- #1

Scott H

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This isn't exactly homework, but I am attempting to learn quantum mechanics on my own and was wondering if this forum could be used for guidance in that direction. What I want to do is familiarize myself, as accurately as I can, with the specific mathematical picture of the universe given to us by quantum theory.

## Homework Statement

I would like to know, in theory, how the potential

**V**generated by a particle whose wavefunction is known is defined.

## Homework Equations

My textbook states that the Schrödinger equation for a two-particle system is:

[tex]i\hbar\frac{\partial \Psi}{\partial t}=H\Psi,[/tex]

where

[tex]H=-\frac{\hbar^2}{2m_1}{\nabla_1}^2-\frac{\hbar^2}{2m_2}{\nabla_2}^2+V(\mathbf{r}_1, \mathbf{r}_2,t).[/tex]

Two questions:

1. Does this equation apply to

*any*two particles in the theory of quantum mechanics?

2. Am I correct in assuming that the generalized Schödinger equation will take on the form

[tex]i\hbar\frac{\partial \Psi}{\partial t}=V(\mathbf{r}_1,\mathbf{r}_2,\cdots,\mathbf{r}_n,t)\Psi-\sum_{j=0}^n \frac{\hbar^2}{2m_j}\nabla_j^2\Psi?[/tex]

The "purely Coulombic" equation for an atom is given in my textbook as,

[tex]H=\sum_{j=1}^Z\left(-\frac{\hbar^2}{2m}\nabla_j^2-\left(\frac{1}{4\pi\epsilon_0}\right)\frac{Ze^2}{r_j}\right)+\frac{1}{2}\left(\frac{1}{4\pi\epsilon_0}\right)\sum_{j\ne k}^Z\frac{e^2}{|\mathbf{r}_j-\mathbf{r}_k|},[/tex]

where the first sum represents the kinetic plus the potential energy of the electron in the electric field of the nucleus, and the second sum represents the potential energy associated with the mutual repulsion of electrons. (

*Introduction to Quantum Mechanics: Second Edition*, Griffiths, p. 223)

## The Attempt at a Solution

The potential used to derive the equation of a hydrogen atom assumes that the proton is stationary and invokes Coulomb's law. My question is: what do we do when a particle

*isn't*stationary?

The first thing that leaps out at me in the equation above is the appearance of

**r**

_{j}and

**r**

_{k}. I am guessing that these position vectors refer to the locations of the electrons and are not independent variables. In that case, since the

**r**

_{j}are constantly changing -- and indeed, if I am correct, becoming

*uncertain*in their positions and velocities -- I do not see an obvious way to relate the Schrödinger equation for the electrons back to a specified potential.

My intuitive guess is that each possible position of a particle, such as an electron, must be associated with its own potential, in which case we would take into account every possible potential at every possible location in calculating the wavefunction for another particle in its vicinity. However, I may be wrong. Could anyone help explain this?