Thursday 31 December 2015

quantum mechanics - What would be the Slater's determinant representation for an excited state?


Setup


Introducing this spinorbital notation:


\begin{align} \Psi_1=\chi_{(r1)}\alpha_{(\omega1)} = 1 \\ \Psi_1=\chi_{(r1)}\beta_{(\omega1)} = \bar{1} \end{align}


and the Slater's determinant, for a system of two electrons, is:


\begin{align} |\Psi\rangle = \frac{1}{\sqrt{2}} \Big| \begin{array}{cc} 1 & \bar{1} \\ 2 & \bar{2} \\ \end{array} \Big| = \frac{|1\bar{2}\rangle-|\bar{1}2\rangle}{\sqrt2} \end{align}


Question


Based on the example, what would be the Slater's determinant representation for an excited state like $\Psi_a^r$ (i.e. with parallel spins like: $|\bar{1}\bar{2}\rangle$) and $\Psi_{ab}^{rs}$ (feel free to use any number) ?




Answer



The slater determinant is only a "trick" to get a total antisymmetric wave function. This is required by the Pauli principle.


For understand this you need to think in indistinguishability of particles. So a any allowed state of a particle need to be assigned equally at each indistinguishable particle in your system. So if you have to state allowed: $|1\rangle$ and $|2\rangle$, then your state of the system is:


$$ |\psi\rangle=\frac{|1\rangle|2\rangle - |2\rangle|1\rangle}{\sqrt2} $$


For a $N$ allowed states is something like:


$$ |\psi \rangle= \frac{1}{\sqrt{N}}\sum_{\sigma \in S_n} p(\sigma) \bigotimes_{i=1}^N |\sigma_i\rangle $$


where $\sigma$ is a permutation of $(1,2,...,N)$, $p(\sigma)$ is the parity of the permutation $\sigma$, and $\sigma_i=\sigma(i)$ is the i-th number of the permutation $\sigma$. The big "O" with an "x" inside is:


$$ \bigotimes_{i=1}^n |\sigma _i\rangle= |\sigma _1\rangle \otimes |\sigma _2\rangle \otimes ...\otimes |\sigma _n\rangle = |\sigma _1,\sigma _2,...,\sigma _n \rangle $$ where $\otimes$ is a tensor product.


Note that the tensor product is not commutative. Then, if you fixing some basis as the position ones, you get:


$$ \psi (x_1,x_2,...x_n)= \frac{1}{\sqrt{N}}\sum_{\sigma \in S_n} p(\sigma) \prod_{i=1}^n \psi_i(x_{\sigma_i}) $$



The non-commutativity goes to $i$ labels, that maintains the order of the permutation $\sigma$. This is precise the formula of the determinant as you can see.


If you want to deal with extra state, you need to do the enumeration $(n,l,m_l,s)\rightarrow i$. e.g. the states allowed is $n=p_1,p_2$ and $s=\pm \frac{1}{2}$. We can enumerate: $(p_1,+\frac{1}{2})\rightarrow 1$, $(p_2,+\frac{1}{2})\rightarrow 2$, $(p_1,-\frac{1}{2})\rightarrow 3$ and $(p_2,-\frac{1}{2})\rightarrow 4$. Some properties of determinant can help you to understanding the structure of this antisymmetric state.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...