This equation determines the wave function $\psi \left(x\right)$ and, through $\psi \left(x\right)$, the probabilities of finding a particle of mass $m$ with potential energy $U\left(x\right)$.

• $\psi \left(x\right)$ is a continuous function.

• $\psi \left(x\right)\to 0$ as $x\to \pm \infty$.

• $\psi \left(x\right)=0$ in a region where it is physically impossible for the particle to be.

• $\psi \left(x\right)$ is normalized.

• The wave function oscillates in the region between the classical turning points.

• State $n$ has $n$ antinodes.

• Node spacing and amplitude increase as kinetic energy $K$ decreases.

• $\psi \left(x\right)$ decays exponentially in a classically forbidden region.

Quantum-mechanical models are characterized by the particle’s potential-energy function $U\left(x\right)$.

• Wave-function solutions exist for only certain values of $E$. Thus energy is quantized.

• Photons are emitted or absorbed in quantum jumps.

A wave function can penetrate into a classically forbidden region with

where the penetration distance is

The probability of tunneling through a barrier of width $w$ is

The correspondence principle says that the quantum world blends smoothly into the classical world for high quantum numbers. This is seen by comparing $\left|\psi \left(x\right)\right|{}^2$ to the classical probability density

$P_{\text{class}}$ expresses the idea that a classi-cal particle is more likely to be found where it is moving slowly.