density matrix Definition and Topics - 17 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. J

    I Density matrix help

    Hi, I am wanting to confirm my understanding of the density matrix in quantum mechanics. Is it the wave function co-efficients squared - in other words the wave amplitudes squared which in turn are the probabilities and then these turn out to be placed into a matrix form with the squared wave...
  2. N

    I What is the meaning of coherent states of mean photon number

    I am studying Quantum Cryptography and I am quite new in Quantum area. I have read an article and I found this confusing statement: My questions: 1. The three stage protocol implementing multiphoton. What is the meaning of coherent states of mean photon number? 2. How to describe the quantum...
  3. AwesomeTrains

    Density matrix for a mixed neutron beam

    Homework Statement A beam of neutrons (moving along the z-direction) consists of an incoherent superposition of two beams that were initially all polarized along the x- and y-direction, respectively. Using the Pauli spin matrices: \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\...
  4. D

    I Indices of a Density Matrix

    I am reading Leonard Susskind's Theoretical Minimum book on Quantum Mechanics. Excercise 7.4 is as follows: Calculate the density matrix for ##|\Psi\rangle = \alpha|u\rangle + \beta|d\rangle##. Answer: $$ \psi(u) = \alpha, \quad \psi^*(u) = \alpha^* \\ \psi(d) = \beta, \quad \psi^*(d) =...
  5. Tspirit

    Quantum Textbook for density matrix and trace?

    I want a QM textbook which introduces detail knowledge of density matrix and trace (i.e. the average), who can recommend one for me? Thank you.
  6. G

    Expectation values as a phase space average of Wigner functions

    Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
  7. L

    Partial traces of density operators in the tensor product

    Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...
  8. G

    I Decomposing a density matrix of a mixed ensemble

    I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...
  9. P

    Density Operator to Matrix Form

    Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...
  10. V

    I Density matrix on a diagonal by blocks Hamiltonian.

    If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?
  11. D

    A Expectation values and trace over the environment

    I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...
  12. entropy1

    I Expectation value in terms of density matrix

    It says in Susskind's TM: ##\langle L \rangle = Tr \; \rho L = \sum_{a,a'}L_{a',a} \rho_{a,a'}## with ##a## the index of a basisvector, ##L## an observable and ##\rho## a density matrix. Is this correct? What about the trace in the third part of this equation?
  13. entropy1

    I Density matrices, pure states and mixed states

    I got (very) confused about the concept of states, pure states and mixed states. Is it correct that a linear combination of pure states is another pure state? Can pure (and mixed) states only be expressed in density matrices? Is a pure state expressed in a single density matrix, whereas mixed...
  14. S

    Decoherence in the long time limit of density matrix element

    For a state |\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle , the density matrix elements in the energy basis are \rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar} How is it that in the long time limit, this reduces to \rho_{ab}(t) \approx |c_a|^2 \delta_{ab} ? Is there some...
  15. K

    Density Matrix for Spin 1/2 particle in a magnetic field

    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...
  16. S

    Microcanonical ensemble density matrix

    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...
  17. M

    Density matrix in the canonical ensemble

    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...