Expectation value Definition and 347 Threads

In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.

View More On Wikipedia.org
Recent contents Top users
  1. facenian

    Why Does My Calculation of <x^2> Yield a Negative Result?

    Homework Statement Evaluate <x^2> for the wave function \psi(x)=\int_{-\infty}^{\infty}dk exp(-|k|/k_0)exp(ikx) My calculation yields a negative answer and I can't find my error Homework Equations...
  2. T

    Is the Expectation Value Always 1 for Normalized State Vectors?

    Homework Statement |O> = k |R1> + 1/9 |R2> a) Find k if |O> has already been normalized, and b) then the expectation value. The Attempt at a Solution a) To Normalise: |(|O>)|2 = (1/9 |R2> + k |R1>).(1/9 |R2> - k|R1>) = 1/81|R2>2 - k2|R1>2 = 1 I just assumed that |k| = (1-(1/81))0.5, but...
  3. T

    Calculating Expectation Value for z component of angular momentum

    Homework Statement Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix). Homework Equations I think the only one relevant is the expectation value: <a> = integral[psi*(a)psi] / integral[psi*psi] where psi* is...
  4. J

    I need the dirac notation expectation value explaining to me please?

    Hi, I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable...
  5. M

    Difference between expectation value and probabilty

    Homework Statement Psi(x) = Ax -a<x<a I am trying to find the probability that my measured momentum is between h/a and 2h/a Homework Equations I have normalized A= sqrt(3/(2a^3)) I know that if I was finding the expected momentum I would use \int\Psi * p \Psi dx The...
  6. B

    Expectation value of Coulomb potential depends on relative spin

    Homework Statement Show that the expectation value of the Coulomb potential v(\vec{r_1},\vec{r_2})=\frac{e^2}{|\vec{r_1}-\vec{r_2}|}, between two electrons depends on the relative orientation of spin of the two electrons. Assume each electron is in the product state form...
  7. H

    Expectation value of momentum in discrete states

    Is there any way of proving <p> = 0 for a discrete (bound) state given it's wave function? I've seen proofs using the hermitian properties of p but I'm interested in proving that the integral of Psi*(x) Psi'(x) is identically zero regardless of Psi(x) as long as it's a solution of Schroedinger's...
  8. J

    Simple quantum problem - find eigenvalues, probabilities, expectation value?

    hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...
  9. I

    Is the expectation value of this commutator zero?

    If I have H=p^2/2m+V(x), |a'> are energy eigenkets with eigenvalue E_{a'}, isn't the expectation value of [H,x] wrt |a'> not always 0? Don't I have that <a'|[H,x]|a'> = <a'|(Hx-xH)|a'> = <a'|Hx|a'> - <a'|xH|a'> = 0 ? But if I calculate the commutator, I get: <a'|[H,x]|a'> = <a'|-i p \hbar /...
  10. D

    Understanding Expectation Values in Quantum Mechanics

    Let A be an observable (opeator), and we're assuming that for a given state psi(x), the value of A is given by A acting on psi(x), namely - A|psi>. Also we assume that - P(x) = |psi(x)|^2 So, I'de expect the Expectation value of A to be defined like so: <A> = Integral[-Inf:+Inf]{ P(x) A...
  11. D

    How to Calculate the Expectation Value of H'?

    Homework Statement Calculate the expectation value of \hat{H}' in the state \psi(x,t=0). \hat{H}'=k(\hat{x}\hat{p}+\hat{p}\hat{x}) \psi(x,t=0)=A(\sqrt{3}i\varphi_{1}(x)+\varphi_{3}(x)), where A=\frac{1}{2} Homework Equations The Attempt at a Solution I know it's found by...
  12. I

    Expectation value for angular momentum

    Homework Statement A wavefunction of angular momentum states is given: \psi = \frac{1}{\sqrt{7}}|1,-1\rangle + \frac{\sqrt{35}}{7}|1,0\rangle+\sqrt{\frac{1}{7}}|1,1\rangle Calculate \langle \psi| L_{\pm} |\psi \rangle and \langle 1,1|L_+^2|\psi\rangle3. Attempt at a solution. If the...
  13. M

    Derivation of the velocity of an expectation value

    Homework Statement I am trying to derive for myself the velocity of the expectation value from the information given, specifically that <x> = \int_{-\infty}^{\infty}x|\Psi (x,t)|^2 dt (1) Eq (1) can be transformed into, \frac{d<x>}{dt} =...
  14. D

    QM Measurements - probability, expectation value

    Homework Statement What are the possible results and their probabilities for a system with l=1 in the angular momentum state u = \frac{1}{\sqrt{2}}(1 1 0)? What is the expectation value? ((1 1 0) is a vertical matrix but I can't see how to format that) Homework Equations The...
  15. C

    Expectation Value For a Given Wave Function

    Homework Statement Find the expectation value of x (Find <x>) given the wave function: \psi(x)=[sqrt(m*alpha)/h_bar]e^[(-m*alpha*|x|)/(h_bar)^2] This wave function represents the single bound state for the delta-function potential. It's the solution to the shrodinger equation given the...
  16. S

    Calculating Expectation Value of Angular Momentum Squared for Hydrogen Atom

    Homework Statement Consider a hydrogen atom whose wave function at time t=0 is the following superposition of normalised energy eigenfunctions: Ψ(r,t=0)=1/3 [2ϕ100(r) -2ϕ321(r) -ϕ430(r) ] What is the expectation value of the angular momentum squared? Homework Equations I know...
  17. H

    Expectation value of spin operators.

    Homework Statement If an electron is in an eigen state with eigen vector : [1] [0] what are the expectation values of the operators S_{x}, and S_{z} Interpret answer in terms of the Stern-Gerlach experiment. The Attempt at a Solution Im not too sure how to calculate the...
  18. M

    Quantum mechanical expectation value

    I'm trying to calculate the expectation value of the momentum squared (p^2) of the harmonic oscillator ground state. The integral involves the second derivative of a Gaussian (exponential of a negative squared term) Then the integral involves, after working it out, an x^2 term times...
  19. LarryS

    Probability Density or Expectation Value?

    In a paper in Physical Review A, the author discusses a wave function for one particle, Ψ(r,t), where r is the position vector. He writes "The probability distribution for one-particle detection at a point r is given by |<r|Ψ >|2 ". Is that correct? The above expression looks, to me...
  20. L

    Calculating Expectation Value of Kinetic Energy in 3D Bound State

    ok. this is an easy enough thing to prove in one dimension but my question seems to be 3 dimensional and it's causing me some hassle: show the expectation value of the kinetic energy in a bound state described by the spherically symmetric wavefunction \psi_T(r) may be written \langle...
  21. W

    Expectation value of position of wavepacket

    Hello, this is just a general question, how is <x^2> evaluated, if <x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket) Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ? I'm only wondering how the squared works...
  22. Q

    Time-dependence of expectation value <x> in a quantum harmonic oscillator?

    Find the time dependence of the expectation value <x> in a quantum harmonic oscillator, where the potential is given by V=\frac{1}{2}kx^2 I'm assuming some wavefunction of the form \Psi(x,t)=\psi(x) e^{-iEt/\hbar} When I apply the position operator, I get: <x>=\int_{-\infty}^\infty...
  23. V

    Expectation Value of Momentum Squared

    Homework Statement A particle of mass m is in the state Psi(x,t) = Ae^(-a[(mx^2)+it]) where A and a are positive real constants. a) Find A b) For what potential energy function V(x) does Psi satisfy the Shrodinger equation? c) Calculate the expectation values of x, x^2, p, and...
  24. I

    The reciprocal expectation value

    I am aware of the expectation value \left\langle\ r \right\rangle. But I was wondering what is physically meant by the expectation value: \left\langle\frac{1}{r}\right\rangle The reason I am asking is because calculating this (reciprocal) expectation value for the 1s state of hydrogen, one...
  25. Q

    Expectation value of raising/lowering operators

    Homework Statement This has been driving me CRAZY: Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle and \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle Homework Equations Raising/lowering eigenvalue equations: a |n...
  26. P

    Formula for expectation value of raidous in Hydrogen atom

    Let's consider eigenstates |nlm\rangle of hamiltonian of hydrogen atom. Can anyone prove that \langle r \rangle = \langle nlm|r|nlm\rangle = \frac{a}{2}(3 n^2-l(l+1)). Where a - bohr radious. I've been trying to prove it using some property of Laguerre polynomials (which are radial part...
  27. B

    Expectation value r^2 for a radial wave function

    Homework Statement The ground state (lowest energy) radial wave function for an electron bound to a proton to form a hydrogen atom is given by the 1s (n=1, l=0) wave function: R10 = (2 / a3/2) exp(-r / a) where r is the distance of the electron from the proton and a is a constant. a)...
  28. V

    Missing 'x' in Expectation Value Formula

    Homework Statement In my textbook, the formula for the expectation value is written as: <x> = \int \Psi^{*}\Psi dx Shouldn't there be an x next to |\Psi|^{2} ? Thanks. Homework Equations The Attempt at a Solution
  29. H

    Expectation value for a spin-half particle.

    Homework Statement Calculate the expectation value of the operator _{}Sz for a spin-half particle known to be in an eigenstate of the operator _{}Sz Homework Equations The Attempt at a Solution I know the eigenvalues for the _{}Sz but how can I find the expectation values...
  30. R

    Quantum Mechanics expectation value problem

    Homework Statement An electron is in the spin state in the Sz representation |ψ> = A (1-2i 2)T <- this is a 2 X 1 matrix If Sx is measured, what values and probabilities do you get? What is the expectation value of Sx? Homework Equations The Attempt at a Solution...
  31. R

    What is the Expectation Value Problem?

    [b/]/
  32. J

    Angular momentum and expectation value

    My teacher said that angular momentum doesn't have orientation in space - but how can that be? Isn't cos(theta) = L_z / |L vector| ? Also (an unrelated question) could somebody give an example of how the integration process goes when you are trying to get an expectation value for something...
  33. K

    Quantum numbers of a field acquiring vacuum expectation value

    Why should symmetries require a field that acquires vacuum expectation value to have the same quantum numbers as the vacuum? Please give me a reference also..if possible...
  34. B

    Expectation value of a wave function

    Homework Statement The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0. Homework Equations The Attempt at a Solution The only thing I'm having a problem...
  35. A

    How do I find the variance of p for a given wave function?

    Homework Statement I am trying to find the variance of p for a wave function \Psi(x,0)=A(a^2-x^2) I'm confused about how to set up the integral. it should be something like -i^2h^2\int_{-a}^a A(a^2-x^2) (\frac{\partial\Psi}{\partial x})^2 dx I'm confused about the partial...
  36. U

    Hydrogen atom 1/r^2 expectation value

    Homework Statement Using the Feynman-Hellman theorem, determine the expectation values of 1/r and 1/r^2 for the hydrogen atom. Homework Equations Hamiltonian: H=-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} energy...
  37. G

    Is this right? Re: Finding expectation value of L_z

    Okay, so I'm now reviewing ladder operators (no, not homework). While reviewing a quantum problem involving the L_z operator at this website (http://quantummechanics.ucsd.edu/ph130a/130_notes/node219.html#example:expectLz"), I found myself confused. Okay, here's my question: don't we need to...
  38. R

    Expectation Value of x: (2a+b)/4

    Homework Statement Find the expectation value <x> if: from 0 <= x <= a, psi = A x/a from a <= x <= b, psi = A(b-x)/(b-a) Normalizing gives me that A = sqrt(3/b) (verified correct)Homework Equations The Attempt at a Solution <x> = \int_0^b x \psi^2 dx = \int_0^a \frac{A^2 x^3}{a^2}dx + \int_a^b...
  39. N

    How to show that expectation value is always positive?

    Homework Statement In quantum mechanics, how to show that the expectation value is always positive? Homework Equations The Attempt at a Solution
  40. A

    Expectation value of two annihilation operators

    Hello, I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following, c = \frac{(a + ib)}{\sqrt{2}}...
  41. K

    QM: expectation value of a harmonic oscillator (cont.)

    Thanks for all the help on the first question but now I have to solve for <T>. I have no idea how to do this, and I could use some help for a kick start. thanks!
  42. K

    QM: expectation value of a harmonic oscillator

    first post! but for bad reasons lol Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2 i solved for x: x=sqrt(h/2mw)((a+)+(a-)) so <x> integral of si x ((a+)+(a-)) x si. therefor the integral of si(n+1) x si + si(n-1) x si. si(n+1)...
  43. F

    What is the Expectation Value of Momentum for a Wave Function?

    Homework Statement Consider a wave function \psi (x,t) = R(x,t) exp(i S(x,t)) what is the expectation value of momentum?Homework Equations <f(x)> = \int^{\infty}_{-\infty} \psi^* f(x) \psi dx \hat{p} = -i \hbar \frac{\partial}{\partial x} The Attempt at a Solution This is for an intro to...
  44. L

    Need help with proof for expectation value relation.

    Homework Statement I have to prove the following: \hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle Edit: L = Angular Momentum & N = Torque Homework Equations I used Ehrenfest's theorem, and I've got the equation in the following form: \frac{1}{i} \left(\left[L,H\right]\right) +...
  45. K

    Finding Expectation Value of Electric Dipole Moment Matrix Form

    Homework Statement I we know the eigenstates of the system be |\psi_1\rangle and |\psi_2\rangle. Current state of the system is |\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle Try to find the expectation value of electric dipole moment \mu (assume it is real) and write it in...
  46. E

    Expectation Value in Inf. Box in an Eigenstate

    Homework Statement Obtain an expression for the expectation value <Pxn>n N=1,2... of a particle in an infinite box ( V=\infty for x<0 and x>L ; V=0 for 0<X<L) which is in an eigenstate of the energy. Homework Equations Pn =+- \sqrt{2*m*En } = +- (n*pi*Hbar) / L The Attempt at a...
  47. J

    Quantum problem - Calculating the expectation value of energy?

    Homework Statement Hi all, i have a problem: i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). Ψ(x,t) = (1/sqrt2)[Ψ0(x).e-[i(E0)t/h] + Ψ1(x)e-[i(E1)t/h]], where Ψ0,1(x) are the ground and...
  48. S

    Find the expectation value of the linear momentum

    Homework Statement For a given wave function Psi(x,t)=Aexp^-(x/a)^2*exp^-iwt*sin(kx) find the expectation value of the linear momentum. Homework Equations <p>=integral(-inf,inf) psi* p^ psi dx p^=-ih(bar) d/dx sin x = (exp ix - exp -ix)/2i cos x = (exp ix + exp -ix)/2 The Attempt...
  49. K

    How Does Time Dependence Influence Expectation Values in Quantum Mechanics?

    If \Psi (x,t) = \psi (x) g(t), should I then use \Psi or \psi when calculating <p> and <p ^2>?
  50. K

    What is the Expectation Value Problem in Quantum Mechanics?

    Homework Statement Calculate \Delta x = \sqrt{\left\langle(x - \left\langle x \right\rangle )^2 \right\rangle} if \left\langle x \right\rangle = 0 and \left\langle x^2 \right\rangle = a^2(\frac{\pi - 6}{12 \pi^2}) 2. The attempt at a solution \left\langle(x - \left\langle x...
Back
Top