Quantum mechanical equation of motion?

In summary: 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.
  • #1
jeebs
325
4
I have this question that gives me a density matrix [tex] \rho (t) = \Sigma_a |\Psi_a(t)\rangle P_a \langle \Psi_a(t)| [/tex] 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 [tex] |\Psi_a(t)\rangle [/tex]

I am asked to first show that the trace [tex] Tr(\rho) = 1 [/tex] and that the expectation value of some operator K is [tex] \langle K \rangle = Tr(\rho K) [/tex], 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 [tex] i\hbar\frac{\partial \rho(t)}{\partial t} = [H,\rho (t)] [/tex] ".

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 [tex] m\frac{d^2x}{dt^2} = -c\frac{dx}{dt} - kx [/tex] 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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5


I can understand the confusion you may have with this question. The quantum mechanical equation of motion is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is also known as the von Neumann equation, named after the mathematician who first derived it.

In order to understand this equation, we need to first understand the density matrix \rho(t) that is given to us. The density matrix is a mathematical representation of a quantum system that takes into account the probabilities of the system being in different states. In this case, the different states are represented by the wavefunctions |\Psi_a(t)\rangle and the probabilities of the system being in these states are given by Pa. The trace of the density matrix, Tr(\rho), represents the total probability of the system being in any state, which in quantum mechanics must always be equal to 1.

Moving on to the second part of the question, we are asked to calculate the expectation value of an operator K using the density matrix. This is a common practice in quantum mechanics, where operators are used to represent physical observables such as energy, momentum, or position. The expectation value of an operator is the average value of that observable in a given quantum state, and it is calculated by taking the trace of the product of the density matrix and the operator, as shown in the equation \langle K \rangle = Tr(\rho K).

Now, let's focus on the main part of the question, which is to show that the equation of motion is i\hbar\frac{\partial \rho(t)}{\partial t} = [H,\rho (t)]. Here, H represents the Hamiltonian operator, which is a fundamental operator in quantum mechanics that describes the total energy of a system. The equation of motion shows that the time evolution of the density matrix is related to the Hamiltonian operator through a commutator. This means that the change in the density matrix with respect to time is related to the difference between the Hamiltonian and the density matrix itself.

To understand this equation further, we need to consider the Schrödinger equation, which is another fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The Schrödinger equation is i\hbar\frac{\partial |\Psi(t)\rangle}{\partial t} = H|\Psi(t)\rangle, where |\Psi(t)\rangle is the wavefunction of the system. If
 

What is the quantum mechanical equation of motion?

The quantum mechanical equation of motion describes the behavior and evolution of a quantum system over time. It is a mathematical equation that takes into account both the position and momentum of a particle.

How is the quantum mechanical equation of motion derived?

The quantum mechanical equation of motion is derived from the Schrödinger equation, which is the fundamental equation of quantum mechanics. It describes how the wave function of a quantum system changes over time.

What is the significance of the quantum mechanical equation of motion?

The quantum mechanical equation of motion is significant because it allows us to make predictions about the behavior of quantum systems. It helps us understand how particles move and interact at a microscopic level, and has been instrumental in the development of technologies such as transistors and lasers.

Can the quantum mechanical equation of motion be applied to macroscopic objects?

No, the quantum mechanical equation of motion is only applicable to microscopic particles. It does not accurately describe the behavior of larger objects, which are governed by classical mechanics.

Are there any limitations to the quantum mechanical equation of motion?

Yes, the quantum mechanical equation of motion is limited in its ability to accurately predict the behavior of particles at very high speeds or in very strong gravitational fields. In these extreme conditions, the principles of quantum mechanics and general relativity must be combined to create a more comprehensive theory.

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