I am confused about pure state or in mixed states. I've seen several threads on this forum, but I still can't get the grasp of it. I only have very little quantum chemistry to know what these means. So instead, I want to know the answer for specific examples so that I can get an idea.(adsbygoogle = window.adsbygoogle || []).push({});

So I would like to know "Yes" or "No" to each numbered question in the following text where I attempt to derive wavefunctions and energies of new states produced by excitonic interaction between two identical molecules.

Say for example, I have two identical molecules aligned close to each other. I can have three possible state:

1) When both molecule is in ground state: [itex] \left | \phi _{0} \right \rangle = \left | \varphi _{1} \right \rangle \otimes \left | \varphi _{2} \right \rangle [/itex]

2) When molecule 1 is in excited state: [itex] \left | \phi _{1} \right \rangle = \left | \varphi _{1}^{*} \right \rangle \otimes \left | \varphi _{2} \right \rangle [/itex]

3) When molecule 2 is in excited state: [itex] \left | \phi _{2} \right \rangle = \left | \varphi _{1} \right \rangle \otimes \left | \varphi _{2}^{*} \right \rangle [/itex]

Where [itex] \left | \varphi _{1} \right \rangle [/itex] and [itex] \left | \varphi _{2} \right \rangle [/itex] represents molecule 1 and 2 in ground state, respectively. The star means excited state. (Q1. Are all of these wavefunctions pure state?) So the total wavefunction of the entire system is:

[tex]\left | \Phi \right \rangle = a_{0}\left | \phi _{0} \right \rangle + a_{1}\left | \phi _{1} \right \rangle + a_{2} \left | \phi _{2} \right \rangle [/tex]

(Q2. Is this a pure state or mixed state?) The "true" hamiltonianHfor deriving the energy of this system is given by:

[tex] \textbf{H} = \textbf{H}_{0} + \textbf{H}_{J} [/tex]

Where [itex] \textbf{H}_{0} [/itex] is the Hamiltonian in the absence of excitonic interaction, and [itex] \textbf{H}_{J} [/itex] is the excitonic interaction Hamiltonian. In the case of 2 identical molecules with excitonic interactions strength ofJ, ground state energy ofE_{gr}and excited state energy ofE_{ex}, the Hamiltonian is:

[tex]\textbf{H} = \begin{pmatrix}

E_{gr} & 0 & 0\\

0 & E_{ex} & J \\

0 & J & E_{ex}

\end{pmatrix}[/tex]

Solving the Schrodinger equation:

[tex]\textbf{H}\left | \Phi \right \rangle = E\left | \Phi \right \rangle[/tex]

through diagonalization gives eigenvalues of:

[itex]E_{0} = E_{gr}[/itex]

[itex]E_{1} = E_{ex}-J[/itex]

[itex]E_{2} = E_{ex}+J[/itex]

and eigenvectors of:

[itex]\left | \Phi _{0} \right \rangle = \left | \phi _{0} \right \rangle[/itex]

[itex]\left | \Phi _{1} \right \rangle = \frac{1}{\sqrt{2}} \left ( \left | \phi _{1} \right \rangle - \left | \phi _{2} \right \rangle \right )[/itex]

[itex]\left | \Phi _{2} \right \rangle = \frac{1}{\sqrt{2}} \left ( \left | \phi _{1} \right \rangle + \left | \phi _{2} \right \rangle \right )[/itex]

(Q3. Are these pure state or mixed state?) Finally, the wavefunction for the "new" entire system that takes account of excitonic interactions is given by:

[tex]\left | \Psi \right \rangle = b_{0} \left | \Phi _{0} \right \rangle + b_{1} \left | \Phi _{1} \right \rangle + b_{2} \left | \Phi _{2} \right \rangle[/tex]

(Q4. Is this pure state or mixed state?)

Thank you.

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# B Still confused about superposition and mixed states

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