# What is Eigenstates: Definition and 192 Discussions

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector

(
α
,
β
)

{\displaystyle (\alpha ,\beta )}
, with a length of one; that is, with

|

α

|

2

+

|

β

|

2

=
1
,

{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}
where

|

α

|

{\displaystyle |\alpha |}
and

|

β

|

{\displaystyle |\beta |}
are the absolute values of

α

{\displaystyle \alpha }
and

β

{\displaystyle \beta }
. A mixed state, in this case, has the structure of a

2
×
2

{\displaystyle 2\times 2}
matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:

|
ψ

=

1

2

(

|

↑↓

|

↓↑

)

,

{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\big (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\big )},}
which involves superposition of joint spin states for two particles with spin 1⁄2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.

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1. ### I Eigenstates of particle with 1/2 spin (qbit)

A very basic doubt about a QM system (particle) with spin 1/2 (qbit). From the Bloch sphere representation we know that a qbit's pure state is represented by a point on the surface of the sphere. Picking a base, for instance the pair of vector/states ##\ket{\uparrow}## and ##\ket{\downarrow}##...

12. ### Operator with 3 degenerate orthonormal eigenstates

With this information I concluded that the diagonal elements of ##\hat{A}## are equal to the eigenvalue ##a##, so ##\hat{A} = \begin{bmatrix} a & A_{12} & A_{13} \\ A_{21}& a & A_{23}\\A_{31} & A_{32} & a \end{bmatrix}## but I can't see how to go from this to the commuting relation, since I...

26. ### QM: Writing time evolution as sum over energy eigenstates

Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates. I've previously shown that ##\hat{H} = \sum_j...
27. ### Eigenstates of Rashba Spin-Orbit Hamiltonian

Homework Statement I am given the Rashba Hamiltonian which describes a 2D electron gas interacting with a perpendicular electric field, of the form $$H = \frac{p^2}{2m^2} + \frac{\alpha}{\hbar}\left(p_x \sigma_y - p_y \sigma_x\right)$$ I am asked to find the energy eigenvalues and...
28. ### A Mass Eigenstates: Definite Mass States Explained

Does saying "states of definite mass" is the same as saying "mass eigenstates"?
29. R

### What are the most common eigenstates of molecules in chemistry?

What are the different eigenstates of molecules that are most often used in chemistry?
30. ### I Neutrino flavour eigenstates and expansion of the universe

Neutrinos were flavor eigenstates at the time of their decoupling from baryonic matter. Since they were not pure mass eigenstates, how do you take this fact into account if you try to study how they evolved as the universe expanded? Could we determine if the heaviest neutrino could be non...
31. ### I How are eigenstates and eigenvalues related in quantum mechanics?

Hi, I have come across two definitions of eigenstates (and eigenvalues), both of which I understand but I don't understand how the two are related: 1) An eigenstate is one where you get the original function back, usually with some multiple, which is called the eigenvalue. 2) An eigenstate...
32. ### I Examples where mixed states are eigenstates

I have actually read so much about density matrix and eigenstates today. I just want to know what particular situations when mixed states are eigenstates. Can this occur? Mixed states and eigenstates have one thing in common.. they have a value.. but I know mixed states aren't eigenstates...
33. ### I Gravitational Effect on Electron Eigenstates

As a hydrogen atom approaches a Neutron star, is the probability distribution of eigenstates of the electron in that atom influenced by the gravitational field of the star?
34. ### I What is the physical interpretation of eigenstates in quantum mechanics?

Hey everyone, I've been doing some quantum mechanics but I think I have yet to fully grasp the meaning of eigenstate. What I mean is, I understand that an eigenstate ##x## is such that, if we have an operator ##\hat{A}##, it satisfies ##\hat{A} x=\lambda x## and so ##\hat{A}## represents a...
35. ### I Are Eigenstates of operators always stationary states?

Hello everyone, I am wondering if the eigenstates of Hermitian operators, which represent possible wavefunctions representing the system, are always stationary wavefunctions, i.e. the deriving probability distribution function is always time invariant. I would think so since these eigenstates...
36. ### Pauli Spin Matrices - Lowering Operator - Eigenstates

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...
37. ### A Field operator eigenstates & Fock states(Hatfield's Sch rep)

In chapter 10 of his book "Quantum Field Theory of Point Particles and Strings", Hatfield treats what he calls the Schrodinger representation of QFT. He starts with a free scalar field and introduces field operators ## \hat \varphi(\vec x) ## and its eigenstates ## \hat \varphi(\vec...

47. ### Why Do Boundary Conditions in a 1D Quantum Box Lead to Different Quantum States?

Problem: The particle in a 1D box [0, a] Eqs.: The general solution of the time-independent Schrödinger eq. may be written as ψ(x) = Acos(kx) + Bsin(kx), E = ħ2k2/2m. Imposing the boundary conditions ψ(0) = ψ(a) = 0 , we get immediately A = 0, ka = nπ (for any positive integer n). Using x' = x...
48. ### A Asymptotic momentum eigenstates in scattering experiments

In a typical collider experiment, two particles, generally in approximate momentum eigenstates at ##t=-\infty##, are collided with each other and we measure the probability of finding particular outgoing momentum eigenstates at ##t=\infty##. Firstly, what does it mean for the particles to be in...
49. ### Commuting Operators: Understanding How M1,M2 & M3 Work Together

Homework Statement It is known that ##M_1,M_2, M_3## commute with each other but I don't see how the second line is achieved even though it says that it's using that ##M_1## and ##M_2## commute?
50. ### Eigen-energies and eigenstates of a tri-atomic system

Homework Statement An extra electron is added to one atom of a tri-atomic molecule. The electron has equal probability to jump to either of the other two atoms. (a) Find the eigen-energies for the system. Assume that the new electron energy ##\bar{E_{0}}## is close to the non-hopping case...