yxgao
Dec30-04, 04:03 PM
Hi!
Two (distinguishable) non-interacting electrons are in an infinite square well with hard walls at x=0 and x=a, so that the one particle states are
\phi_n(x)=\sqrt{\frac{2}{a}} sin(\frac{n\pi}{a}x), E_n=n^2K where K=\pi^2\hbar^2/(2ma^2)
My question is what are the spins and space-spin wavefunctions for the state with the lowest two energy eigenvalues?
The answer for the ground system is:
S=0
\phi_1(x_1,x_2)=\frac{2}{L} sin(\frac{\pi x_1}{L}) sin(\frac{\pi x_2}{L}) * \frac{1}{\sqrt{2}} (\chi_{up}(1)\chi_{down}(2) - \chi_{down}(1) \chi_{up}(2))
(the first part denotes the space wave function, the second part denotes the spin wavefunction)
E = 2 K where
K = \frac{\pi^2\hbar^2}{2mL^2}
Is this becuase since the wavefunction is symmetric, the spin wavefunction must be antinsymmetric (singlet). Therefore, the electrons are in opposite spins and S=0. However, I'm confused about how to find the spin-state of the non-ground system. I know that S=1 and it must be a symmetric configuration but there are three of them. How do I know which one it is in?
Thanks.
Two (distinguishable) non-interacting electrons are in an infinite square well with hard walls at x=0 and x=a, so that the one particle states are
\phi_n(x)=\sqrt{\frac{2}{a}} sin(\frac{n\pi}{a}x), E_n=n^2K where K=\pi^2\hbar^2/(2ma^2)
My question is what are the spins and space-spin wavefunctions for the state with the lowest two energy eigenvalues?
The answer for the ground system is:
S=0
\phi_1(x_1,x_2)=\frac{2}{L} sin(\frac{\pi x_1}{L}) sin(\frac{\pi x_2}{L}) * \frac{1}{\sqrt{2}} (\chi_{up}(1)\chi_{down}(2) - \chi_{down}(1) \chi_{up}(2))
(the first part denotes the space wave function, the second part denotes the spin wavefunction)
E = 2 K where
K = \frac{\pi^2\hbar^2}{2mL^2}
Is this becuase since the wavefunction is symmetric, the spin wavefunction must be antinsymmetric (singlet). Therefore, the electrons are in opposite spins and S=0. However, I'm confused about how to find the spin-state of the non-ground system. I know that S=1 and it must be a symmetric configuration but there are three of them. How do I know which one it is in?
Thanks.