This article summarizes equations in the theory of quantum mechanics.
A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant.
ℏ m
Im
ℏ
{\displaystyle {\begin{aligned}\mathbf {j} &={\frac {-i\hbar }{2m}}\left(\Psi ^{*}\nabla \Psi -\Psi \nabla \Psi ^{*}\right)\\&={\frac {\hbar }{m}}\operatorname {Im} \left(\Psi ^{*}\nabla \Psi \right)=\operatorname {Re} \left(\Psi ^{*}{\frac {\hbar }{im}}\nabla \Psi \right)\end{aligned}}}
star * is complex conjugate The general form of wavefunction for a system of particles, each with position ri and z-component of spin sz i.
Sums are over the discrete variable sz, integrals over continuous positions r. For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary).
Following are general mathematical results, used in calculations.
in bra–ket notation:
for non-interacting particles:
Time-independent case:
For momentum and position;
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions.
Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.
where the position of particle n is xn.
There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum):[1]
for non-interacting particles
where the position of the particle is r = (x, y, z).
where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is
for non-interacting particles
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.
where the position of particle n is xn.
This last equation is in a very high dimension,[2] so the solutions are not easy to visualize.
The De Broglie relations give the relation between them:
The De Broglie relations give:
σ ( n ) σ ( ϕ ) ≥
Orbital magnitude:
Total magnitude:
In what follows, B is an applied external magnetic field and the quantum numbers above are used.