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**Identical Particles -- Silly question**

**1. The problem statement, all variables and given/known data**

Reviewing for final, can someone check this really quick?

Two non-interacting particles are in an infinite cube, each side of length L. Determine the two-particle wave function and also the energy of the ground state and the first excited state:

a) The particles are distinguishable

b) The particles are identical bosons of spin 0

c) The particles are identical fermions of spin 1/2 (also: identify the singlet and triplet states)

**2. Relevant equations**

For a 3-D infinite potential box:

[tex]\psi(x,y,z)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_x \pi x}{L}\right)Sin\left(\frac{n_y \pi y}{L}\right)Sin\left(\frac{n_z \pi z}{L}\right)[/tex]

[tex]E_{n_xn_yn_z}=\frac{\hbar^2 \pi^2}{2 m L^2} (n_x^2+n_y^2+n_z^2)[/tex]

And also:

[tex]\psi(\vec{r_1},\vec{r_2})_{\pm}=A[\psi_a(\vec{r_1})\psi_b(\vec{r_2})\pm\psi_a(\vec{r_2})\psi_b(\vec{r_1})][/tex]

And don't forget about spin:

[tex]\psi(x)=\psi(x)\chi^{\pm}[/tex]

**3. The attempt at a solution**

a) For distinguishable

Ground state:

[tex]E_{111}=\frac{3\hbar^2 \pi^2}{2 m L^2} [/tex]

First excited state (degeneracy exists):

[tex]E_{112}=E_{121}=E_{211}=\frac{3\hbar^2 \pi^2}{ m L^2} [/tex]

[tex]\psi(\vec{r_1},\vec{r_2})=\Psi_a(x_1,y_1,z_1)\Psi_b(x_2,y_2,z_2)[/tex]

Where

[tex]\psi_a(x_1,y_1,z_1)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_1} \pi x_1}{L}\right)Sin\left(\frac{n_{y_1} \pi y_1}{L}\right)Sin\left(\frac{n_{z_1} \pi z_1}{L}\right)[/tex]

[tex]\psi_b(x_2,y_2,z_2)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_2} \pi x_2}{L}\right)Sin\left(\frac{n_{y_2} \pi y_2}{L}\right)Sin\left(\frac{n_{z_2} \pi z_2}{L}\right)[/tex]

b)For identical bosons of spin 0

Ground state:

[tex]E_{111}=\frac{3\hbar^2 \pi^2}{2 m L^2} [/tex]

First excited state (no degeneracy)

[tex]E_{112}=\frac{3\hbar^2 \pi^2}{ m L^2} [/tex]

[tex]\psi(\vec{r_1},\vec{r_2})_{+}=A[\psi_a(x_1,y_1,z_1)\psi_b(x_2,y_2,z_2)+\psi_a(x_2,y_2,z_2)\psi_b(x_1,y_1,z_1)][/tex]

Where

[tex]\psi_a(x_1,y_1,z_1)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_1} \pi x_1}{L}\right)Sin\left(\frac{n_{y_1} \pi y_1}{L}\right)Sin\left(\frac{n_{z_1} \pi z_1}{L}\right)[/tex]

[tex]\psi_b(x_2,y_2,z_2)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_2} \pi x_2}{L}\right)Sin\left(\frac{n_{y_2} \pi y_2}{L}\right)Sin\left(\frac{n_{z_2} \pi z_2}{L}\right)[/tex]

[tex]\psi_a(x_2,y_2,z_2)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_1} \pi x_2}{L}\right)Sin\left(\frac{n_{y_1} \pi y_2}{L}\right)Sin\left(\frac{n_{z_1} \pi z_2}{L}\right)[/tex]

[tex]\psi_b(x_1,y_1,z_1)=\left(\frac{2}{L}\right)^{\frac{2}{3}}Sin\left(\frac{n_{x_2} \pi x_1}{L}\right)Sin\left(\frac{n_{y_2} \pi y_1}{L}\right)Sin\left(\frac{n_{z_2} \pi z_1}{L}\right)[/tex]

c) Identical fermions of spin 1/2

Ground state:

[tex]E_{112}=\frac{3\hbar^2 \pi^2}{m L^2} [/tex]

First excited state:

[tex]E_{122}=\frac{9\hbar^2 \pi^2}{2 m L^2} [/tex]

[tex]\psi(\vec{r_1},\vec{r_2})_{-}=A[\psi_a(x_1,y_1,z_1)\chi^{+}\psi_b(x_2,y_2,z_2)\chi^{+}-\psi_a(x_2,y_2,z_2)\chi^{+}\psi_b(x_1,y_1,z_1)\chi^{+}][/tex]

Wave functions are the same as in part b).

How do I distinguish between a singlet and triplet state? I know singlet is S=0, and triplet is S=1, but not quite sure what it wants.

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