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


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











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




{\displaystyle |\alpha |}




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


{\displaystyle \alpha }


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


{\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:












{\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. cianfa72

    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}##...
  2. K

    Quantum Well Centred at the Origin Doubled in Size at time t=0. Symmetry seems to render the question incorrect

    Firstly I have found the eigenstates for both the original well and the new well as the following $$\psi_{n,\frac{L}{2}} = \begin{cases} \sqrt{\frac{2}{L}} \cos{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ odd} \\ \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ even}...
  3. strangerep

    QM Eigenstates and the Notion of Motion

    I'll risk a quick off-topic answer here, since I think it's straightforward QM, not vague "interpretation" stuff. :oldbiggrin: In QM (e.g., Ballentine p81), for a free particle, ##H = \frac12 \, M\, V\cdot V + E_0##. So in the ground state ##|E_0\rangle## we have ##H|E_0\rangle = E_0...
  4. snoopies622

    B Are electrons in atoms always in eigenstates?

    Going back to high school chemistry, i remember being taught that the electrons in an atom can each be identified with four quantum numbers - one for energy, two for angular momentum and one for spin. These numbers are integers except for the spin quantum numbers, which are either 1/2 or -1/2...
  5. A

    I "Different" energy eigenstates - clarification of meaning?

    Apologies for an additional thread (could not delete the previous one which was not coherent). After reading this paper: https://link.springer.com/article/10.1007/s10701-021-00464-7 "Fast Vacuum Fluctuations and the Emergence of Quantum Mechanics" Gerard ’t Hooft I was struck by a general...
  6. P

    Quantum exam practice, operators and eigenstates

    I'm really not sure what the question expects me to do here but here is what I do know. If the state is an eigenstate it should satisfy the eigenvalue equation for example; $$\hat{H} f_m^l = \lambda f_m^l$$ but is the question asking me to use each operator on each state? How do I know if...
  7. S

    I Multiple questions about eigenstates and eigenvalues

    I have multiple questions about eigenstates and eigenvalues. The Hilbert space is spanned by independent bases. The textbook said that the eigenvectors of observable spans the Hilbert space. Here comes the question. Do the eigenvectors of multiple observables span the same Hilber space? Here...
  8. Kaguro

    A Degenerate Perturbation Theory: Correction to the eigenstates

    Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate: $$ H^0 \ket a = E^0 \ket a$$ $$ H^0 \ket b = E^0 \ket b$$ Let's make them...
  9. F

    I Momentum/angle values of eigenstates in a graphene system

    I am trying to differentiate between states in a quantum system of graphene with open boundaries in one direction and periodic boundaries in the other direction. The eigenstates have been simulated with the eigenenergies, but how would I proceed in calculating the expectation value of the...
  10. docnet

    A Relationship between Larmor precession and energy eigenstates

    Quantum mechanically, a spin 1/2 particle in a uniform magnetic field has two energy eigenstates ##\ket{up}## and ##\ket{down}## and rotational degrees of freedom (distinct from the energy eigenstates) about the axis of the magnetic field. this can be derived from the Pauli matrix commonly...
  11. S

    I Why is this 3D operator with degeneracies only giving me 2 eigenstates

    The matrix representation of a certain operator in a certain basis is $$\begin{bmatrix} 1 & 0 & 0 \\0 & 0 & -i \\ 0 & i & 0 \end{bmatrix} .$$ The eigenvalue problem leads to this equation $$0=det\begin{bmatrix} 1-\lambda & 0 & 0 \\0 & -\lambda & -i \\ 0 & i & -\lambda \end{bmatrix}...
  12. Celso

    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...
  13. M

    The propagator of eigenstates of the Total Angular Momentum

    To show that when ##[J^2, H]=0 ## the propagator vanishes unless ##j_1 = j_2## , I did (##\hbar =1##) $$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$ $$ = e^{iHt}[jm_H - jm] $$ So we have $$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$ $$ =...
  14. N

    I Parity Eigenstates: X Basis Explanation

    On page 298 of Shankar's 'Principles of Quantum Mechanics' the author makes the statement : ""In an arbitrary ##\Omega## basis, ##\psi(\omega)## need not be even or odd, even if ##| \psi \rangle ## is a parity eigenstate. "" Can anyone show me how this is the case when in the X basis...
  15. L

    I Hydrogen atom: Energies and eigenstates

    When we say energy levels of the hydrogen atom. Are that energies of the atom or of an electron in the atom? Also corresponding states? http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html Why energies are negative? E_n \propto \frac{-1}{n^2}
  16. H

    A Eigenstates of ##\phi^4## theory

    What is known about the eigenstates of the ##\phi^4## theory in QFT? Is there an informal understanding of how these states are organized in the nonperturbative regime? For example, are there known to be any bound states in any dimensions? How does the energy of a multiparticle eigenstate (if...
  17. I

    Why can we always choose energy eigenstates to be purely real function

    I couldn't quite answer, so looked at the solution. I just want to ensure I am undertsanding the answer correctly. The answer is given here on page 3. Q2a: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps4_sol.pdf Am I right in concluding that...
  18. Zala Jaydevsinh

    I Momentum Eigenstates: Writing & Conservation

    Momentum eigenstates can be written in form of e^(2*pi*x) how?? and also i have question how momentum is conserved as consequences of periodicity of wave function.
  19. J

    I Finding Normalization Constants for a Set of Energy Eigenstates

    I do not know what I'm doing wrong but I'm working on the problem of finding the normalization constants for the energy eigenstate equation for a 1D plane wave that is traveling from the left into a potential barrier where E < V at the barrier. This is from Allan Adams' Lecture 12 of his 2013...
  20. Z

    Eigenstates of a free electron in a uniform magnetic field

    I started with the first of the relevant equations, replacing the p with the operator -iħ∇ and expanding the squared term to yield: H = (-ħ^2 / 2m)∇^2 + (iqħ/m)A·∇ + (q^2 / 2m)A^2 + qV But since A = (1/2)B x r (iqħ/m)A·∇ = (iqħ / 2m)(r x ∇)·B = -(q / 2m)L·B = -(qB_0 / 2m)L_z and A^2 =...
  21. J

    I Eigenstates and repeated measurements

    A measurement X collapses the wave function randomly into an eigenstate of X. Then if a different measurement Y is made the wave function will randomly collapse into an eigenstate of Y. So for example if you measure position, the wave function will collapse into a narrow peak. Now if you measure...
  22. P

    Verify Eigenstates: Solving s_i & m_i Equations

    I simply use the equation above, and the eigenvalus whish yield: ##\hbar^2 [ s_1(s_1+1) + s_2(s_2+1) + m_1m_2 + \sqrt{s_1(s_1+1) - m_1(m_1+1)}\sqrt{s_2(s_2+1) - m_2(m_2-1)} + \sqrt{s_2(s_2+1) - m_2(m_2+1)}\sqrt{s_1(s_1+1) - m_1(m_1-1)}## Very straight forward. My issue is that I don't know...
  23. J

    Position wave function of energy eigenstates in 1D box

    Homework Statement Consider a particle which is confined in a one-dimensional box of size L, so that the position space wave function ψ(x) has to vanish at x = 0 and x = L. The energy operator is H = p2/2m + V (x), where the potential is V (x) = 0 for 0 < x < L, and V (x) = ∞ otherwise. Find...
  24. DaveC426913

    Eigenstates, Block equations, Regge-theories

    PF has gotten so far over my head, I keep expecting to see this: :frown:
  25. CharlieCW

    Eigenstates of helicity operator

    Homework Statement For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form $$L(p)=exp(i\phi J^{(21)})exp(i\theta...
  26. M

    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. K

    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

    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. C

    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. J

    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. O

    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. Jim Lundquist

    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. G

    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. F

    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. C

    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. ShayanJ

    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...
  38. S

    I Phase factors of eigenstates in time-dependent Hamiltonians

    For a time-dependent Hamiltonian, the Schrodinger equation is given by $$i\hbar\frac{\partial}{\partial t}|\alpha;t\rangle=H(t)|\alpha;t\rangle,$$ where the physical time-dependent state ##|\alpha;t\rangle## is given by $$|\alpha;t\rangle =...
  39. H

    I The Wavefunction and eigenstates

    Suppose I want to measure the momentum of a quantum system. What I do is I take the momentum operator and expand my wavefunction in term of the eigenfunctions of that operator, then I operate on the wavefunction with the operator and the reusult of the measurment is that the wavefunction...
  40. B

    B Can Everett Branches Break Quantum Laws?

    Are Everett branches (or relative states) the eigenstates or the Hilbert subspaces (or others?)? Once in a branch (or world), what law of QM would be broken if you can cut off the branch you are sitting on and revert back to the global state vector (isn't the quantum eraser, etc. about...
  41. H

    Eigenfunctions, eigenstates and eigenvalues

    Homework Statement The problem states consider A_hat=exp(b*(d/dx)). Then says ψ(x) is an eigenstate of A_hat with eigenvalue λ, then what kind of x dependence does the function ψ(x) have as x increases by b,2b,...? Homework EquationsThe Attempt at a Solution Started out by doing...
  42. Philethan

    Prove the energy eigenstates are degenerate

    Homework Statement Two observables ##A_{1}## and ##A_{2}## which do not involve time explicitly, are known not to commute, ## [A_{1},A_{2}]\neq0, ## yet we also know that ##A_{1}## and ##A_{2}## both commute with the Hamiltonian: ## [A_{1},H]=0\text{, }[A_{2},H]=0. ## Prove that the energy...
  43. E

    I Eigenstates of two Coupled Harmonic Oscillator

    Hello everyone! For my quantum mechanics class I have to study the problem of two quantum oscillator coupled to each other and in particular to find the eigenstates and eigenergies for a subspace of the Fock space. I know that, in general, to solve this kind of problem I have to diagonalize the...
  44. tomdodd4598

    I Physical eigenstates of systems of n particles of spins sᵢ?

    I am relatively well versed when it comes to systems of spin, or doing the maths for them at least, but am unsure whether all of the {L2, Lz, (other required quantum numbers)} basis eigenstates for a general system of n particles of spins si, where si is the spin of the ith particle, can...
  45. A

    I Field eigenstates and path integral formulation

    Recently I have reviewed by reference books to get a better understanding of the fundamentals of QFT and there is one thing I still do not understand. In the QFT derivation of the path integral formula, it seems that every book and online resource makes the assumption that for the field operator...
  46. Xico Sim

    I Conservation of strangeness and eigenstates

    Hi, guys. In Povh's book, page 198, he says: "The strong force conserves the strangeness S and so the neutral kaons are in an eigenstate of the strong interaction." I do not see why this must be the case. My atempt to understand it: $$ŜĤ_s |K_0 \rangle = Ĥ_sŜ |K_0 \rangle$$ So $$Ŝ(Ĥ_s |K_0...
  47. itssilva

    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. S

    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. Raptor112

    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. R

    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...