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Homework Statement:
 Spatial wave function for two particles
Relevant Equations:

##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt2}[\psi_{n}(x_{1})\psi_{k}(x_{2})+\psi_{k}(x_{1})\psi_{n}(x_{2}))##
##V_{A}(x) = \sqrt\frac{2}{L} sin (\frac{\pi x}{L})##
##V_{B}(x) = \sqrt\frac{2}{L} sin (\frac{2\pi x}{L})##
Hi,
I just need someone to check over my work. I am having trouble with the next part of this question and I just wanted to check that this part was correct first.
I have two particles in an infinite square well (walls at x=0 and x=L). I need write an expression for the spatial wave function. From the question I know that the spatial part is symmetric (spin is antisymmetric and the particles are fermions) so will be given by
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt2}[\psi_{n}(x_{1})\psi_{k}(x_{2})+\psi_{k}(x_{1})\psi_{n}(x_{2}))##
One of the particles is in the ground state and the other is in the first excited stat. Normalised eigenfunctions given in the question are:
##V_{A}(x) = \sqrt\frac{2}{L} sin (\frac{\pi x}{L})##
##V_{B}(x) = \sqrt\frac{2}{L} sin (\frac{2\pi x}{L})##
Therefore for the spatial part I have:
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt2} [\sqrt\frac{2}{L} sin (\frac{\pi x_{1}}{L})\sqrt\frac{2}{L} sin (\frac{2\pi x_{2}}{L}) + \sqrt\frac{2}{L} sin (\frac{2\pi x_{1}}{L}) \sqrt\frac{2}{L} sin (\frac{\pi x_{2}}{L})]##
Which gives the spatial wavefunction:
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt L} [ sin (\frac{\pi x_{1}}{L}) sin (\frac{2\pi x_{2}}{L}) + sin (\frac{2\pi x_{1}}{L})sin (\frac{\pi x_{2}}{L})]##
Does this make sense or am I missing something? Can I simplify this further?
Thanks in advance for help. This forum has been amazing.
Thanks
I just need someone to check over my work. I am having trouble with the next part of this question and I just wanted to check that this part was correct first.
I have two particles in an infinite square well (walls at x=0 and x=L). I need write an expression for the spatial wave function. From the question I know that the spatial part is symmetric (spin is antisymmetric and the particles are fermions) so will be given by
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt2}[\psi_{n}(x_{1})\psi_{k}(x_{2})+\psi_{k}(x_{1})\psi_{n}(x_{2}))##
One of the particles is in the ground state and the other is in the first excited stat. Normalised eigenfunctions given in the question are:
##V_{A}(x) = \sqrt\frac{2}{L} sin (\frac{\pi x}{L})##
##V_{B}(x) = \sqrt\frac{2}{L} sin (\frac{2\pi x}{L})##
Therefore for the spatial part I have:
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt2} [\sqrt\frac{2}{L} sin (\frac{\pi x_{1}}{L})\sqrt\frac{2}{L} sin (\frac{2\pi x_{2}}{L}) + \sqrt\frac{2}{L} sin (\frac{2\pi x_{1}}{L}) \sqrt\frac{2}{L} sin (\frac{\pi x_{2}}{L})]##
Which gives the spatial wavefunction:
##\psi^{+}(x_{1}, x_{2})=\frac{1}{\sqrt L} [ sin (\frac{\pi x_{1}}{L}) sin (\frac{2\pi x_{2}}{L}) + sin (\frac{2\pi x_{1}}{L})sin (\frac{\pi x_{2}}{L})]##
Does this make sense or am I missing something? Can I simplify this further?
Thanks in advance for help. This forum has been amazing.
Thanks