# Quantum mechanical equation of motion?

• jeebs
The time derivative of a density operator is just the time evolution of the summed squared projection operators on the space of all possible states of the system. This problem involves the time evolution of the density operator, which is a specific case of a broader result called Erenfest's theorem, which describes the time evolution of a general operator. The time derivative of a density operator is just the time evolution of the summed squared projection operators on the space of all possible states of the system.

#### jeebs

I have this question that gives me a density matrix $$\rho (t) = \Sigma_a |\Psi_a(t)\rangle P_a \langle \Psi_a(t)|$$ and all I am told is that it is just for some "quantum system" and that Pa is the probability of the system being in the state $$|\Psi_a(t)\rangle$$

I am asked to first show that the trace $$Tr(\rho) = 1$$ and that the expectation value of some operator K is $$\langle K \rangle = Tr(\rho K)$$, which was fairly straightforward. Then the part I am having problems with is that is asks me to "show that the equation of motion is $$i\hbar\frac{\partial \rho(t)}{\partial t} = [H,\rho (t)]$$ ".

I am told nothing about what situation this "equation of motion" might apply to. This is literally everything the question tells me and I have never been asked to do this before in a quantum sense. The only kind of equation of motion I have ever had to set up was in a classical sense like, say, a damped spring, say $$m\frac{d^2x}{dt^2} = -c\frac{dx}{dt} - kx$$ and clearly this is done in terms of forces, which don't seem to get used in quantum mechanics.
So in short, I have no idea what I am supposed to make of this, or what use calculating that trace and expectation value was, if that has any relevance. Any suggestions on how to approach this?

Thanks.

jeebs said:
I am told nothing about what situation this "equation of motion" might apply to. This is literally everything the question tells me and I have never been asked to do this before in a quantum sense.

In this case, the term "equation of motion" is being used to refer to the time evolution of the density matrix. The precise situation or system would be encoded in the Hamiltonian. However, the details are unnecessary, since the result you're asked to prove follows directly from the time-dependent Schrodinger equation.

fzero said:
In this case, the term "equation of motion" is being used to refer to the time evolution of the density matrix. The precise situation or system would be encoded in the Hamiltonian. However, the details are unnecessary, since the result you're asked to prove follows directly from the time-dependent Schrodinger equation.

so what, I'm basically just being asked to work out what the commutator is, and what density operator's time derivative is, and if I see that these two things are equal then I've done what I was being asked?

anyway why do we say that the equation of motion is the time derivative of this density operator anyway? I'm no expert in QM but that doesn't immediatley scream "motion" to me.
All I know of a density operator is that it is a statistical mixture of projection operators. What has taking the time derivative of a set of projection operators got to do with motion?

Like I said, the term "equation of motion" is being used here in a broader sense to label an equation that describes the time evolution of some quantity. In your example, you had an equation of motion that described the time evolution of the displacement of a spring. In QM, the time-dependent Schrodinger equation describes the time evolution of the wavefunction, so in that sense it is also an equation of motion. In a certain sense, the "motion" involved is that in the space of solutions to the time-independent Schrodinger equation.

This problem involves the time evolution of the density operator, which is a specific case of a broader result called Erenfest's theorem, which describes the time evolution of a general operator.