So far, we have used Schrödinger’s equation to see how a single particle, usually an electron, behaves in a variety of potentials. If we are going to think about atoms other than hydrogen, it is neces...So far, we have used Schrödinger’s equation to see how a single particle, usually an electron, behaves in a variety of potentials. If we are going to think about atoms other than hydrogen, it is necessary to extend the Schrödinger equation so that it describes more than one particle. All elementary particles are either fermions, which have antisymmetric multiparticle wavefunctions, or bosons, which have symmetric wave functions. Electrons, protons and neutrons are fermions; photons are bosons.
Since \(\hat{a}\) and \(\hat{a}^{\dagger}\) are the quantum-mechanical operators of the complex amplitude \(a = A\text{exp}\{i\varphi \}\) and its complex conjugate \(a^* = A\text{exp}\{–i\varphi \}\)...Since \(\hat{a}\) and \(\hat{a}^{\dagger}\) are the quantum-mechanical operators of the complex amplitude \(a = A\text{exp}\{i\varphi \}\) and its complex conjugate \(a^* = A\text{exp}\{–i\varphi \}\), where \(A\) and \(\varphi\) are real amplitude and phase of the wavefunction, Equation (\ref{82}) yields the following approximate uncertainty relation (strict in the limit \(\delta \varphi << 1\)) between the number of particles \(N = AA^*\) and the phase \(\varphi \):
