# How do I construct a Density Matrix corresponding to a Hamiltonian?

I have a Hamiltonian and I want to know the corresponding density matrix. The matrix I'm interested in is the one in this question.

There's many different density matrices that can correspond to a given Hamiltonian.

For the 8x8 matrix in your question, there's 8 different "eigenstate" density matrices that can be obtained, one for each of the 8 eigenvectors. The density matrices are constructed by doing the outer product of the eigenvectors. For the $$i^{\rm{th}}$$ eigenstate of the Hamiltonian, the density matrix $$\rho_i$$ is:

$$\rho_i = |\psi_i\rangle_ \langle \psi_i|$$.

A system can also be in a "pure" superposition of eigenstates, for example:

$$|\psi \rangle = \frac{1}{\sqrt{2}}|\psi_1\rangle + \frac{1}{\sqrt{2}}|\psi_2\rangle$$.

Then the density matrix is once again made by doing the outer product of the pure wave function $$|\psi\rangle$$ with itself.

A system can also be in a "mixed" state, which means it's a linear combination of "pure" states.

In this case you would construct the density matrix like this (for example):

$$\rho = 0.5 \rho_1 + 0.5\rho_2$$,

which descrbes a state which is a 50% mixture of $$\rho_1$$ and a 50% mixture of $$\rho_2$$.

• Thanks a mil thats very helpful. So what you're saying is I need to construct all 8 density matrixes, add them together to get the sum density matrix? – Oisin Brannock Sep 30 '18 at 11:48
• It depends what you need the density matrix for :) – user1271772 Sep 30 '18 at 11:51
• I need it to get the purity of my qubit states and the internal energy of the system vs. the magnetisation factor h – Oisin Brannock Sep 30 '18 at 11:53
• The Hamiltonian doesn't tell you the state of the system. It just tells you how the current state of your system changes with respect to time. – user1271772 Sep 30 '18 at 11:54

Your question remains very unclear as to what it actually is that you want to calculate.

There is no direct correspondence between a system Hamiltonian and the quantum state of the system. No matter what the Hamiltonian, any quantum state is a valid state of the system.

Where a Hamiltonian comes in useful is, if you know the state at some time (say, $$t=0$$), you can find out what the state is at any later time via the Schroedinger equation $$i\frac{\partial |\psi\rangle}{\partial t}=H(t)|\psi\rangle.$$ If $$H$$ does not change in time, you get $$|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle$$ or, if your initial state is a mixed state, $$\rho(t)=e^{-iHt}\rho(0)e^{iHt}.$$

Now, there are two reasonable things that might be relevant in terms of a state derived from a Hamiltonian - the thermal state and the ground state (which is the thermal state at 0 temperature). At temperature $$T$$, the thermal state is $$\rho_{\text{thermal}}=\frac{e^{-H/(k_BT)}}{\text{Tr}(e^{-H/(k_BT)})},$$ while the ground state is simply the eigenstate of $$H$$ with the smallest energy. You can (crudely) think of the thermal state as the best guess about what the state would be if you cooled it to a temperature $$T$$.

Purity has nothing to do with the Hamiltonian. If you know the density matrix $$\rho$$ of your system, purity is just $$\text{Tr}(\rho^2)$$. The Hamiltonian will help you with the expected internal energy: $$\text{Tr}(\rho H)$$ but, again, the state has to be provided from elsewhere, not from the Hamiltonian.
• @DaftWullie: I think the question is completely fine the way it is. "What is the density matrix corresponding to a given Hamiltonian?" the answer is that when the system has Hamiltonian H, there's many different $\rho$ that are valid. I've given some examples and you've given 2 examples. Not any density matrix is okay (for example if the size is different from the size of the Hamiltonian). Further questions about the user's problem can be asked in a new question. – user1271772 Oct 1 '18 at 14:38