So, I'm doing some undergraduate research in quantum spin systems, looking at the ground states of the Heisenberg Hamiltonian, [tex]H=\sum{J_{ij}\textbf{S}_{i}\textbf{S}_{j}}[/tex]. But I think I have a critical misunderstanding of some fundamental quantum mechanics concepts. (I'm a math major, only had one introductory QM course...)(adsbygoogle = window.adsbygoogle || []).push({});

Say you just have two interacting spin-1/2 particles. The Hamiltonian can be written as [tex]\textbf{S}_{1}\textbf{S}_{2}[/tex] (simplified by letting [tex]J_{1,2}=1[/tex]) which is equal to:

[tex]\left( \begin{array}{cccc}

1 & 0 & 0 & 0 \\

0 & -1 & 2 & 0 \\

0 & 2 & -1 & 0 \\

0 & 0 & 0 & 1 \\\end{array} \right)[/tex]

which has one ground state, [tex]\frac{1}{\sqrt{2}}(\mid\downarrow\uparrow\rangle - \mid\uparrow\downarrow\rangle)[/tex], with energy eigenvalue -3.

So let's say I have a system set up in this state. I make a measurement. I find the system is in state [tex]\mid\downarrow\uparrow\rangle[/tex] (which I would find half the time). Now, this is a basis state in my Hilbert space, but its not an eigenvector of my Hamiltonian.

What does that mean, if the state after a measurement is no longer an eigenvector of the Hamiltonian? Is the energy of the system the same as before the measurement? I feel like I'm missing something important here.

I guess I just don't understand what it means to have a superposition of states as an eigenvector of some observable. Like, I want to look at ground states, and I'm getting mostly linear combinations of basis states (especially for higher number of interacting particles) but physically I can't wrap my head around what this means.

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# Simplified Heisenberg Hamiltonian; Linear combinations of basis states

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