Chapter 16: Dependence of Amplitudes on Position

Dependence of Amplitudes on Position transitions from previously discussed simplified, discrete base states (like those used for ammonia or hydrogen molecules) to a complete, formal description of quantum mechanics where probability amplitudes depend continuously on position. This shift requires moving away from working with two base states to defining an amplitude for finding an electron at every location in space. The foundation begins by reviewing amplitudes for an electron moving along a line of atoms (a lattice), where the relationship between the energy of the state and the square of the wave number suggests a connection between wave packets and classical particle behavior, introducing the concept of effective mass (m sub eff). This continuous description is formalized using the wave function, ψ(x), which is defined as the amplitude (x | psi) for finding a particle in state (|psi) at a specific position x. Crucially, the probability of locating the electron within a small positional interval (Delta x) is proportional to the absolute square of the wave function, ∣ψ(x)∣ 2 Delta x. When moving to continuous systems, new mathematical tools are needed, specifically the Dirac delta function, δ(x), which facilitates the necessary normalization condition (x | x prime) = δ(x−x ′ ) and allows integrals to replace sums. The formalism extends to describing the particle's momentum, linking the spread of the wave function in position space to its spread in momentum space, illustrating the core concept of the Heisenberg uncertainty principle (Delta p Delta x must be (greater than) or equal to h-bar/2). The culmination of this development is the derivation of the fundamental equation governing the time evolution of the wave function: the Schrödinger equation. Starting from the time rate of change of amplitudes in discrete space, the chapter demonstrates how this transforms into a differential equation where the time derivative of the amplitude is proportional to its second spatial derivative for a free particle. By incorporating a potential energy function V(x), the complete, time-dependent Schrödinger equation is established in both one and three dimensions. Finally, analyzing time-independent solutions to the Schrödinger equation, particularly for a particle confined within a potential well, reveals that physically acceptable, bound solutions exist only for specific, discrete values of the energy E. This graphical analysis of wave function behavior provides a clear explanation of the origin of quantized energy levels in quantum mechanics.