text: The concept of macroscopically occupied quantum states is introduced by Fritz London. In this section it will be explained what it means if a single state is occupied by a very large number of particles. We start with the wave function of the state written as
  with Ψ the amplitude and  the phase. The wave function is normalized so that
  The physical interpretation of the quantity
  depends on the number of particles. Fig. 1 represents a container with a certain number of particles with a small control volume ΔV inside. We check from time to time how many particles are in the control box. We distinguish three cases:
  # There is only one particle. In this case the control volume is empty most of the time. However, there is a certain chance to find the particle in it given by Eq. (). The probability is proportional to ΔV. The factor ΨΨ is called the chance density.
  # If the number of particles is a bit larger there are usually some particles inside the box. We can define an average, but the actual number of particles in the box has relatively large fluctuations around this average.
  # In the case of a very large number of particles there will always be a lot of particles in the small box. The number will fluctuate but the fluctuations around the average are relatively small. The average number is proportional to ΔV and ΨΨ is now interpreted as the particle density.
  In quantum mechanics the particle probability flow density J (unit: particles per second per m), also called probability current, can be derived from the Schrödinger equation to be
  with q the charge of the particle and  the vector potential; cc stands for the complex conjugate of the other term inside the brackets. For neutral particles , for superconductors  (with e the elementary charge) the charge of Cooper pairs. With Eq. ()
  If the wave function is macroscopically occupied the particle probability flow density becomes a particle flow density. We introduce the fluid velocity v via the mass flow density
  The density (mass per volume) is
  so Eq. () results in
  This important relation connects the velocity, a classical concept, of the condensate with the phase of the wave function, a quantum-mechanical concept.