# What is Density operator: Definition and 41 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. ### A Statistical ensemble interpretation done right

Concerning statistical ensemble interpretation (SEI), a lot of confusion has been created and lot of nonsense has been said, especially in a recent thread on "missed opportunities in Bohmian mechanics". Even Ballentine himself said many things about it that do not always seem perfectly clear and...
2. ### I Derivation of two-electron density operator

Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
3. ### I Density Operators of Pure States

Quantum states are most often described by the wavefunction ,##\Psi##. Variable ,##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)## defines probability density function of the system. Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is...
4. ### I The interpretation of probability

I am looking for a way to compare the handling of probability in QT with how it's done in classic PT (probability theory) - and their interpretations. QT does have it's own formalism that works, so there isn't much motivation to bring it into a usual representation which makes it hard to find...
5. ### I Derivatives for a density operator

Hi. Suppose I have a state ##\left | \psi (0)\right >=\sum_m C_m \left | m\right >## evolving as $$\left | \psi (0+dz)\right>=\left | \psi (0)\right >+dz \sum_iD_i\left | i\right >=\sum_m C_m \left | m\right >+dz \sum_iD_i\left | i\right >=\sum_m( C_m+dz D_m)\left |m\right >.$$ Then the density...

27. ### Complex coefficents in density operator expansion?

Hey, I recently had an exam where the quantum state were on the form |\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle ) Here I formed the density operator for the pure state \rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle...
28. ### The definition of the density operator in Pathria

Hello Everybody, I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator: \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, where N is the number of systems in the ensemble and the...
29. ### Trace and its square of mixed state density operator using integral

Homework Statement I want to show that tr\left(\hat{\rho}_{mixed}\right)=1 tr\left(\hat{\rho}_{mixed}^{2}\right)<1 when \hat{\rho}_{mixed}=\frac{1}{2\pi}\int_{0}^{2\pi}d \alpha \hat{\rho}(\psi) Homework Equations tr\left(\psi\right)= \sum_{n}\langle n|\psi|n\rangle...
30. ### Density operator for one part of a two-party state

Homework Statement Let |\psi\rangle_{AB}=\sum_{i}\sum_{j}c_{ij}|\varphi_{i}\rangle_{A}\otimes|\psi_{j}\rangle_{B} be a normalized two party state, being \{|{\varphi_{i}}\rangle_{A}\} and \{|{\psi_{j}}\rangle_{B}\} basis of H_{A} and H_{B} respectively, with dimensions N and M. Find \rho_{A}...
31. ### Commutator of density operator with kinetic energy operator

Hey guys, maybe you can help me with the following problem. I have to calculate the commutator relations in position representation: a) [V,ρ] b) [p,ρ] c) [p^2,ρ] Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator I already solved the first one. You just have to...
32. ### Entropy as a function of a density operator?

I've got this problem about a harmonic oscillator with energy eigenstates |n>, which is prepared in the state |\psi> = \frac{1}{cosh r}\sum_n tanh^nr|n\rangle where r is any real number and the sum is from n=0 to infinity. I'm asked to calculate the entropy S(\rho) = -Tr(\rho ln\rho) if...
33. ### Understanding the Two Types of Density Operators in Quantum Mechanics

I'm confused about the two density operators: \rho=\sum_{i}\delta(r-r_{i}) and \rho=\sum_{i}|\psi_{i}>\rho_{ii}<\psi_{i}| Is there anyone explaining this question to me? Thanks very much.
34. ### Construct Density Operator from Ensemble Average of Sx, Sy and Sz

Homework Statement I have been given a problem. The density matrix can be constructed if the ensemble average of Sx, Sy and Sz are given. But I have no idea on how to construct the density matrix from these Si's. Any help is most welcome. Homework Equations Ensemble...
35. ### Microcanonical ensemble, density operator

hi, usually the density operator for the microcanonical ensemble is given by \rho = \sum_n p_n|n><n| where |n> are energy eigenstates and p_n is the probability that our system is in this state. p_n = const. if the energy corresponding to |n> is in the energy inteval (E,E+∆E)...
36. ### Commutator of the density operator

Hello all! I hope some of you are more proficient in juggling with bra-kets... I am wondering if/when the density operator commutes with other operators, especially with unitaries and observables. 1. My guess is, that it commutes with unitaries, but I am not sure if my thinking is correct...
37. ### Why Must the Density Operator Be Hermitian? Exploring Its Physical Significance

Hi there, In all text of QM I have, they tells that the density operator is hermitian. But without considering the math, from the physics base, why density operator must be hermitian? What's the physical significane of the eigenvalue of density matrix? Thanks
38. ### Density Operator: Probability of Transition |0,e> to |1,g>

Assume there is a two level system, two eigenstates are written as |\psi_1\rangle = \cos\theta |1, g\rangle + \sin\theta |0, e\rangle and |\psi_2\rangle = -\sin\theta |1, g\rangle + \cos\theta |0, e\rangle For the density operator of the system is written as \rho =...
39. ### Question about number operator and density operator

In quantum harmonic oscillator, we define the so called number operator as \hat{N} = \hat{a}^\dagger\hat{a} Apply \hat{N} to the state with n number of particles, it gives \hat{N}|n\rangle = n |n\rangle so \langle n| \hat{N}|n\rangle = \langle n| n |n\rangle = n But in other...
40. ### Questions about the Density Operator Formulation of Quantum Mechanics

so I've been reading about the density operator formulation of quantum mechanics and I have some questions what is the density operator analog of the schrodinger equation that determines the time evolution? and how do you perform a projection measurement on a quantum system in the density...
41. P

### Density Operator for 2 Systems: Pure State

We are given that 2 systems can only be found in the states |00\rangle, |01\rangle, |10\rangle, |11\rangle. We are also given that the density operator is \rho=\frac{1}{2}\left(|00\rangle \langle 00|+|11\rangle \langle 00|+|00\rangle \langle 11|+|11\rangle \langle 11|\right). a)Write the...