# 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.

Created: 2023-06-25 Sun 02:32

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