Expectation value Definition and 346 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.

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  1. L

    Expectation value of Normal ordered Stress-energy tensor

    Hi, In Birrel and Davies ch4 they write: \langle \psi|:T_{ab}:|\psi \rangle =\langle \psi|T_{ab}|\psi \rangle -\langle 0|T_{ab}|0 \rangle this is for the usual Mink field modes and vac state. Why does normal ordering reduce to this expression, could anybody point me the way to...
  2. S

    Expectation Value for a Function with Cusp

    Homework Statement In my homework assignment I have a wavefunction defined as \Psi(x)=N\exp(-|x|/a) and I am asked to find the expectation value of momentum squared in configuration space. Homework Equations \int\Psi*(x)\hat{p^2}\Psi(x)dx The Attempt at a Solution N is 1/\sqrt{a}...
  3. S

    How Does Hermitian Property Lead to \langle A^2 \rangle in Quantum Mechanics?

    I know that <Aψ|Aψ> = <A2>. Can anyone tell please tell me how you get one line to the another?
  4. M

    Expectation Value Question with Unknown Operator

    Homework Statement Consider an observable A associated to an operator A with eigenvalues an. Using the formula <A> = ∫ψ*Aψ compute the expectation value of A for the following wave function: \Psi=\frac{1}{\sqrt{3}}\phi_{1}+\frac{1}{\sqrt{6}}\phi_{2}+\frac{1}{\sqrt{2}}\phi_{3} where...
  5. J

    Expectation value of P for an infinite-square well

    I just have a simple question to get me started. If I am given an initial value wavefunction ψ(x,0) and I am asked to find <P> at t = 0 can I use this: <P> = -ih∫ψ*(x,0)\frac{∂}{∂x}ψ(x,0)dx or do I need to find ψ(x,t) before I find <P>?
  6. B

    How to Find the Expectation Value in the Stern-Gerlach Experiment?

    Homework Statement An electron starts in a spin state |\psi(t=0)\rangle = |z \uparrow \rangle and evolves in a magnetic field B_0(\hat{x} + \hat{z}). The Hamiltonian of the system is \hat{H} = \alpha \vec{B}\cdot\vec{S}. Evaluate \langle \psi (t_{1/2}) | S_x | \psi(t_{1/2}) \rangle...
  7. L

    When Is Measured Value of Observable A Always the Same?

    Homework Statement Given an observable quantity A, when will it happen that the same value for A will be measured every time? What is the relationship between the operator \hat{A} and \Psi for this case? and What is the relationship between \widehat{A} and \widehat{H}, the...
  8. J

    How to Calculate the Expectation Value of x^2 in Quantum Mechanics?

    Homework Statement I am given ψ(x), want to calculate <x^{2}>. Homework Equations \psi(x) = a\exp(ibx-(c/2)(x-d)^2) <x^2> = \int\limits_{-∞}^∞ \psi^*x^2\psi \mathrm{d}x The Attempt at a Solution Well, I normalized the wave function and found a = (\frac{c}{\pi})^{1/4}. So, the...
  9. S

    Expectation value in a linear superposition

    In a linear superposition, what is the relationship between expectation value of, say, energy and the amplitude coefficients of the eigenfunctions?
  10. F

    Express moment / expectation value in lower order expectation values

    Hello everybody, I'm looking for a proof of the following equation: <x6> = <x>6+15s2<x>4 where the brackets denote an expectationvalue and s is the standard deviation. Thanks in advance!
  11. M

    Scalar Field Theory-Vacuum Expectation Value

    Homework Statement I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...
  12. A

    How Do You Calculate the Expectation Value of Momentum in Quantum Mechanics?

    I've managed to get myself confused on a seemingly simple point of mathematics. When I calculate the expectation value of momentum in quantum mechanics <p>=\int{\psi* \frac{\hbar}{i} \frac{d}{dx} \psi dx} To what should I be applying the derivative? \psi?
  13. J

    Expectation Value of a component of Orbital Angular Momentum

    Homework Statement given a certain state |ψ> that is an eigenstate of L^2 and Lz Calculate <Lx^2> and <Lx> Homework Equations L^2|ψ> = l(l+1)h^2 Lz|ψ> = mh|ψ> Lx = YPz - ZPy The Attempt at a Solution <Lx^2> = (1/2)(h^2)[l(l+1)-(m1)^2] for Lx i got <Lx> = ∫ψ(YPz-ZPy)ψ dx
  14. J

    Higgs vacuum expectation value imples what energy density?

    I understand that the the Higgs field has a vacuum expectation value of 246 GeV. I think that means that the expectation of the Hamiltonian energy operator applied to the vacuum state is 246 GeV. What does this imply for the energy density of the Higgs field in the vacuum (i.e. in Joules /...
  15. P

    Expectation value of: energy, angular momentum

    Hi all! Homework Statement If we consider the hydrogen atom as a spinless particle. Let this system in the state \Psi ( \vec{r} )= \frac{1}{6} [4 \Psi_{100} ( \vec{r} )+ 3 \Psi_{211}- \Psi_{210} ( \vec{r} ) + \sqrt{10}\Psi_{21-1} ( \vec{r} )] Calculate: 1) Expectation value of...
  16. J

    Show that the expectation value of angular momentum <Lx> is zero

    Homework Statement Show that the expectation value of angular momentum <Lx> is zero Homework Equations L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩ L± = Lx ± iLy The Attempt at a Solution I'm supposed to use ladder operators here to show <Lx> is zero. I start with...
  17. R

    Expectation value of a product of operators

    Just to check something: If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ? Or can you not split up the operators like <a|A (B|a>) ? Thanks.
  18. K

    Calculate Expectation Value of Hamiltonian using Dirac Notation?

    Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...
  19. B

    Obtain an expression for the expectation value in terms of energy

    Homework Statement The ground state wave function for a particle of mass m moving with energy E in a one-dimensional harmonic oscillator potential with classical frequency omega is: u(subscript 0) (x)= N(subscript 0) exp((-alpha^2)(x^2)/2) and alpha=sqrt (m *omega/h-bar) where...
  20. P

    Is the Expectation Value of Momentum for a Superposition of Sine Functions Zero?

    Ψ(x)=(2/a)^(1/2) [csin(nπx/a)] The Expectation value of momentum <P>=∫Ψ＊(x)[-ih d/dx ] Ψ(x) dx = 0 the average momentum is zero.It means the particle is moving equally in the +x and -x. And if Ψ(x)=(2/a)^(1/2) {csin(πx/a)+dsin(2πx/a)} I calculate the average momentum is also...
  21. H

    Quantum Mechanics Spin Expectation Value

    Homework Statement What is the expectation valueof the Sχ for a system in the time-dependent state |Ψ> = 2e-2iωt |z+> -ieiωt |z-> Homework Equations maybe the state must be normalised first i.e 1/√5 times the initial ψ The Attempt at a Solution And then say<ψ|Sχ|ψ> where ψ...
  22. G

    Find the expectation value of momentum squared for a simple harmonic oscillator

    Find the expectation value of (px)2, keeping in mind that ψ0(x) = A0e−ax2 where A0 = (2mω0/h)^1/4, and <x2> = ∫x2|ψ|2dx = h_bar / 2mω0 <ψ(x)|px2|ψ(x)> = ∫ψ(x)(pop2)ψ(x) dx pop = [hbar / i] (\delta/\deltax) I'm not going to attempt to type out me solving the integral because it...
  23. fluidistic

    Expectation value , a few questions

    "Expectation value", a few questions I've read that in quantum mechanics we use the term "expectation value" for example for the energy of a system. Despite its name, the expectation value of the energy of for instance the quantum harmonic oscillator is not the most probable measured energy of...
  24. J

    Expectation Value of Angular Momentum

    Homework Statement 1. For the ground-state of the 2D rigid rotor what is the expectation value of the angular momentum? And what is the corresponding uncertainty in this value? 2. Describe in words what the uncertainty in position is. 3. Explain why the rigid rotor can have vanishing...
  25. fluidistic

    Quantum mechanics, expectation value

    Homework Statement Show that for the expectation values the following relations hold: d \langle x \rangle /dt =\langle p \rangle /m and d \langle p \rangle /dt = - \langle d V/dx \rangle. Homework Equations \langle x \rangle = \int _{- \infty}^{\infty} \Psi ^* x \Psi dx. The Attempt...
  26. A

    Expectation Value/ Evaluating Gaussian Integral

    Homework Statement I'm re-hashing a problem from my notes; given the wave function \psi(x)=Ne^{-(x-x_0)/2k^2} Find the expectation value <x>. Homework Equations The normalization constant N for this is in my notes as N^2=1/\sqrt{2 \pi k^2} N=1/(2\pi k^2)^{(1/4)} It should be...
  27. teroenza

    What Does <x^2> Represent in Quantum Mechanics?

    Homework Statement 1. What is <x^{2}>, in terms of position and expectation values. 2. How can I use the correspondence principal to explain the quantum vs classical results (below). My textbook (Serway, Modern Physics) uses <x> as the expectation value, meaning the average position of a...
  28. C

    Is the Expectation Value of Uncertainty Constant in Time?

    Homework Statement Show that, if [H,A] = 0 and dA/dt = 0, then <&Delta;A> is constant in time. Homework Equations d<A>/dt = <i/ℏ[H,A] + dA/dt> The Attempt at a Solution I am trying to use the above equation to show that d<&Delta;A>/dt is 0, and I can get to d&Delta;A/dt = 0, but I...
  29. A

    Expectation Value for Gaussian Wave Packet.

    Homework Statement Calculate <x> for the Gaussian wave packet \psi(x)=Ne-(x-x0)/2k2 Homework Equations \left\langle x \right\rangle = \int dx x|\psi(x)|2 The Attempt at a Solution So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm...
  30. M

    The expectation value for the radial part of the wavefunction of Hydrogen.

    The wavefunction of hydrogen is given by \psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta, \phi) If I am only given the radial part, and asked to find the expectation value of the radial part I integrate the square of the wavefunction multiplied by r cubed allowing r to range from 0 to...
  31. jinksys

    Expectation value of the expectation value

    Homework Statement The expectation value of the position observable x is <x> = ∫ψ*xψdx. The expectation value of the expectation value, <<x>>=<x>, is still the expectation value...why? Homework Equations The Attempt at a Solution All I can think of is that the expectation value...
  32. B

    Quantum statistics expectation value

    1. What is the expectation value, <x>, for the given distribution over the interval from – to + infinity of the function: f(x)=e^(-.5(x-mu)^2(sigma^-2)) 2. This is a statistics problem i think. I just need to know how this type of problem is worked out because it is relevant to my...
  33. N

    Quantum expectation value (griffiths)

    Homework Statement If x is the position of a particle then the expectation value of x, <x> is : (I got lazy and just copied an image from Wiki, just pretend <x> is on the lhs of the eqn) When Griffith derives an expression for d<x> / dt, he uses the fact that dx/dt is zero, since "the wave...
  34. K

    Quantum Mechanics - Finding expectation value

    Homework Statement Find the expectation value of position as a function of time. Homework Equations This is in the latter half of a multi-part question, previously we were given that: Eqn 1: Ψ(x, t) = A(ψ1(x)e−iE1t/h¯ + iψ2(x)e−iE2t/h¯) and in an even earlier part: Eqn 2: ψn(x) =...
  35. F

    Proof: Operators with same expectation value

    Given some state \left|\psi\right\rangle, and two operators \hat{A} and \hat{B}, how do you prove that if \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle then \hat{A} = \hat{B} ?
  36. P

    Is the expectation value of momentum of a stationary state zero?

    Given a stationary state H \psi = E \psi \Rightarrow \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right)\psi = E\psi Firstly is it true that \left<p\right> = \frac{\hbar}{i}\int\psi^* \frac{\partial \psi}{\partial x} dx= 0 ?? If it is, how do we prove it?
  37. S

    What is the expectation value of Sz for the ground state?

    Homework Statement Kindly take a look at the attachment for the problem statement. Homework Equations Schrodinger Equation: H psi = E psi The Attempt at a Solution For Part A: H psi = E psi S= 1/2 So energy of ground state is -1/2.K.H.hcross Could you tell me if I am on the...
  38. L

    Can posts be deleted on online forums?

    Homework Statement how do you delete a post?
  39. T

    Position expectation value in harmonic oscillator

    Hello, I want to find <xftf|x(t)|xiti> in harmonic oscillator. I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff. Thank you
  40. C

    How to Find the Expectation Value of Total Energy?

    Homework Statement given a wavefuntion \Psi = (1/sqrt50) (3\mu1 + 5\mu2 - 4\mu3) what is the expectation value of the total energy? My thoughts were to calculate <\Psi|\hat{}H|\Psi> but the previous part to the question asks for the probability of each outcome(which I know how to...
  41. P

    Expectation value of spin in an Ising lattice

    Homework Statement I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0) \left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0 and \left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0...
  42. N

    Expectation Value of O†: Is It Equal to Complex Conjugate of <O>?

    Homework Statement Hi Say I have an operator O, and I find its expectation value <O>. Now, if I wish to find the expectation value of O† († denoting Hermitian conjugate), then will this just equal the complex conjugate of <O>? Niles.
  43. J

    Complex conjugate of the expectation value of momentum

    Homework Statement Compute the complex conjugate of <p> using eq 1.35 (<p>=∫ψ*(h/i)∂/∂x ψ dx) and prove that <p> is real (<p>=<p>*) Homework Equations equation 1.35 is given above The Attempt at a Solution to take the c.c. don't i just add a minus to the i and switch the stars like...
  44. C

    Expectation value of the angular momentum operator

    Homework Statement Hey forum, I copied the problem from a pdf file and uploaded the image: http://img232.imageshack.us/img232/6345/problem4.png What is the probability that the measurement of L^{2} will yield 2\hbar^{2} Homework Equations \left\langle L^{2} \right\rangle = \left\langle \Psi...
  45. P

    Can a Vector Have an Expectation Value?

    Homework Statement Prove that for a particle in a potential V(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque: d/dt <L> = <N> Where N = r x(-del V) N, r, and L are vectors. Homework Equations...
  46. X

    Expectation value of X and Y component of angular momentum

    Homework Statement Show: <Jx>=<Jy>=0 Homework Equations Jx=1/2(J++J-) Jy=1/2(J+-J-) The Attempt at a Solution <jm l Jx l jm> = < jm l 1/2 J+ l jm> + < jm l j- l jm > = < jm l h/2 sqrt [(j-m)(j+m+1)] + h/2sqrt[(j+m)(j+m+1) l jm > i am not sure how to apply the next step
  47. J

    Expectation value of the sum of two random variables

    Homework Statement The expectation value of the sum of two random variables is given as: \langle x + y \rangle = \langle x \rangle + \langel y \rangle My textbook provides the following derivation of this relationship. Suppose that we have two random variables, x and y. Let p_{ij}...
  48. D

    Expectation Value: My Understanding vs. Prof.

    My understanding was that the expectation value of an observable H for a state |a> is just <a|H|a>. But in a homework problem, my prof. used <H> = <a|H|a>/<a|a>. I'm a little confused by the discrepancy, why the discrepancy?
  49. T

    Calculating expectation value of angular kinetic energy

    Homework Statement the origial question given is: Show that the difference in energies between 2s and 2p radial wavefunctions is equal to the energy of the angular part of the 2p wavefunction, and thus that they have the same overall energy. hints given:a)use virial theorem to determine...
  50. 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...
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