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.

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  1. K

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

    Homework Statement A particle is in a infinite square poteltian well between x=0 and x=a. Find <p> of a particle whose wave function is \psi(x) = \sqrt{\frac{2}{a}}sin\frac{n \pi x}{a} (the ground state). 2. The attempt at a solution <p> = \frac{2 \hbar k}{\pi} \int^{a}_{0}sin^{2}...
  2. G

    [Q]Time deviation of expectation value

    Hi, You know famous equation, \frac{d<A>}{dt} = <\frac{i}{\hbar}[\hat{H},\hat{A}] + \frac{\partial\hat{H}}{\partial t} > But liboff said if \frac{\partial \hat{A} }{\partial t} = 0 then, \frac{d<\hat{A}>}{dt} = 0 this is the proof \frac{d<A>}{dt} =...
  3. M

    Expectation Value of Nsub.1 for Rare Species: What is <N> & DeltaN?

    the number of hairs Nsub.1 on a certain rare species can only be the number 2sup.l(l=0,1,2...) The probability of finding such an animal with 2sup.l hairs is exp-1/l ! what is the expectation,<N>? what is deltaN?
  4. P

    Expectation Value of x: Definition & Meaning

    How does this follow from the defintion of the expectation value of x
  5. P

    Integration help (expectation value)

    I'll skip the format because this isn't for a course, just a textbook I'm reading. Also because it shows the steps but I'm unsure about one of them. It might be a dumb question, but here goes: It's for calculating \frac{d<p>}{dt} Using the momentum operator we have: \frac{d}{dt}<p> =...
  6. O

    Expectation value of aharmonic oscillator

    Homework Statement I need to find the expectation value of x of an aharmonic oscillator of a given potential: V_{(x)} = c x^2 - g x^3 - f x^4 Homework Equations Two relevant equations: First: I'm using the partition function to find the expectation value <x>= \frac { \int x Z...
  7. K

    What is the expectation value for p in the given quantum mechanics problem?

    Homework Statement First off, this is my first time posting here so please excuse any editing mistakes or guidelines I may have overlooked. This is problem 1.17(c) from Griffiths, Introduction to Quantum Mechanics 2nd edition. It reads: \Psi(x, 0) = A(a^2 - x^2), -a\leqx\leqa. \Psi(x, 0)...
  8. D

    How to Find the Expectation Value of an Operator with a Constant Commutator?

    Problem Consider an operator \hat{A} whose commutator with the Hamiltonian \hat{H} is the constant c... ie [\hat{H}, \hat{A}] = c. Find \langle A \rangle at t > 0, given that the system is in a normalized eigenstate of \hat{A} at t=0, corresponding to the eigenvalue a. Attempt Solution We...
  9. B

    Confusion: deriving momentum expectation value in QM

    On pages 16-17 of Griffith's Intro to QM, he writes the following: \frac{d\left\langle x \right\rangle}{dt}= \int x \frac{\partial}{\partial t}|\Psi|^{2} dx = \frac{i\hbar}{2m}\int x \frac{\partial}{\partial x} \left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial...
  10. G

    Understanding Expectation Value in Quantum Mechanics: A Closer Look

    We all know the concept of expectation value,it is the average of all possible outcomes of an experiment. Mathematically average of x is written as (Σnkxk / Σnk ). Quantum-mechanically nk is represented by probability density(P), where P = ∫Ψ*Ψ d3r, then <r> = ∫ r P(r) d3r -----------(1)...
  11. 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...
  12. B

    Need help with expectation value

    Homework Statement I have a random, uniformly distributed vector with Cartesian components x,y,z. I should calculate the expectation value of the products of the components, e.g. <x\cdot x>, <x\cdot y>, ..., <z\cdot z>. Homework Equations In spherical coordinates the x,y,z components...
  13. T

    In an experiment, do we measure the eigenvalue or expectation value?

    In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?
  14. 3

    Expectation value of P^2 for particle in 2d box

    [SOLVED] expectation value of P^2 for particle in 2d box I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral...
  15. M

    Finite expectation value <-> finite sum over Probabilties

    Homework Statement If X is a real valued random variable with E[|X|] finite. <-> \sum(P(|X|>n)) finite , with the sum over all natural numbers from 1 to infinity. Homework Equations As a tip I am given that for all integer valued X>0 E(X) = \sum(P(X)>k , where the sum goes over all k =1 to...
  16. C

    Calculating Expectation Value of Particle in Square Potential Well

    I'm confused re a particle of energy E < V inside a square potential of width 'a' centered at x = 0 with depth V. They give the wavefunction for outside the well as \Psi(x) = Ae^{k|x|} for |x| > a/2 and k^2 = -\frac{2ME}{\hbar^2} => k = i\frac{\sqrt{2ME}}{\hbar} ? I need the probability that...
  17. T

    Expectation value using ladder operators

    I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...
  18. T

    Expectation value for a superposition

    Homework Statement u(x) = \sqrt{\frac{8}{5}}\left(\frac{3}{4}u_{1}(x)-\frac{1}{4}u_{3}(x)\right) Determine the time-dependent expectation value of position of this wave function (the particle is in an infinite potential well between x = 0 and x = a). The Attempt at a Solution I...
  19. W

    What is the proof for the expectation value of a quantum system?

    I am trying to show that \frac{d}{dt}<x^2>=\frac{1}{m}(<xp>+<px>)....(1) With the wavefunction \Psi being both normalized to unity and square integrable Here is what I tried... <xp> = \int_{-\infty}^{\infty}{\Psi}^*xp{\Psi}dx <px> =...
  20. C

    Expectation Value Homework: Integrating Gaussian Distribution

    Homework Statement Can somebody help me integrate \int{x\cdot p(x)} where p(x) is the Gaussian distribution (from here http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html) The Attempt at a Solution I can't really get anywhere. It's true that \int{e^{x^2}} has no analytical...
  21. E

    Find Expectation Value of x for \psi(x,t)

    I am trying to find <x> for \psi(x,t) = A exp\left(-|x|/L - i*E*t/\hbar\right) I found the normalization factor of 1/L and I took \int_{-\infty}^{\infty}\left( x * exp(|x|/L) \right) in two integrals however I got as a final result: L * -\infty * exp(-\infty/ L) - L *...
  22. P

    Expectation value for Hydrogen radius

    Homework Statement Find the expectation value for a hydrogen atom's radius if n=25 and l=0. Homework Equations expectation value = <f|o|f> where f=wavefunction and o=operator The Attempt at a Solution So I know that to find an expectation value, you integrate over all relevant...
  23. quantumdude

    Expectation Value of Momentum in H-Atom

    Here's a silly question. I'm sure I should know the answer, but alas my most recent QM course was 9 years ago. I sat down to calculate the expectation value of momentum in the H-atom today, because some kid on another forum wanted to know how fast an electron in an atom is. I was going to...
  24. M

    What Are the Expectation Values of Observables A and B in State |b2>?

    Homework Statement We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates...
  25. V

    Expectation value of the square of the observable

    Homework Statement I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square? Homework Equations \langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx \langle Q^2 \rangle = \int_{-\infty}^{\infty}...
  26. V

    Expectation value of an operator (not its corresponding observable value)

    1. Problem statement This isn't a homework question itself, but is related to one. More specifically, I'm computing the time-derivative of \langle x \rangle using the correspondence principle. One side simplifies to \left\langle \frac{\hat{p}}{m} \right\rangle, but what is the physical meaning...
  27. S

    How Do You Find <r^2> for Uncertainty in Position?

    When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate? Also as I am...
  28. D

    Expectation value of 1s state of hydrogen driving me absolutely nuts

    My question says: "Evaluate the expectation value <1/r> of the 1s state of hydrogen. How does this result compare to the result found using the Bohr theory?" Firstly, I have been told that <1/r> does not mean <1/r> but rather that it means 1/<r>. Having made it this far I now do the 1/<r>...
  29. L

    Find Expectation Value of Wavefunction in 1-D Box

    OK, here is the problem: An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is): \phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}]...
  30. H

    Hermiticity and expectation value

    is there a better way to check for hermicity than doing expecation values? for example, what if you had xp (operators) - px (operators), or pxp (operators again); how can you tell if these combos are hermetian or not, without going through the clumsy integration (that doesn't give a solid...
  31. K

    Expectation value of Energy Quantum

    I'm still really confused on how to go about calculating this for non eigenstates. I'm trying to do the problem below, and am wondering how to go about it. \Psi (x,0) = A (1-2 \sqrt {\frac{m \omega}{\hbar}} x)^2 e^ {-\frac{m \omega x^2}{2 \hbar}} So I can't calculate the expectation...
  32. Repetit

    Position expectation value of a particle in a box

    I have calculated the expectation value of a particle in a box of width a to be a/2. The wavefunction of the particle is: N Sin(k_n x) Exp[-i \frac{E_n t}{\hbar}] Now, in the first excited state with k_n equal to 2\pi / a the position probability density peaks at a/4 and 3a/4 but is zero...
  33. S

    QM expectation value relation <x^n>, <p^n>

    I need to calculate <x^n> and <p^n> for psi(x)=exp(-ax^2/2) for n even. For <x^n>: <x^n>=integral(exp(-ax^2)*x^n )dx from -inf to +inf then i use integration by parts to get an infinite series and i use a formula to find the finite sum of the series =[exp(-ax^2)*x^(n+1)/((n+1-2a*(n+1)^2)]...
  34. G

    How can I find the square of expectation value for a particle in a box?

    hi all can sombody show me the way I could get the square expectation value http://06.up.c-ar.net/03/fd4f.jpg for a particle in a box where the answer is given to us : http://06.up.c-ar.net/03/87d0.jpg
  35. Reshma

    Expectation value of momentum of wavefunction

    I have a wavefunction given by: \psi = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} With boundary conditions 0<x<L. When I compute the expectation value for the momentum like this: \overline{p_x} = \int_0^L \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} \left(-i\hbar \frac{\partial}{\partial...
  36. A

    Find Expectation Value for 1st 2 States of Harmonic Oscillator

    how do you find the expectation value <x> for the 1st 2 states of a harmonic oscillator?
  37. R

    Expectation value of continuous random variable

    Hi.. i am doing this question for Probability Theory, to find E[x] of a continuous random variable E[x] = the integral from (0 to infinity) of 2x^2 * e^(-x^2) dx So I used integration by parts... u = x^2 du = 2xdx dv = e^(-x^2) <--- ahh... how do you integrate that. (it dosn't look like...
  38. B

    Finding expectation value using Heisenberg picture

    We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator. I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do...
  39. K

    Expectation value of following function

    I need to find the momentum expectation value of the function in the attached picture. It is the function of the harmonic oscillator (first excited state). :confused: I know that the expectation value is the value that we measure with the highest probability if we measure the system. But...
  40. B

    Expectation value, harmonic oscillator

    Hi, I have to find the expectation values of xp and px for nth energy eigenstate in the 1-d harmonic oscillator. If I know <xp> I can immediately find <px>since [x,p]=ih. I use the ladder operators a_{\pm}=\tfrac1{\sqrt{2\hslash m\omega}}(\mp ip+m\omega x) to find <xp>, but I get a complex...
  41. M

    Expectation value of an anti-Hermitian operator

    Hi, could anyone tell me how one would show that the expectation value of a anti-Hermitian operator is a pure imaginary number? Thanks.
  42. C

    Time deritivate of the expectation value of p

    This is the problem: Calculate: \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \frac {d \mean{p}}_{dt} Here's a few more points to keep in mind... (A) The assumption is that <p> is defined as: \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \mean{p} = -i \hbar \int \left( \psi^* \frac...
  43. B

    I want to find the expectation value [tex]\langle x^2 \rangle[/tex] in

    I want to find the expectation value \langle x^2 \rangle in some problem. To do this I make a change-of-variable, \xi = \sqrt{\frac{m\omega}{\hslash}}x, and compute the expectation value \langle \xi^2 \rangle like this: \langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi...
  44. P

    X's expectation value in quantum physics

    When I'm in a dimension higher than 1, do I need to integrate over all space (V) or only the x axis? Thanks in advance.
  45. C

    Why Does the Expectation Value Depend on the Space Used?

    I had thought that the expectation value would be the same...whether you did it in momentum space or position space. Could someone explain what is going on in this particular problem? \psi (x) = \sqrt{b} e^{-b |x| + i p_0 x / \hbar } Taking the Fourier transform, I can get this...
  46. C

    How Do You Calculate the Expectation Value <x²> for a Particle in a Box?

    If the expectation value <x> of a particle trapped in a box L wide is L/2, which means its average position in the middle of the box. Find the expectation value <x squared>. How do I go about doing this? I am really confused.
  47. I

    Quantum Mechanics: How Do I Find Expectation Values for Position and Momentum?

    Expectation value problem pleasezzz help ASAP Hi Everyone, I have a problem on one of my problems in the quantum course. I need tofind the expectation values <x>,<x^2>, <p> & <p^2> for the function e^(-(x-xo)^2/2k^2) please email me if you need theformulaes.. i have them but i...
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