Multiple Particles in a Box: How Does the Schrodinger Equation Change?

[pivoxa15]

The Schrödinger equation is for 1 particle in a system? If so what happens if there are two or more in a system such as a box?

Would you use two SEs or one with the mass as the combined mass of the particles. Or something else? I have a feeling the latter is the case as the two particles could be very different in terms of physical characteristics from each other.

 

[jostpuur]

If it is specified that the particles are identical fermions or bosons, then stuff get difficult to understand. But if you ignore this statistical stuff temporarily, and just assume the particles are not identical, and also assume that the particles don't interact, then here's how it goes:

When you have one particle in a one dimensional box, the wave function is

$$\psi:\mathbb{R}\times[-R,R]\to\mathbb{C}$$, $$\psi(t,x)$$

and the shrodinger's equation is

$$i\hbar\frac{d}{dt}\psi = (-\frac{\hbar^2}{2m}\partial^2 + U(x))\psi$$

where the potential is constant on interval from -R to R, and infinite outside.

When you have two particles in a one dimensional box, the wave function is

$$\psi:\mathbb{R}\times[-R,R]\times[-R,R]\to\mathbb{C}$$, $$\psi(t,x_1,x_2)$$

and the shrodinger's equation is

$$i\hbar\frac{d}{dt}\psi = (-\frac{\hbar^2}{2m_1}\partial^2_1 - \frac{\hbar^2}{2m_2}\partial^2_2 + U(x_1,x_2))\psi$$

where the potential is some constant when $$(x_1,x_2)\in[-R,R]^2$$, and infinite when both or just other parameter is outside the interval.

Does this make sense?

 

[pivoxa15]

What happens if I assume the particles are identitical?

 

[Haelfix]

You need to add spin terms to the equation, and antisymmetrize the wave functions.

I do not know if this example has a closed form solution or not in this case