Density matrix Definition and 125 Threads

Hi, How do eigenvalues and eigenvectors relate to the density operator. Given the eigenvalues of a matrix, can they help to find the density operator ? I have seen the formula within http://www.quantiki.org/wiki/Density_matrix but given a matrix how do I work it out to get the density operator...

Hello, In the density matrix formalism I have read in numerous places that coherence is identified with the off-diagonal components of the density matrix. The motivation for this that is usually given is that if a state interacts with the environment in such a way that the basis state...

Hi everyone! I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part. Before starting I...

Homework Statement density matrix : ρ(t)=0.5+0.5*a(t)⋅σ (a is a 3 dimensional vector and σ is paul victor) H=-μ*σ⋅B (B is a three dimensional magnetic field ) and also assume that in t=0 , ρ(0)=0.5+0.5*a(0)⋅σ whats is the motion equation of a(t)? Homework Equations whats is the motion equation...

Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...

Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...

Homework Statement We have a quantum rotor in two dimensions with a Hamiltonian given by \hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2} . Write an expression for the density matrix \rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle Homework Equations...

Suppose I have a two level system with the states labeled ##|0\rangle## and ##|1\rangle##. In this basis, these correspond to density matrices: ## \rho_0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \rho_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} ## I can create a coherent...

Hey guys! In an n-electron system, The second order reduced DM is defined as \Gamma (x_{1},x_{2}) = \frac{N(N-1)}{2}\int{\psi(x_{1},x_{2}...,x_{n})\psi^{*}(x_{1},x_{2}...,x_{n})}dx_{3}...dx_{n} It can be intepreted as the probability of finding two electrons at...

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space ##\mathcal{H}##. Composition is defined through the tensor product and reduction through partial trace. Operations on the system are...

Homework Statement Hello people, I am trying to understand a problem statement as well as the density operator, but I still don't get it, desperation is making me posting here. The problem comes as The problem then continues with other questions but I'm having troubles with the very first one...

Hello everybody, this is my first time being here. I am a beginner learning some introductions on Bose-Einstein Condensation (BEC) on my own. Often times in the literature (say, [1], [2] (p.409) ) it comes the one-body(single-particle) density matrix, as...

Hey I am currently studying Quantum Mechanics and I have difficulty grasping a concept. I don't understand the following step in the derivation: \langle X_{A} \rangle=tr\left[\left(X_{A}\otimes I_{B}\right)\rho_{AB}\right] =tr_{A}\left[X_{A} tr_{B}\left[\rho_{AB}\right]\right] Thanks

Homework Statement Find condition for which ##\hat{\rho}## will be pure state density operator? ##\hat{\rho} = \begin{bmatrix} 1+a_1 & a_2 \\[0.3em] a_2^* & 1-a_1 \end{bmatrix}## Homework Equations In case of pure state...

Is it not possible to deduce quantum states from a density matrix?

Homework Statement Hi The density matrix evolves as \dot \rho = -\frac{i}{\hbar}[H,\rho] but is this equation written in the Schrödinger or Heisenberg picture? I'm not entirely sure how to figure this out. In my book it just mentions the equation, not how it is derived (which may have given...

I'm re-reading some course notes on quantum teleportation, and something isn't making sense. In the description my instructor gave, we used the Bell state ##|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)## for the entangled pair. So, suppose the state we want to teleport is...

Homework Statement Hi there. just working on a problem from sakurai's modern quantum mechanics. it is: A) Prove that the time evolution of the density operator ρ (in the Schrodinger picture) is given by ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0}) B) Suppose that we have a pure ensemble at...

I have an one-body density matrix in a Sine wave basis set (Thus psi = psi*). Unfortunately, these are not the natural orbitals (I have correlated particles), so I have off-diagonal elements. I believe I know how to extract the charge density from this density matrix \rho(x;x') = \sum_{ij}...

Hi I have three states (I believe bell states) and want to find the density matrix, am I right in thinking: 1) \frac{|00> + |11>}{\sqrt{2}} \rightarrow \rho = \left( \begin{array}{cc} \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} \\ \end{array} \right) (because it is pure) 2)...

The possible values of the diagonal elements of a density matrix are restricted by the condition \mathrm{Tr}~\rho = 1 . Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious \mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}, \mathrm{Im}~\rho_{nm} = -...

I have a density matrix in one basis and need to change it to another. I know the eigenvectors and eigenvalues of the basis I want to change to. How do I do this? Any help really appreciated- thanks!

Hi, I am trying to work out the atomic inversion of the Jaynes cummings model using the density matrix. At the moment i have a 2x2 matrix having used the Von neumann equation (technically in Wigner function in x and y). Each of my matrix elements are 1st order pde's describing the...

I am unsure how to (mathematically) do the partial trace of a density matrix so that I can find the expectation value of an observable. I am working on a model similar to the Jaynes cummings model. My density matrix is of the form; \rho = [\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22}]...

Hi, I am trying to solve a modified Jayne's cummings model using the Von Neumann equation and Wigner function but am having a problem with Density matrix hermicity; I am trying to solve in Schrodinger picture. I have my system Hamiltonian as; H_{0} = \frac{1}{2}\hbar \Omega \sigma_{z}...

Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could...

How would you define density matrix for an ensemble of identical harmonic oscillators in thermal equilibrium? For example, consider N atoms in a crystalline lattice. I would like to find density matrix to calculate the average dipole moment of the ensemble and also its standard deviation...

Hi, I'm working on a modified version of the Jayne's Cummings model and am a little confussed. I have: -Taken modified version of JCM Hamiltonian in Schrodinger picture. -Used Von Neumann equation to get evolution of density matrix -Converted to Wigner function. I want to run...

Hi all, I've been reading the seminal Zurek papers on decoherence but there is one (crucial) point on which I am confused. I understand the mathematical demonstrations that over very short timescales the superpositions of states represented as off-diagonal terms in the density matrix can be...

Hello everyone, I'm having some trouble, that I was hoping someone here could assist me with. I do hope that I have started the topic in an appropriate subforum - please redirect me otherwise. Specifically, I'm having a hard time understanding the matrix elements of the density matrix...

Dear all, Could anyone please explain to me for a 3 level syste, coupled with 2 lasers, what is the equation describing the disspative term of the equation of motion of the density matrix? I am looking for a non-radiative decay, namely the overall population should conserved. I have found...

I have a quick question. I've been trying to search for an answer, but I'm probably looking in the wrong places. Is it valid to have negative off-diagonal elements in a density matrix? Thanks!

I have worked out a problem on reduced density matrix, but my final expression looks kind of weird to me. I would appreciate someone can point out where I have gone wrong. Homework Statement The problem goes like this: Consider a two spin 1/2 particles. Their Hilbert space is spanned bu...

Firstly, I have been able to find almost nothing on this kind of question in textbooks or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could...

Hi, The Quantum Liouville Equation is \dot{\rho} = \frac{i}{\hbar}[\rho, H] where the dot denotes the partial derivative with respect to time t. We take \hbar = 1 hereafter for convenience. Tr(\dot{\rho}) = 0 Consider Tr(\rho^2) Differentiating with respect to time...

I'm really excited to get this as a homework problem. I have wanted to feel good about this formalism is quantum mechanics for a while now but my own stupidity has been getting in the way... With this homework problem hopefully I can move on to a new level. Homework Statement The most...

Homework Statement What are the conditions so that the matrix \hat{p} describes the density operator of a pure state? Homework Equations http://img846.imageshack.us/img846/2835/densitymatrix.png p=\sum p_{j}|\psi_{j}><\psi_{j} The Attempt at a Solution I know that tr(\rho)=1 for...

Homework Statement Given a matrix M(a) = (a -(1/4)i ; (1/4)i a) (semicolon separates rows) a) Determine a so that M(a) is a density matrix. b) Show that the system is in a mixed state. c) Purify M(a) The Attempt at a Solution a) from conditions for a density matrices...

1. Show that an arbitrary density operator for a mixed state qubit may be written as 2. \rho = \frac{I+r^i\sigma_i}{2}, where ||r||<1 (Nielsen and Chuang pg 105) 3. So my attempt was as follows Given that a \rho is hermitian it may be written as a linear combination of the pauli...

Density matrix and von Neumann entropy -- why does basis matter? I'm very confused by why I'm unable to correctly compute the von Neumann entropy S = - \mathrm{Tr}(\rho \log_2{\rho}) for the pure state | \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2 Now, clearly the simplest...

I have a question regarding the slide: http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture3.pdf On page 18-21 it gives the proof of the theorem that | \psi_i^{~} \rangle and |\phi_{i}^{~}\rangle generate the same density matrix iff |\psi_{i}^{~}\rangle = \sum_{j} u_{ij}...

Hello, I am looking for a guide to quantum mechanics and the density matrix formalism which uses the Einstein summation convention. Does such a guide exist?

Homework Statement I have in my textbook (QM by Auletta) the example of the polarization-entangled state of a pair of photons given by: |\psi\rangle_{12}=\frac{1}{\sqrt{2}}\left(|h\rangle_1\otimes|v\rangle_2+|v\rangle_1\otimes|h\rangle_2\right) The density matrix associated to such a...

Hello. I need some help to prove the first property of the density matrix for a pure state. According to this property, the density matrix is definite positive (or semi-definite positive). I've been trying to prove it mathematically, but I can't. I need to prove that |a|^2 x |c|^2 +...

What are the conditions for some matrix to be a density matrix ? I know of these conditions: 1.) \rho=\rho^{2} 2.) Tr(\rho)=1 (for pure state) Is this all ?

This is probably my misunderstanding of the notation... The definition of a density matrix is in the attached file. (Sorry, the latex editor is not rendering properly when I preview my post). This definition is a sum over only one index 'j', which will invariably lead to a diagonal matrix...

Does anyone kown how to apply time dependent perturbation theory to densities matricies (I'm interested in first order)? Thanks.

My lecturer keeps telling me that if a density matrix describes a pure state then it must contain only one non-zero eigenvalue which is equal to one. However I can't see how this is true, particularly as I have seen a matrix \rho_A = \begin{pmatrix} 1/2 & - 1/2 \\ -1/2 & 1/2 \\ \end{pmatrix} for...

Can someone help me prove that tr(\rho^2) \leq 1 ? Using that \rho = \sum_i p_i | \psi_i \rangle \langle \psi_i | \rho^2 = \sum_i p_i^2 | \psi_i \rangle \langle \psi_i | tr(\rho^2) = \sum_{i, j} p_i^2 \langle j | \psi_i \rangle \langle \psi_i | j \rangle Where do I go from here? Thanks guys.

what do the non-diagonal components of a density matrix tell us about the mixed state we are in?