Topics in Quantum Mechanics pp 51-79 | Cite as

# Quantization and the Schrödinger Equation

## Abstract

The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function *Ψ* of a quantum system as an eigenfunction of a quantized Hamiltonian operator *H*: *HΨ* = *λΨ* where the (real) eigenvalue *λ* is the quantum *energy* of the system in the state *Ψ* see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are *quantized*: they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76].

## Keywords

Quantum Mechanic Zeta Function Large Eigenvalue Quantum Energy Potential Energy Function## Preview

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