If the electron is a standing wave, why should it be confined to a circle? After de Broglie's crucial insight that particles are waves of some sort, it took less than three years for the mature quantum theory to be found, not once, but twice. By Werner Heisenberg in 1925 and by Erwin Schrödinger in 1926. If we let the electron be a standing wave in three dimensions, we have all it takes to arrive at the Schrödinger equation, which is at the heart of the mature theory.
Let's keep to one spatial dimension. The simplest mathematical description of a wave of angular wavenumber and angular frequency (at any rate, if you are familiar with complex numbers) is the function
Let's express the phase in terms of the electron's energy and momentum
The partial derivatives with respect to and are
We also need the second partial derivative of with respect to :
We thus have
In non-relativistic classical physics the kinetic energy and the kinetic momentum of a free particle are related via the dispersion relation
This relation also holds in non-relativistic quantum physics. Later you will learn why.
In three spatial dimensions, is the magnitude of a vector . If the particle also has a potential energy and a potential momentum (in which case it is not free), and if and stand for the particle's total energy and total momentum, respectively, then the dispersion relation is
By the square of a vector we mean the dot (or scalar) product . Later you will learn why we represent possible influences on the motion of a particle by such fields as and
Returning to our fictitious world with only one spatial dimension, allowing for a potential energy , substituting the differential operators and for and in the resulting dispersion relation, and applying both sides of the resulting operator equation to we arrive at the one-dimensional (time-dependent) Schrödinger equation:
In three spatial dimensions and with both potential energy and potential momentum present, we proceed from the relation substituting for and for The differential operator is a vector whose components are the differential operators The result:
where is now a function of and This is the three-dimensional Schrödinger equation. In non-relativistic investigations (to which the Schrödinger equation is confined) the potential momentum can generally be ignored, which is why the Schrödinger equation is often given this form:
The free Schrödinger equation (without even the potential energy term) is satisfied by (in one dimension) or (in three dimensions) provided that equals which is to say: However, since we are dealing with a homogeneous linear differential equation — which tells us that solutions may be added and/or multiplied by an arbitrary constant to yield additional solutions — any function of the form
with solves the (one-dimensional) Schrödinger equation. If no integration boundaries are specified, then we integrate over the real line, i.e., the integral is defined as the limit The converse also holds: every solution is of this form. The factor in front of the integral is present for purely cosmetic reasons, as you will realize presently. is the Fourier transform of which means that
The Fourier transform of exists because the integral is finite. In the next section we will come to know the physical reason why this integral is finite.
So now we have a condition that every electron "wave function" must satisfy in order to satisfy the appropriate dispersion relation. If this (and hence the Schrödinger equation) contains either or both of the potentials and , then finding solutions can be tough. As a budding quantum mechanician, you will spend a considerable amount of time learning to solve the Schrödinger equation with various potentials.