What is Density matrix: Definition and 125 Discussions

In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.
Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information.

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  1. B

    Explore the Concept of Density Matrix with Jensa: A Comprehensive Guide

    Thank you jensa.
  2. L

    Inspection of the Density Matrix

    I have a question about density matrices. Is there a way to deduce the purity of the density matrix just by inspection? -L
  3. maverick280857

    Density matrix to represent polarization, what is this? References anyone?

    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...
  4. M

    Is the Square of a Density Matrix Equal to the Density Matrix Itself?

    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...
  5. Fredrik

    Is the determinant of a mixed state density matrix always positive?

    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...
  6. B

    Understanding Spin Density Matrix Invariance

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

    Pertubation and density matrix

    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}...
  8. K

    Spatial linewidth and density matrix

    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...
  9. H

    Density Matrix Doubt: Can Coherance Be Zero?

    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?
  10. K

    Off diagonal element of density matrix

    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| ?
  11. W

    Density Matrix in that DFT bible book

    -- 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...
  12. Y

    Finding Reduced Density Matrix for 2 Spin-Half System

    At the thermal equilibrium, the density matrix of a 2 spin-half system is given by: \begin{displaymath} \mathbf{\rho} = \left(\begin{array}{cccc} e^{-(1+c)/T} & 0 & 0 & 0\\ 0 & cosh[(1-c)/T] & -sinh[(1-c)/T] & 0\\ 0 & -sinh[(1-c)/T] & cosh[(1-c)/T] & 0\\ 0 & 0 & 0 & e^{-(1+c)/T}...
  13. Y

    Finding the Density Matrix of a 4x4 System at Thermal Equilibrium

    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...
  14. D

    Calculating Time Evolution of Density Matrix

    Hi, I am trying to calculate the time evolution of a density matrix. Like if there is a mixed state with 50% of |x, 0> and 50% of |y, 0>. After time t due to time evolution, the kets become: |x,t>= e^(-i/h Ht) |x,0> and so on. Is it ok to use these kets instead of the original ket to...
  15. D

    How Are the Coefficients of a Qubit's Density Matrix Constrained?

    What is the arbitrary density matrix of a mixed state qubit?
  16. D

    What is density matrix of one on two entangled qubits?

    Let we have two qubits A and B. First qubit has eigenstates |A0> and |A1>, and second has |B0> and |B1>. Let them be in the entangled state, described with vector c1 * |A0> * |B0> + c2 * |B0> * |B1>| where c1 and c2 are complex numbers with |c1|^2 + |c2|^2 = 1. Then what is density...
  17. pellman

    Density matrix off diagonal terms - what do they mean?

    A superposition of states such as a_1|\psi_1\rangle+...+a_n|\psi_n\rangle represents a single physical state, a state for which the probability of a measurement finding the system in state |\psi_k\rangle is |a_k|^2. The a_k represent "quantum-type" probabilities. On the other hand the...
  18. C

    Finding Density Matrix of Silver Atoms Sorted by Stern-Gerlach Devices

    Homework Statement Suppose a source emits silver atoms, which are sent through three different types of Stern-Gerlach devices, each of them is either sorting the atoms along the x, y or z axis. If we preform a measurement along the z or y axis, the atoms are sorted in the ” + ”...
  19. E

    Commutator of a density matrix and a real symmetric matix

    Let p1,p2 be two density matrices and M be a real, symmetric matrix. Now, <<p1|[M,p2]>>= <<p1|M*p2>>-<<p1|p2*M>>= Tr{p1*M*p2}-Tr{p1*p2*M}= 2i*Tr{(Im(p1|M*p2))}. Why is it that this works out as simply as (x+iy)-(x-iy)? How is Tr{p1*p2*m}=conjugate(Tr{p1*M*p2})? I can't seem to figure...
  20. W

    Density matrix for QFT from the path integral?

    (1) How does one obtain the density matrix formalism for quantum fields from the path integral? (2) Suppose I have a box containing interacting particles of different kinds. Is it possible to incorporate into the density matrix formalism both a non-zero temperature T as well as a time t...
  21. C

    Is there any approximation to the two particle density matrix

    Let phi(x) and phi_dagger(x) be field operators which satisfy the appropriate commutation relations. Then is there any analytic approximation for the two particle density matrix given by <phi_dagger(x)phi_dagger(x')phi(x')phi(x)> Thanks!
  22. S

    Solving for the Time-Dependent Vector in QM Density Matrix

    We have a spin state described by a time-dependent density matrix \rho(t) = \frac{1}{2}\left(\mathbf{1}+\mathbf{r}(t)\cdot \mathbf{\sigma} \right) Initial condition for the motion is \mathbf{r} = \mathbf{r}_0 at t = 0. We are then asked to give a general expression for \rho(t) in terms of...
  23. CarlB

    Is the density matrix a better way of describing quantum states than spinors?

    I'm starting to convince myself that the density matrix is a better way of describing a quantum state than a spinor, even in the case of pure states. But it seems like very few of my textbooks have much to say about density matrices. Any comments? Carl
  24. S

    Reduced Density Matrix: Explained for Ron

    While reading this article http://citebase.eprints.org/cgi-bin/fulltext?format=application/pdf&identifier=oai%3AarXiv.org%3Aquant-ph%2F9708045 (which is by the way very interesting) i've encountered two unknown terms: "reduced denstiy matrix" --- i have never heard this term before...
  25. K

    Finding the density matrix of an ensemble

    Hi I am just doing an undergraduate degree in physics and currently studying a course in the foundations of QM. The problem I want to solve is this: There is an ensemble in a state corresponding to vector (i, 2) A measurement of Sy (with the operator represented by the 2x2 pauli spin...
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