# What is Eigenstate: Definition and 92 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 Finding ##S_x## eigenstate using experiments

Quantum mechanics, McIntyre, pg 62 For above spin ##1## Stern Gerlach experiment a set of results is "## \begin{array}{c} \mathcal{P}_{1 x}=\left.\left.\right|_{x}\langle 1 \mid 1\rangle\right|^{2}=\frac{1}{4} \\ \mathcal{P}_{0 x}=\left.\left.\right|_{x}\langle 0 \mid...
2. ### I Collapse of wavefunction into a forbidden eigenstate for a free particle

For the free particle in QM, the energy and momentum eigenstates are not physically realizable since they are not square integrable. So in that sense a particle cannot have a definite energy or momentum. What happens during measurement of say momentum or energy ? So we measure some...
3. ### B Do eigenstate probabilities change with time?

To my understanding any quantum system can be describes as a linear combination of eigenstates or eigevectors of any hermetian operator, and that the eigen values represent the observable properties. But how does the system change with time? I suppose big systems with many particles change with...
4. ### I Determining Momentum from Wavefunction

The goal I am trying to achieve is to determine the momentum (2D) in a quantum system from the wavefunction values and the eigenergies. How would I go about this in a general manner? Any pointers to resources would be helpfull.
5. ### Probability of measuring an eigenstate of the operator L ^ 2

Calculate, with a relevant digit, the probability that the measure of the angular momentum $L ^2$ of a particle whose normalized wave function is $$\Psi(r,\theta,\varphi)=sin^2(\theta)e^{-i\varphi}f(r)$$ is strictly greater than ##12(\hbar)^2##...
6. ### I Can a quantum system with internal memory be in an energy eigenstate?

If a system is in an eigenstate of the hamiltonian operator, the state of the system varies with time only with a "j exp(w t)" phase factor. So, the system is in a "stationary state": no variation with time of observable properties. But the system could in theory (for what I understand) be...
7. ### Finding the wave function when given the momentum eigenstate

For part a, I have the following $$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$ but I am totally lost on how to proceed.
8. ### I Are Energy Eigenstate Coefficients Real-valued?

Given any system with discreet energy eigenstates, φn(x)e-iEnt . The φn are functions only of position. But are they also almost always real-valued?Thanks in advance.
9. ### I What is an Eigenstate? Simple Explanation

I found this: Eigenstate: a quantum-mechanical state corresponding to an eigenvalue of a wave equation. would you please some one explain simply? Thanks
10. ### Do Lx and Ly have a same eigenstate?

Certainly, ##\left [ A ,B \right ] \neq 0## does not mean that they do not have a same eigenstate. But how to construct a same eigenstate for ##L_x## and ##L_y## if it exists? Since ##L_x Y_l^m = \frac \hbar 2 \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} + \sqrt...
11. ### A Simultanious eigenstate of Hubbard Hamiltonian and Spin operator in tw

Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...
12. ### To prove that a coherent state is an eigenstate of the annihilation operator

The definition of coherent state $$|\phi\rangle =exp(\sum_{i}\phi_i \hat{a}^\dagger_i)|0\rangle$$ How can I show that the state is eigenstate of annihilation operator a? i.e. $$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$
13. ### Is there an uncertainty in energy measurement in quantum mechanics?

Let's say I have a system whose time evolution looks something like this: This equation tells me that if I measure energy on it, I will get either energy reading ## E_0 ## or energy reading ## E_1 ## , when I do that, the system will "collapse" into one of the energy eigenstates, ## \psi_0 ##...
14. ### Finding the Eigenstate of S2 for a Spin 1 Particle

Homework Statement I'm trying to show the Eigenstate of S2 is 2ħ^2 given the matrix representations for Sx, Sy and Sz for a spin 1 particle Homework Equations Sx = ħ/√2 * \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} Sy = ħ/√2 * \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i...
15. ### I The Mystery of Eigenstate Preference in Quantum Mechanics

If any superposition of quantum states is stable, why the preference for one of the eigenstates of the observable at the measurement? What is the attraction of such state?
16. ### 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...
17. ### Time Evolution of Spin in a Magnetic Field

Homework Statement The Hamiltonian of a spin 1/2 particle is given by: $$H=g\overrightarrow { S }\cdot \overrightarrow { B }$$ where ##\overrightarrow { S }=\hbar \overrightarrow{\sigma }/2## is the spin operator and ##\overrightarrow { B }## is an external magnetic field. 1. Determine...
18. ### A Proton anti-proton pair and CP eigenstate

p\bar{p} pair is a CP eigenstate? As p and \bar{p} are fermions (the pair is assumed to be at S-state), the pair seems to be C's eigenstate with eigenvalue of -1. As they have opposite intrinsic parity, the pair state seems to be P's eigenstate with eigenvalue -1. Then isn't it CP eigenstate...
19. ### I What is an Eigenstate in Relation to Schrodinger Equation?

What is an eigenstate in relation to the Schodinger equation? We've been working with this stuff but I don't exactly understand what that is. I know of linear algebra eigenstates or eigenfunctions but I don't know if they are directly related.
20. ### Simultaneous eigenstate of angular momentum and hamiltonian

Homework Statement The red box only Homework EquationsThe Attempt at a Solution I suppose we have to show L_3 (Π_1) | E,m> = λ (Π_1) | E,m> and H (Π_1) | E,m> = μ (Π_1) | E,m> And I guess there is something to do with the formula given? But they are in x_1 direction so what did they have...
21. ### A Can a Point-Like Particle Defy the Uncertainty Principle?

Imagine a spatial frame of reference attached to a point-like particle. It has x=0 since it is at the origin and p=0 since it is at rest. Having definite position and momentum is normally considered a violation of the uncertainty principle. How would you resolve this paradox? 1. Position frames...
22. ### I What are pure and mixed states in quantum mechanics?

what does it mean that a particle is a pure eigenstate? could someone explain this to me simply ?
23. ### Momentum Eigenstate with a non-zero potential

Homework Statement Prove that if a particle starts in a momentum eigenstate it will remain forever in a eigenstate given the potential c*y where c is a constant and y is a spatial variable. Homework Equations (h/i)d/dx is the momentum operator and a momentum eigenstate when put in the...
24. ### I Eigenstate of two observable operators

Let's say you have two operators A and B such that when they act on an eigenstate they yield a measurement of an observable quantity (so they're Hermitian). A and B do not commute, so they can't be measured simultaneously. My question is this: You have a matrix representation of A and B and...
25. ### I Why Is the Probability of Finding a Particle in a Position Eigenstate Zero?

Why is the probability of finding a particle in an eigenstate of position zero and not one? When we say we have located a particle at a particular position - why is it always in a superposition of position eigenstates about that position. But still the probability should not be zero. I need...
26. ### I Is Pointer State Eigenstate or Mixed State?

What really are Pointer States in Zurek stuff? is it an eigenstate or mixed states Zurek seems to be saying that you can reprepare Pointer States even if they are macroscopic. What can you say?
27. ### I When is a given state an eigenstate of a given operator?

How do I know if some given state is and eigenstate of some given operator?
28. ### Operator, eigenstate, small calculation

Hello :-) I have a small question for you :-) 1. Homework Statement The Operator e^{A} is definded bei the Taylor expanion e^{A} = \sum\nolimits_{n=0}^\infty \frac{A^n}{n!} . Prove that if |a \rangle is an eigenstate of A, that is if A|a\rangle = a|a\rangle, then |a\rangle is an...
29. ### A QFT and transitions between momentum states

Hi, I'm trying to learn some QFT at the moment, and I'm trying to understand how interactions/nonlinearities are handled with perturbation theory. I started by constructing a classical mechanical analogue, where I have a set of three coupled oscillators with a small nonlinearity added. The...
30. ### I Free Particle: Time dependence of expectation values Paradox

It would be really appreciated if somebody could clarify something for me: I know that stationary states are states of definite energy. But are all states of definite energy also stationary state? This question occurred to me when I considered the free particle(plane wave, not a Gaussian...
31. ### I Symmetry of Hamiltonian and eigenstates

Suppose we have an electron in a hydrogen atom that satisfies the time-independent Schrodinger equation: $$-\frac{\hbar ^{2}}{2m}\nabla ^{2}\psi - \frac{e^{2}}{4\pi \epsilon_{0}r}\psi = E\psi$$ How can it be that the Hamiltonian is spherically-symmetric when the energy eigenstate isn't? I was...
32. ### How do I find eigenstates and eigenvalues from a spin operator?

Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...
33. ### Is state an energy eigenstate of the infinite square well

Homework Statement Is state ψ(x) an energy eigenstate of the infinite square well? ψ(x) = aφ1(x) + bφ2(x) + cφ3(x) a,b, and c are constants Homework Equations Not sure... See attempt at solution. The Attempt at a Solution I have no idea how to solve, and my book does not address this type...
34. ### What's the difference between these two equations?

say we have some wavefunction |psi> and we want to find the probability of this wavefunction being in the state |q>. I get that the probability is given by P = |<q|psi>|^2 since we're projecting the wavefunction onto the basis state |q> then squaring it to give the probability density. However...
35. ### Hamiltonian and momentum operator acting on a momentum eigenstate

suppose that the momentum operator \hat p is acting on a momentum eigenstate | p \rangle such that we have the eigenvalue equation \hat p | p \rangle = p| p \rangle Now let's project \langle x | on the equation above and use the completeness relation \int | x\rangle \langle x | dx =\hat I we...
36. ### Lowest eigenstate of hopping matrix

So, I was examining the ground state of a Bose-Hubbard dimer in the negligible interaction limit, which essentially amounts to constructing and diagonalizing a two-site hopping matrix that has the form H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1}, with all other elements zero...
37. ### 'Symmetry argument' for eigenstate superposition

Homework Statement For an infinite potential well of length [0 ; L], I am asked to write the following function ##\Psi## (at t=0) as a superposition of eigenstates (##\psi_n##): $$\Psi (x, t=0)=Ax(L-x)$$ for ## 0<x<L##, and ##0## everywhere else. The attempt at a solution I have first...
38. ### Lorentz transforming a momentum eigenstate

Let's take a quantum state ##\Psi_p##, which is an eigenstate of momentum, i.e. ##\hat{P}^{\mu} \Psi_p = p^{\mu} \Psi_p##. Now, Weinberg states that if ##L(p')^{\mu}\,_{\nu}\, p^{\nu} = p'##, then ##\Psi_{p'} = N(p') U(L(p')) \Psi_{p}##, where ##N(p')## is a normalisation constant. How to...
39. ### How to Calculate Uncertainty for Spin-1/2 Eigenstates?

Homework Statement Calculate ΔSx and ΔSy for an eigenstate S^z for a spin-1/2 particle. Check to see if the uncertainty relation ΔSxΔSy ≥ ħ|<Sz>|/2 is satisfied. Homework EquationsThe Attempt at a Solution I'm confused on where to start. As I am with most of this quantum stuff. From what...
40. ### Finding eigenstate for the annhilation operator

Homework Statement Find the eigenvector of the annhilation operator a. Homework Equations a|n\rangle = \sqrt{n}|{n-1}\rangle The Attempt at a Solution Try to show this for an arbitrary wavefunction: |V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle...
41. ### Eigenstate solution for a spin chain with Hubbard model

Homework Statement I am trying to solve the model analitically just for 2 sites to have a comparison between computational results. The problem is my professor keeps saying that the result should be a singlet ground state and a triplet of excited states, but when I compute it explicitally I...
42. ### Quantum state of system before measurement

Hello! If we consider a single-particle system, I understand that the measurement of an observable on this system will collapse the wave function of the system onto an eigenstate of the (observable) operator. Therefore, we know the state of the system immediately after the measurement. But as...
43. ### Uncertainty Relation for Eigenstate of Spin-1/2 Particle

Homework Statement Calculate ##\triangle S_x## and ##\triangle S_y## for an eigenstate of ##\hat{S}_z## for a spin##-\frac12## particle. Check to see if the uncertainty relation ##\triangle S_x\triangle S_y\ge \hbar|\langle S_z\rangle|/2## is satisfied. Homework Equations ##S_x =\frac12(S_+...
44. ### Checking if state is an eigenstate

Homework Statement For the three-dimensional harmonic oscillator H_{xyz} = \frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}+\frac{1}{2}m \omega^2 x^2 + \frac{1}{2}m\omega^2 z^2 + \frac{1}{2}m\omega^2 z^2 Consider: | \alpha_1 > = \frac{1}{\sqrt{2}} (|n_x = 0, n_y = 0, n_z = 0> + |n_x = 0...
45. ### Probability of finding a system an eigenstate

Homework Statement As the homework problem is written exactly: Consider the quantum mechanical system with only two stationary states |1> and |2> and energies E0 and 3E0, respectively. At t=0, the system is in the ground state and a constant perturbation <1|V|2>=<2|V|1>=E0 is switched on...
46. ### Commutator expectation value in an Eigenstate

Hi, suppose that the operators $$\hat{A}$$ and $$\hat{B}$$ are Hermitean operators which do not commute corresponding to observables. Suppose further that $$\left|A\right>$$ is an Eigenstate of $$A$$ with eigenvalue a. Therefore, isn't the expectation value of the commutator in the eigenstate...
47. ### Finding the Eigenstate for Harmonic Oscillator?

Homework Statement Given the a|n> = α|n-1>, show that α = √n : Homework Equations The Attempt at a Solution <n|a^{+}\hat {a}|n> = \alpha <n|a^{+}|n-1> = | \hat a|n>|^2 \alpha = \frac{<n|a^{+}\hat {a}|n>}{<n|a^{+}|n-1>} Taking the complex conjugate of both sides: \alpha* =...
48. ### Eigenstate and real space representation

Hi All, I was going through a paper on quantum simulations. Below is an extract from the paper; I would be obliged if anyone can help me to understand it: We will use eigenstate representation for transverse direction(HT) and real space for longitudinal direction(HL) Hamiltonians. HL=...
49. ### Is there a frequency eigenstate for light?

I thought it was the coherent state, but since that is an eigenstate of the annihilation operator, and the annihilation operator is not hermitian, then it has no corresponding observable, and I'm assuming that one can observe frequency. Thanks.
50. ### Hydrogen atom eigenstate: proving orthogonality of states

Homework Statement Hey dudes So here's the question: Consider the first excited Hydrogen atom eigenstate eigenstate \psi_{2,1,1}=R_{2,1}(r)Y_{11}(\theta, \phi) with Y_{11}≈e^{i\phi}sin(\theta). You may assume that Y_{11} is correctly normalized. (a)Show that \psi_{2,1,1} is orthogonal...