Density matrix Definition and 111 Threads

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...

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...

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}...

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 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...

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!

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?

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?

Thank you jensa.

I have a question about density matrices. Is there a way to deduce the purity of the density matrix just by inspection? -L

Hi, I have a particle physics exam tomorrow morning (in a few hours from now, in my time zone). I'm trying to figure out the whole reasoning behind pion-nucleon scattering. Please bear with me.. We write the scattering matrix as S = 1 - iT where T is given by T = f + i g...

Homework Statement to prove : square of density matrix= the density matrix itself (for a pure ensemble) Homework Equations density matrix=sum over P(i) ket(i) bra(i) where Pi = probability that random chosen system from ensemble shows state i. summed over i , where P=1 for pure ensemble The...

Why is the determinant of a mixed state density matrix always positive? In the specific case of a 2-dimensional Hilbert space, the density matrix (as well as any other hermitian matrix) can be expressed as \rho=\frac 1 2 (I+\vec r\cdot\vec \sigma) so its determinant is...

Could anyone help me to understand how the spin density matrix is invariant under unitary transformation?

Hi there, I am reading a text by Robert W. Boyd "Nonlinear optics", in page 228, he used pertubation theory on two-level system and let the steady-state solution of the dynamics equation of density matrix as w = w_0 + w_1 e^{-i\delta t} + w_{-1}e^{i\delta t} where w=\rho_{bb} - \rho_{aa}...

Hi there, I am thinking an interesting problem of spatial linewidth of two-level system. Suppose in some way I find out an element of the desinty matrix for the upper state of two-level system, \rho_{ee} and it turns out that \rho_{ee} is a function of a parameter G, which could be space...

In a density matrix, can some coherances (off diagonal terms) be zero while the diagonal terms(populations) aren't? I am confused because rhoij=Ci*Cj'. How can coherance be zero if Ci and Cj aren't?

For two level system , let denotes the ground state as 1 and exctied state as 2, for writing the office off-diagonal matrix element for the density operator, shall it be \rho_{12} = |2\rangle\langle 1| and \rho_{21} = |1\rangle\langle 2| ?

-- i know there were threads about reduced density matrix in this forum, but I am reading "Density-functional theory of atoms and molecules" by Parr R., Yang W., their notation is quite confusing to me... their notation is the same as shown in this page...

How to obtain the density matrix of the following system at thermal equilibrium? Given: Hamiltonian H :(in 4x4 matrix form) Hij = the i-th row and j-th column element of H H11 = (1+c)/2 H22 = -(1+c)/2 H23 = 1-c H32 = 1-c H33 = -(1+c)/2 H44 = (1+c)/2 where c is a parameter and all...