New Interpretations of Wave Functions in Quantum Mechanics

Tian Ma & Shouhong Wang, Quantum Mechanism of Condensation and High Tc Superconductivity, 38 pages, October 8, 2017, hal-01613117

In this post, we describe the new interpretation of quantum mechanical wave functions introduced in the above paper.

In classical quantum mechanics, a micro-particle is described by a complex-valued wave function ${\Psi: \Omega \rightarrow \mathbb C}$, satisfying such a wave equation as the Schrödinger equation with external interaction potential ${V(x)}$: $\displaystyle i\hbar\frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2m}\Delta\Psi+V(x)\Psi,\quad x\in\Omega, \ \ \ \ \ (1)$

where ${\Omega \subset \mathbb R^3}$ is the region that the particle occupies, and ${m}$ is the mass of the particle. The Schrödinger equation conserves the energy, and the wave function ${\Psi}$ can be expressed as $\displaystyle \Psi=e^{-iEt/\hbar}\psi(x), \qquad \psi=|\psi| e^{i \varphi}, \ \ \ \ \ (2)$

where ${E}$ is the energy, and ${\psi}$ is the time-independent wave function, satisfying $\displaystyle -\frac{\hbar^2}{2m}\Delta\psi+V(x)\psi=E\psi. \ \ \ \ \ (3)$

The classical Born statistical interpretation of quantum mechanics amounts to saying that without constraints, the motion of a micro-particle is random and there is no definite trajectory of the motion. Also ${|\psi(x)|^2}$ stands for the probability density of the particle appearing at the particular point ${x}$. The Born interpretation of the wave function is treated as a fundamental postulate of quantum mechanics. This leads to the classical Einstein-Bohr debates, and is the origin of absurdities associated with the interpretation of quantum mechanics.

The key observation for the new interpretation is that $\displaystyle \frac{\hbar}{m}\nabla\varphi(x) \ \ \ \ \ (4)$

can be regarded as the velocity field of the particles, and the wave function ${\psi}$ is the field function for the motion of all particles with the same mass in the same class determined by the external potential ${V(x)}$. More precisely, we have the following new interpretation of quantum mechanical wave functions:

New Interpretation of Wave Functions

1. Under the external potential field ${V(x)}$, the wave function ${\psi}$ is the field function for the motion of all particles with the same mass in the same class determined by the external potential ${V(x)}$. In other words, it is not the wave function of a particular particle in the classical sense;
2. When a particle is observed at a particular point ${x_0\in\Omega}$, then the motion of the particle is fully determined by the solution of the following motion equation with initial position at ${x_0}$: \displaystyle \begin{aligned} &\frac{\mbox{d}x}{\mbox{d}t}=\frac{\hbar}{m}\nabla\varphi(x), \\ & x(0)=x_0, \end{aligned} \ \ \ \ \ (5)

where ${\varphi}$ is the phase of the wave function ${\psi}$ in (2);

3. With ${\psi}$ being the field function, $\displaystyle |\psi(x)|^2=\mbox{distribution density of particles at } x; \ \ \ \ \ (6)$

4. The energy ${E}$ in (3) represents the average energy level of the particles and can be written as $\displaystyle E=\int_{\Omega}\bigg[\frac{\hbar^2}{2m}|\nabla|\psi||^2+\frac{\hbar^2}{2m}|\nabla\varphi(x)|^2 +V(x)|\psi|^2\bigg] {d}x, \ \ \ \ \ (7)$

where in integrand on the right-hand side, the fist term represents the non-uniform distribution potential of particles, the second term is the average kinetic energy, and the third term is the potential energy of the external field. Here ${\nabla|\psi|}$ is characteristic of quantum mechanics and there is no such a term in classical mechanics.

In summary, our new interpretation says that ${ \psi=|\psi| e^{i \varphi} }$ is the common wave function for all particles in the same class determined by the external potential ${V(x)}$, ${|\psi(x)|^2}$ represents the distribution density of the particles, and ${ \frac{\hbar}{m} \nabla \varphi }$ is the velocity field of the particles. The trajectories of the motion of the particles are then dictated by this velocity field. The observed particles are the particles in the same class described by the same wave function, rather than a specific particle in the sense of classical quantum mechanics.

This is an entirely different interpretation from the classical Bohr interpretation. Also this new interpretation of wave functions does not alter the basic theories of quantum mechanics, and instead offers new understanding of quantum mechanics, and plays a fundamental role for the quantum theory of condensed matter physics and quantum physics.

It is worth mentioning that the Landau school of physics was the first who noticed that the relation between the superfluid velocity ${v_s}$ and the wave function ${\psi=|\psi| e^{i \varphi}}$ of the condensate is given by (4); see (26.12) on page 106 of [E. Lifshitz and L. Pitaevskii, Statistical physics Part 2, Landau and Lifshitz Course of Theoretical Physics vol. 9, 1980]. However they fail to make an important connection between (4) and the basic interpretation of quantum mechanics.

Tian Ma & Shouhong Wang