The Schrödinger equation is the basic law of quantum mechanics. It plays a role analogous to Newton’s second law in classical mechanics.

• The solutions to Schrödinger’s equation are the stationary states of the system.

• The Schrödinger equation predicts the quantized energy levels of the system.

• Photons are emitted or absorbed in quantum jumps between energy levels.

《 LOOKING BACK Sections 39.3–39.4 Wave functions and normalization

Classical systems respond to forces. In contrast, a quantum system is described by a potential-energy function $U\left(x\right)$. You’ll learn to use potential wells to model quantum systems. Systems that we’ll model in this chapter include

• The rigid box (an infinite potential well).

• The nucleus (a finite potential well).

• The quantum harmonic oscillator.

• The covalent bond.

《 LOOKING BACK Section 10.5 Energy diagrams

Wave-function shapes are determined by the potential-energy function $U\left(x\right)$.

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

• A particle can penetrate into classically forbidden regions of space, where $E<U$, with an exponentially decaying wave function.

A surprising finding of quantum mechanics is that a particle can pass through an energy barrier that would reflect a classical particle, emerging with no energy loss on the other side. This is called tunneling. This totally nonclassical behavior is the basis of the scanning tunneling microscope.

Much of modern technology, including semiconductors, rests on the quantum properties of atoms and solids. Examples we’ll look at in this chapter include

• Quantum-well lasers.

• Molecular bonds.

• Nuclear energy levels and radiation.

• The scanning tunneling microscope.