})$ .

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 $V_{m i n}$ , one lip at $V_{-}$ and the other lip at $V_{+}$ with $V_{-} < V_{+}$ .

For $E < V_{m i n}$ , the wavefunction does not exist (it must be zero) - it is forbidden since the solutions intensify.

For $E < V_{-}$ , the wavefunction is bounded by the energy walls, with some decaying behaviour outside. Continuous function and discontinuity in derivative at both walls.

For $E < V_{+}$ , the wavefunction bounces off, scattering, the $V_{+}$ wall. Continuous function and discontinuity in derivative at $V_{+}$ wall.

For $E > V_{+}$ , the wavefuntion is a free particle. Continuous wave function and derivative.