What is observables: Definition and 115 Discussions

In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.

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  1. D

    Set of Commuting Observables for pures states 2p-1,2px and 2s

    Hi, Here I have a question, apparently easy, but that I think it is a bit tricky. Homework Statement Indicate how can a hydrogen atom be prepared in the pure states corresponding to the state vectors ψ2p-1 and ψ2px and ψ2s. It is assumed that spin-related observables are not...
  2. M

    Observables and common eigenvectors

    Homework Statement In a given basis, the eigenvectors A and B are represented by the following matrices: A = [ 1 0 0 ] B = [ 2 0 0 ] [ 0 -1 0] [ 0 0 -2i ] [ 0 0 -1] [ 0 2i 0 ] What are A and B's eigenvalues? Determine [A, B]. Obtain a set...
  3. snoopies622

    Finding the operators for time derivatives of observables

    Looking through this matrix approach to the quantum harmonic oscillator, http://blogs.physics.unsw.edu.au/jcb/wp-content/uploads/2011/08/Oscillator.pdf especially the equations m \hat{ \ddot { x } } = \hat { \dot {p} } = \frac {i}{\hbar} [ \hat {H} , \hat {p} ] I'm getting the impression...
  4. naima

    Can hidden variable theories assign values to observables?

    hi PF I read the no-go Kochen theorem using 18 vectors projection (here). One attribute a value to an observable A: v(A) This function is supposed to be linear and verify v(AB) = v(A) v(B) so v(AB - BA) = v(A)v(B) - v(B)v(A) = 0 (v is a real number function) v must assign v(id) = 1 because v(A...
  5. B

    Simultaneous observables for hydrogen

    Homework Statement Is there a state that has definite non-zero values of E, L^2 and L_xHomework Equations L^2 and L_z commute with the Hamiltonian so we can find eigenfunctions for theseThe Attempt at a Solution I would say that there is a state with simultaneous eigenfunctions of L_x,L_y,L_z...
  6. Q

    Quantum Mechanics and Observables

    Hi guys! I have one question! I know that in quantum mechanics, observables must be in the form of operators. However, does that mean that all observables are quantized in quantum mechanics?
  7. M

    Relation between actual measurement and Mathematical observables

    I'm having a gap in understanding the relation between them, and resolving my confusion is really appreciated. For example, the Hamiltonian operator, why do we call its eigenvalues energies? how do we actually measure it in the laboratory, quantum mechanically? And maybe I need a better...
  8. S

    If decoherence matters in 2 different observables

    ¿Can´t matter that decoherence matters for 2 distinct observables simultaneously ¿ Maybe the measurement apparatus must been defined clasically in this case
  9. E

    Physical observables, locality, and a preferred basis

    Quantum mechanics says that physical observables are self-adjoint operators. Is this correspondence a bijection, ie can we realize any such operator as a physical observable? There are obvious practical concerns with physically realizing certain contrived operators. But are there any...
  10. S

    Observables, Measurements and all that

    Hi Folks, I somehow cannot get the difference and have to admit that I am left confused. For a probability of measuring m with the operator M on state \Psi_i p(m|i) = <\Psi_i| M^{+}_m M_m |\Psi_i> = <\Psi_i| M_m |\Psi_i>. The average of an observable is defined as <O> = <\Psi_i| O...
  11. L

    Good set of commuting observables?

    Homework Statement I keep seeing this crop up throughout my QM course but i still don't understand what a "good" set of commuting observables would be. . Surely any set of observables that commute have to be a good set? I may just be stating the obvious but the way its phrased it makes me feel...
  12. atyy

    Relational Observables in LQG: Gauge Invariance and Incorporation of Matter

    It's often said that gauge invariant local observables in quantum gravity must be relational. In classical gravity, relational observables are constructed with matter. LQG for the most part has not had matter, yet it has been said to have observables such as area or volume. Are these...
  13. A

    Why most observables have real eigenvalues

    I have always been quite confused about the fact that any measurement MUST yield a real number. What says it must so? Don't we modify our measurement apparatus to yield something which is consistent with the theory. So coulnd't we just imagine having complex values for momentum and position. All...
  14. E

    How do you know when a set of observables form a CSCO?

    I understand that the operators have to commute, and therefore that the measurement of one has no bearing on the measurement of the other. I know that H, L^2, and L_z form a CSCO for the H atom. Basically, I conceptually understand CSCO. Often though, in my class, we will be working on a...
  15. C

    Observables commute and time operator

    I just have two questions relating to what I have been studying recently. 1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically). 2) One form of the HUP says that...
  16. G

    How Many Base States Should Incompatible Observable B Have?

    Homework Statement Say I have two incompatible observables A and B. A has a finite number of base states say 4. How many base states should B have? Homework Equations Heisenberg's uncertainty AB - BA <> 0 The Attempt at a Solution I guess the answer is 4 as A and B basis states...
  17. C

    Dirac Notation, Observables, and Eigenvalues, OH MY

    Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet. Okay, so everyone knows that <ψ|Aψ> = <a>...
  18. S

    Quantum Physics: observables, eigenstates and probability

    Homework Statement Observable \widehat{A} has eigenvalues \pm1 with corresponding eigenfunctions u_{+} and u_{-}. Observable \widehat{B} has eigenvalues \pm1 with corresponding eigenfunctions v_{+} and v_{-}. The eigenfunctions are related by: v_{+} = (u_{+} + u_{-})/\sqrt{2} v_{-} =...
  19. C

    Understanding the Deterministic Evolution of Wave Functions in Quantum Mechanics

    One of the postulates of QM is that if the system is isolated from external interaction that its wave function will evolve deterministically. So just the measurements of observables are not deterministic. What is our reason for assuming that the wave function will evolve deterministically?
  20. T

    Measuring Observables in 3+1 Formalism

    Hello, lets say I have Hamiltonian aproximation H(\vec{x}_a,\vec{p}_a) of gravitational interaction that can be used for n-body simulation of mass particles and photons. Spacetime curved by simulated particles would be asymptoticly flat. But I don't have a metric etc. All I have is evolution...
  21. S

    Why don't all observables commute in QM?

    In classical physics, all observables commute and the commutator would be zero. However this is not true in Quantum Mechanics, observables like position and momentum (time and frequency/energy) don't commute. Why? Is it because the (probability) wave functions/forms of position and momentum...
  22. B

    Transformation of observables by permutation

    Hi. Consider two isomorphic state spaces \mathcal{E}(1) and \mathcal{E}(2). The first belongs to a proton, the second to an electron and they both have the same spin. #Let B(1) be an observable defined on the first space, spanned by |1,u_{i}\rangle, eigenvectors of B(1) with eigenvalues b_i...
  23. C

    Question about measuring physcial observables

    So my measurement of a quantum system is an eigenvalue of that operator. And we are not able to predict what eigenvalue we will get, even if we knew the precise state vector before we make our measurement. But QM allows us to calculate the average of all these eigenvalues, if we made an...
  24. J

    Observables and Commutation (newbie questions?)

    Some questions. Am I getting this basically right? What does a "state vector" look like? It looks like |α> or |β> But more than that... It is a complex vector in Hilbert space? Now, you get "observables" from state-vectors by performing operators on them. So the state-vector...
  25. H

    All states are stationary, all observables are constant.

    The following lecture from University of Oxford contains an explanation of the constancy of probability distributions for all observables when a system is in a stationary state: http://www.youtube.com/watch?v=0yvX4jhzblY#t=15m35s. However, the derivation of the vanishing amplitude does not...
  26. F

    Eigenvalues and eigenvectors of observables

    Homework Statement Calculate the Eigenvalues and eigenvectors of H= 1/2 h Ω ( ]0><1[ + ]1><0[ ) Homework Equations I know H]λ> = λ]λ> The Attempt at a Solution I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?
  27. G

    Eigenvalues of commuting observables (angular momentum)

    Homework Statement Is z|lm\rangle an eigenstate of L^{2} ? If so, find the eigenvalue.Homework Equations L_{z}|lm\rangle = \hbar m|lm\rangle L^{2}|lm\rangle = \hbar^{2} l(l+1)|lm\rangleThe Attempt at a Solution So since L_{z} and L^{2} are commuting observables, they have are...
  28. U

    Do the observables for d/dx and x^2 commute?

    Homework Statement Are the observables corresponding to the d/dx and x^2 operators complementary? Homework Equations none The Attempt at a Solution I know that if the operators do not commute then their corresponding observables are complementary. I just don't know how to show...
  29. I

    Why does gravity forbid local observables?

    I heard in a conference that gravity forbids to construct local gauge invariants like Tr-\frac14 F^{\mu\nu}_aF_{\mu\nu}^a and only allows non-local gauge invariant quantities like Wilson Loops: Tr P e^{\oint_{\gamma} A_a dx^a}. Could someone explain me where does it come from? I have a basis...
  30. L

    Why are observables represented by operators in Hilbert space?

    i have been trying to learn a bit of quantum mechanics,this is some thing that has been bothering me , if the states of a system can be expressed as vectors in the Hilbert space,what is the motivation behind saying that physical observables can be given by operators?even then ,how can we say...
  31. M

    Maximal set of commuting observables

    Hi, I am having a little trouble with the concept of finding out the maximal set of commuting observables. Suppose I have n commuting operators. Then the wavefunction I use must have n parameters also. For instance, L_{3}, L^{2} and H where H is the Hamiltonian and L is the angular momentum...
  32. V

    Only Observables can be in Superposition?

    Hi, is it true that only Observables can be in Superposition? Meaning superposition of dead and live cats is invalid by some unknown mathematical reasoning? Jambaugh stated thus in the other forum that only observables can be in superposition. What's the mathematical or quantum logical proof...
  33. W

    Subsequent Measurements of two observables, compatible and incompatible pairs

    Homework Statement On an arbitrary state, the observable \hat{A} is measured returning the result a. A compatible observable \hat{B} is then measured returning b. If \hat{A} is then measured again, is the same result a obtained? How about if \hat{A} and \hat{B} are not compatible...
  34. A. Neumaier

    Boundedness of quantum observables?

    I don't like the C^*-algebraic foundations of quantum mechnaics since it assumes that every observable must be bounded and self-adjoint. But most physical observables are not bounded. This came up in another thread, from which I quote some context: You are only missing implicitly...
  35. D

    "Understanding Commuting Observables Proof

    Homework Statement In the proof that two observables \hat{O} and \hat{O}' commute iff they admit a common basis of eigenvectors, I'm not understanding one part. Homework Equations If {|a_k\rangle} is basis in Hilbert space we have: (OO')_{jk}=\langle...
  36. Z

    Associativity problem with observables

    let be A , B and C three non-commuting observables my question is how can one solve the problem with associative property ? i mean (AxB)xC will in general be different from Ax(BxC) and if we had 4 A, B ,C ,D instead of three the problem is even worse , how can anyone deal with it ...
  37. F

    Questions on Quantum Mechanics: Observables, State Functions & More

    some questions... => How are observables related to operators in quantum mechanics? => what is the physical significance of state funtion in quantum mechanics? => why are hermition operators associated with observables in quantum mechanics? => what is the physical interpretation of J...
  38. I

    Physical observables determined by quantum numbers: n, l, ml, ms

    exam practise question: what are the physical observables determined by the quantum numbers n, l, ml and ms of the electron in a hydrogen atom. most places just give the equation or the name not the physical observable determined. so is this right?:- n = principal number determines the...
  39. A

    Do Commuting Observables Imply Commuting Projectors?

    Homework Statement Consider two observables A and B such that [A,B]=0. Given the spectral resolutions of each operator: A = \sum_k a_k P_k B= \sum_j b_j Q_j where P_k and Q_j are projectors onto the eigenstates of their respective operator. Show that [P_k,Q_j]=0 for every k and jThe...
  40. Fredrik

    Understanding Observables: Exploring AB Correspondence in Measuring Devices

    I have to ask a question that looks very simple, and perhaps is very simple, but for some reason I can't answer it in a way that I'm satisfied with. If A and B are bounded self-adjoint operators that correspond to two different equivalence classes of measuring devices in the real world, and...
  41. F

    Degeneracy and commuting observables

    Homework Statement Theorem 5 of a text I've been reading that I downloaded from online (for interested parties, the link (a pdf) is http://bohr.physics.berkeley.edu/classes/221/0708/notes/hilbert.pdf) says that "If two observables A and B commute, [A, B] = 0, then any nondegenerate eigenket...
  42. C

    Is it Possible to Measure Q or P Without Determining x1 and x2 or p1 and p2?

    Question about measuring observables. If have 2 particle system the particle seperation Q=x1-x2 and total momentum P=p1+p2 are observables of the system as a whole and are commuting. How do you measure these observables. It would seem the only way to measure the separation is to measure the...
  43. B

    Energy Eigenstates and Degeneracy: Proving Non-Commuting Observables

    Homework Statement Two observables A_1 and A_2, which do not involve time explicitly, are known not to commute, \left[A_1,A_2\right]\ne 0 Yet they both commute with the Hamiltonian: \left[A_1,H\right]=0 \left[A_2,H\right]=0 Prove that they energy eigenstates are, in general, degenerate...
  44. N

    Hamiltonian problem: observables

    Homework Statement A particle that moves in 3 dimensions has that Hamiltonian H=p^2/2m+\alpha*(x^2+y^2+z^2)+\gamma*z where \alpha and \gamma are real nonzero constant numbers. a) For each of the following observables , state whether or why the observable is conserved: parity , \Pi; energy...
  45. G

    Quantum Mechanics - Measurements and Observables

    Homework Statement Consider a state |psi>, and two non-commuting observables A and B. Now study the following chain of measurements: (i) On |psi> a A [sic] measurement gives the result a1, and a subsequent measurement of B gives the result b2. (ii) On |psi> a B measurement gives the result...
  46. P

    Finding averages of observables of Bell states

    Hi I am looking at Quantum Computation by Neilsen and Chuang at the CHSH inequality. Looking at the spin singlet state they make measurements of for example the observable Z1 and Z2-X1 and then find the expectation value of the product. I am slightly confused here because a) Z and X are...
  47. Peter Morgan

    Quantum Field Theory Observables: Distinguishing Two Types in Free Fields

    For free quantum fields, there are two types of observables indexed by wave-number, \tilde{\hat{\phi}}(k), the Fourier transform of the local field, which can be written as a(-k)+a^\dagger(k), and projection operators such as a(k)^\dagger\left|0\right>\left<0\right|a(k), a(k_1)^\dagger...
  48. Fra

    Diff invariant (measurement theory and observables)

    The recent threads makse we want to as a simple question. How many considers the notion of a _fundamental_ "diff invariant observables" as a clear and unquestionable requirement of the future theory of QG? To me this far from clear from the conceptual point of view. It's not even clear...
  49. I

    Spin Observables: Defining Reference Axes

    What is the criterion to define the reference axes, since $S_x$, $S_y$ and $S_z$ correspond to different Pauli matrices?
  50. R

    Can expectation value of observables be imaginary?

    I am quite new to Quantum Mechanics and I am studying it from the book by Griffiths, as kind of a self-study..no instructor and all... For a gaussian wavefunction \Psi=Aexp(-x^{2}), I calculated <p^{2}> and found it to be equal to ah^{2}/(1-2aiht/m) (By h I mean h-bar..not so good at...
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