Two non-interacting particles in a 1D box

Question

I need to find the wave function for two non-interacting particles of mass m_1 and m_2  in 1D infinite box (well) of length $L$, where the positions of the particle is given by $x_i$ ($i$ being 1,2).
I have found the wave function using the method of separation of variables:
$$Psi(x_1,x_2) = C sinleft({frac{n_1 pi x_1}{L}}right) sinleft({frac{n_2 pi x_2}{L}}right)$$
How should I  normalise this to obtain $C$? Should I integrate with respect to to $text{d}x_1 text{d}x_2$ as if this were a 2D problem or is there some other way?

Ruslan · Accepted Answer

Positions of individual particles are just orthogonal degrees of freedom of the system. In general, to obtain a scalar product of two states from the appropriate product of wavefunctions you should just integrate over all of the degrees of freedom (and sum over discrete DoF's)—not forgetting about Jacobian determinant in curvilinear coordinates.
Thus, whether $Psi(x_1,x_2)$ represents a state of two 1D particles or of a single 2D particle, doesn't matter for the purpose of normalization or scalar multiplication. The square of norm is simply
$$langlePsi|Psirangle=int|Psi(x_1,x_2)|^2,mathrm{d}x_1mathrm{d}x_2.$$

Two non-interacting particles in a 1D box

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