What is Expectation value: Definition and 346 Discussions

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. Nate810

    Show that the expectation value of momentum is zero

    Homework Statement Demonstrate that the expectation value of momentum (p) for the wave function: ψ(x)∝e^(-γx) when x>0, ψ(x)=0 when x<0. Hint: Pay special attention to the discontinuity at x=0.[/B] Homework Equations <p>=<ψ|p|ψ>=∫dxψ*(x)[-iħ∂/∂x]ψ(x) from -∞ to ∞. [/B]The Attempt at a...
  2. I

    Expectation value of a real scalar field in p state

    Hello, I've been trying to find <p'|φ(x)|p> for a free scalar field. and integral of <p'|φ(x)φ(x)|p> over 3d in doing the space In writing φ(x) as In doing the first, I get the creation and annihilation operators acting on |p> giving |p+1> and |p-1> which are different from the bra state |p>...
  3. T

    Maximization of an Uncertainty Product

    Homework Statement [/B] Sakurai problem 1.20: find the linear combination of spin-up and spin-down S_z eigenkets that maximizes the uncertainty product \langle(\Delta S_x)^2\rangle\langle(\Delta S_y)^2\rangle. Homework Equations [/B] In general, we can write a normalized spin-space ket as...
  4. blue_leaf77

    Expectation value of 1/(1+x)

    I have an inner product ## \langle \alpha|f| \beta \rangle## where ##f## is an operator that is a function of position ##x## operator (1D). According to the book I read (and I'm sure in any other book as well), that inner product can be written in position representation as ## \int...
  5. D

    Expectation value of a combination of operators

    Homework Statement I will denote operators by capital letters. The question is calculate <p | XXPP | x> / <p | x > Homework Equations X |x> = x |x> P |p> = p |p> P |x> = -i(hbar)d/dx X |p> = i(hbar)d/dp The Attempt at a Solution If I start on the RHS and take PP out I get...
  6. Milsomonk

    Expectation value of the square of Momentum

    Homework Statement The expectation value of <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx For the Guassian wave-packet ψ(x)=(1/(π^1/4)(√d))e^-((x^2)/(2d^2)) Limits on all integrals are ∞ to -∞. Homework Equations <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx ψ(x)=(1/(π^1/4)(√d))e^(iKx)-((x^2)/(2d^2)) The Attempt at a Solution Ok...
  7. BUI TUAN KHAI

    Measured result is equal to expectation value

    Can I ask a basic question. This was a question in a test, I could not solve this. When is it true that the result of a single measurement for a dynamical variable is equal to the expectation value of the operator corresponding to that dynamical variable? Thank you for your help. Sincerely...
  8. G

    Expectation value of operator derivation

    Where one can find a proof of the expectation value of operator expression. <A> = < Ψ | A | Ψ > or <A> = integral( Ψ* A Ψ dx ) Thanks.
  9. FadeToBen

    Expectation value of total energy in QM

    Homework Statement The problem asks me to find the expectation value of W. Homework Equations The given ψ[x,t] is Asin(πx/a) e^((-i Eot)/ħ). By QM postulate 2 the QM operator of W is: iħ δ/δt or equivalently -ħ/i δ/δt. The Attempt at a Solution <w>=∫ψ*iħδ/δtψ= iħδ/δt 1/(2e^(-iEot/)ħ)...
  10. blue_leaf77

    Expectation value of momentum for bound states

    Homework Statement I'm curious in proving that expectation value of momentum for any bound state is zero. So the problem is how to prove this.Homework Equations $$ \langle \mathbf{p_n} \rangle \propto \int \psi^*(\mathbf{r_1}, \dots ,\mathbf{r_N}) \nabla_n \psi(\mathbf{r_1}, \dots...
  11. S

    Probability integral, Expectation Value and Square of Psi

    I have come across a bit of conflict in wording of some physics and chemistry textbooks about the probability of finding particles in certain places. To be more specific, I have come across 3 different statements: 1. $$\int_a^b {| \psi(x) |^2 dx}$$ The above integral is said to give the...
  12. A

    Find Expec. Value of x for Mass M Moving in 1D: Wave Funct. at t=0

    Homework Statement It's an old assignment for exam, but the solution manual gives little help: Describing a particle of mass m moving in one dimension (x) the wave function at time t=0 is: ## \Psi(x,t=0) = A \frac{1}{\sqrt{(x-x_0)^4 + l^4}} ## ##x_0## and ##l## are positive constants...
  13. R

    Expectation value of a SUM using Dirac notation

    Homework Statement Consider a one-dimensional particle subject to the Hamiltonian H with wavefunction \Psi(r,t) =\sum_{n=1}^{2} a_{n}\Psi _{n}(x)e^{\frac{-iE_{n}t}{\hbar}} where H\Psi _{n}(x)=E_{n}\Psi _{n}(x) and where a_{1} = a_{2} = \frac{1}{\sqrt{2}}. Calculate the expectation value of the...
  14. I

    Expectation value in quantom mechanics (a general question)

    Homework Statement Hello, I'm a bit confused about the calculation of the expectation values. Normally, when I have a wave function of sort and I want to calculate the expectation value of some operator, I just insert it into the braket <ψ|A|ψ>, where ψ for example is a wave function composed...
  15. DrClaude

    Expectation value of a product of hermitian operators

    I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...
  16. D

    Is the expectation value of momentum always zero for real wavefunctions?

    When calculating the expectation value of momentum of a real wavefunction is it always zero ? The momentum operator introduces an i into the integral and with real wavefunctions there is no other i to cancel and all Hermitian operators have real expectation values.
  17. P

    Energy Uncertainty and expectation value of H

    Homework Statement A particle at time zero has a wave function Psi(x,t=0) = A*[phi_1(x)-i*sin(x)], where phi_1 and phi_2 are orthonormal stationary states for a Schrodinger equation with some potential V(x) and energy eigenvalues E1, E2, respectively. a) Compute the normalization constant A. b)...
  18. R

    Calculating Expectation Value of Momentum with Fourier Transform

    we have a wavefunction \psi (x) the question asks for \psi (p) and says to use this to calculate the expectation value of momentum. The problem is the expectation value of momentum is integrated over dx so after transforming how do you get the integral to be over dp? thanks for any help with...
  19. S

    Expectation Value of Composite System

    Homework Statement System of 2 particles with spin 1/2. Let \vert + \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \\ \vert - \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} singlet state \vert \Phi \rangle = \frac{1}{\sqrt{2}} \Big( \vert + \rangle \otimes \vert - \rangle - \vert - \rangle...
  20. N

    Expectation value of semi conductors

    CAn someone help me calculate expectation value of energy of electron in conduction band for semi conductor
  21. J

    Expectation Value vs Probability Density

    I know the difference between the expectation value and probability density, but how do you calculate the probability density of an observable other than position? For position, the probability of the particle being in a particular spot is given by |\Psi|^2, which is the probability density, and...
  22. D

    3-D harmonic oscillator expectation value

    Homework Statement The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction ##ψ=e^(-αr) ## show that Homework Equations ##<ψ|H|ψ>/<ψ|ψ> = (\hbarα)^2/2m + 3mω^2/2α^2## The following...
  23. D

    Expectation value of energy in infinite well

    Homework Statement Given the following normalised time-independent wave function the question asks for the expectation value of the energy of the particle. The well has V(x)=0 for 0<x<a Homework Equations ψ( x ) = √(1/a) ( 1+2cos(∏x/a) )sin(∏x/a) The Attempt at a Solution I...
  24. O

    Expectation value of composite observable in singlet state

    Homework Statement I've been reading Leonard Susskind's Theoretical Minimum volume on QM, and enjoying it quite a bit - the book doesn't include exercise solutions at the end though, and if they exist online for this volume I haven't been able to find them. (Perhaps if such solutions...
  25. hellsteiger

    Expectation Value of Position for Even Wavefunction

    Homework Statement Hello, I need to calculate the expectation value for position and momentum for a wavefunction that fulfills the following relation: ψ0(-x)=ψ0(x)=ψ*0(x) The wave function is normalised. Homework Equations There is also a second wave equation that is orthogonal...
  26. fluidistic

    Expectation value of angular momentum

    Homework Statement A particle is under a central potential. Initially its wave function is an eigenfunction ##\psi## such that ##\hat {\vec L ^2} \psi = 2 \hbar ^2## , ##\hat L_3 \psi =0##. Calculate the expectation value of ##\hat {\vec L}## for all times. Homework Equations...
  27. B

    Integration by Parts To Derive Expectation Value of Velocity

    Homework Statement Why can't you do integration-by-parts directly on the middle expression in equation 1.29--pull out the time derivative over onto x, note that \displaystyle \frac{\partial x}{\partial t} = 0, and conclude that \displaystyle \frac{d \langle x \rangle }{dt} = 0Homework Equations...
  28. U

    Expectation value of Lz angular momentum

    Homework Statement Find ##\langle L_z \rangle##. What is ##\langle L_Z \rangle## for one atom only? Homework Equations The Attempt at a Solution Using ##L_z = -i\hbar \frac{\partial }{\partial \phi}##, I get: \langle L_z\rangle = \frac{32}{3} \pi k^2 \hbar a_0^3 Not...
  29. K

    Expectation Value of Gaussian Wave Function: Position & Momentum Zero?

    Why, in a Gaussian wave function the position and momentum expectation value coincide to be zero? Does it have any physical interpretation? I had an idea that expectation value is the average value over time on that state. But, for Gaussian it tells that it vanishes. Can you please explain.?
  30. S

    Expectation value of an operator

    When we say expectation value of an operator like the pauli Z=[1 0; 0 -1], like when <Z> = 0.6 what does it mean? What is difference between calculating expectation value of Z and its POVM elements{E1,E2}? thanks
  31. G

    Why is the expectation value of an observable what it is (the formula)

    Homework Statement I really do not understand why the expectation value of an observable such as position is <x> = \int\Psi*(x)\Psi Homework Equations If Q is an operator then <Q> = = \int\Psi*(Q)\Psi cn = <f,\Psi> The Attempt at a Solution What I understand this is saying is...
  32. Matterwave

    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...
  33. A

    Nuclear force tensor operator expectation value.

    Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...
  34. S

    Expectation value for momentum operator using Dirac Notation

    Question and symbols: Consider a state|ε> that is in a quantum superposition of two particle-in-a-box energy eigenstates corresponding to n=2,3, i.e.: |ε> = ,[1/(2^.5)][|2> + |3>], or equivalently: ε(x) = [1/(2^.5)][ψ2(x) + ψ3. Compute the expectation value of momentum: <p> = <ε|\widehat{}p|ε>...
  35. C

    Easy way to get the expectation value of momentum squared?

    Hello, I've been trying to define <p2> in terms of <x2>, much the same way that you can write <p> = m d<x>/dt, because it would be easier in my calculations. Is this possible, or am I on a fools errand? Edit: For Gaussian distributions.
  36. A

    Condition for expectation value of an operator to depend on time

    Homework Statement A particle is in a 1D harmonic oscillator potential. Under what conditions will the expectation value of an operator Q (no explicit time dependence) depend on time if (i) the particle is initially in a momentum eigenstate? (ii) the particle is initially in an energy...
  37. pellman

    Expectation value for first success in a binomial distribution?

    This is not a homework problem. Just a curiosity. But my statistics is way rusty. Suppose a binomial probability distribution with probability p for a success. What is the expected number of trials one would have to make to get your first success? In practice, this means if we took a large...
  38. T

    Expectation Value of Momentum for Wavepacket

    Homework Statement What is the average momentum for a packet corresponding to this normalizable wavefunction? \Psi(x) = C \phi(x) exp(ikx) C is a normalization constant and \phi(x) is a real function. Homework Equations \hat{p}\rightarrow -i\hbar\frac{d}{dx}The Attempt at a Solution...
  39. H

    What is the expectation value of x for a wave function equal to Ax^3?

    Homework Statement This is a question I had in my Quantum Mechanics class but my problem is with the calculus which is why i am posting it here. The question is to find the expectation value of x given the wave function equals Ax^3 where 0 ≤ x ≤ a, 0 otherwise. The solution given in class is...
  40. L

    What is the expectation value of ψ = x3 when 0≤ x ≤a and 0 otherwise?

    Homework Statement ψ = x3 when 0≤ x ≤a and 0 otherwise find <x> Homework Equations ∫ψ*ψdx=1The Attempt at a Solution So first I multiplied x3 times A, to get Ax3, then plugging that into the equation, I get ∫A2x6dx=1 Then I solve that for A, getting A = \sqrt{\frac{7}{a^{7}}} So I plug that...
  41. R

    When calculating the momentum expectation value

    when calculating the momentum expectation value the term i(h-bar)d/dx goes inbetween the complex PSI and the 'normal' PSI, so do you differentiate the normal PSI and then multiply by the complex PSI? or do you differentiate the product of the two PSI's i.e. the modulus of PSI? thanks for any...
  42. S

    Expectation value for a position measurement

    Homework Statement Given the wave function psi(x,0) = 3/5 sqrt(2/L) sin(xpi/L) + 4/5 sqrt(2/L) sin(5xpi/L) in an infinite potential well from 0 to L, what is the expectation value <x> and rms spread delta E = sqrt(<E^2>-<E>^2) Homework Equations <x> = integral from 0 to L of psi*xpsi dx...
  43. W

    Expectation value of kinetic energy

    Homework Statement Given the following hypothetic wave function for a particle confined in a region -4≤X≤6: ψ(x)= A(4+x) for -4≤x≤1 A(6-x) for 1≤x≤6 0 otherwise Using the normalized wave function, calculate the...
  44. D

    Why is there no help: momentum expectation value 2D particle in a box

    Is there anyone out there that knows how to define the p operator for a 2-d box. Please can you give a full answer, and not only a hint. I think that no one on this planet knows what it is. I have looked all over the internet. If there is no answer. Why don't people just say it? I think nobody...
  45. C

    Expectation value of the time evolution operator

    This problem pertains to the perturbative expansion of correlation functions in QFT. Homework Statement Show that \langle0|T\left[exp\left(i\int_{-t}^{t}dt' H_{I}^{'}(t')\right)\right]|0\rangle = \left(\langle0|T\left[exp\left(-i\int_{-t}^{t}dt'...
  46. C

    Expectation value of energy for a quantum system

    Homework Statement Let ##\Psi(x,0)## be the wavefunction at t=0 described by ##\Psi(x,0) = \frac{1}{\sqrt{2}}\left(u_1(x) + u_2(x)\right)##, where the ##u_i## is the ##ith## eigenstate of the Hamiltonian for the 1-D infinite potential well. The energy of the system is measured at some t -...
  47. D

    Expectation value of a hermitian operator prepared in an eigenstate

    Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...
  48. P

    Expectation value for non commuting operator

    if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?
  49. D

    Expectation value of an operator in matrix quantum mechanics

    Homework Statement Hey everyone. Imma type this up in Word as usual: http://imageshack.com/a/img577/3654/q9ey.jpg Homework Equations http://imageshack.com/a/img22/3185/pfre.jpg The Attempt at a Solution http://imageshack.com/a/img703/8571/xogb.jpg
  50. K

    How do I find this expectation value?

    Homework Statement A hydrogen like ion (with one electron and a nucleus of charge Ze) is in the state ψ = ψ_{2,0,0} - ψ_{2,1,0} What's the expectation value of \hat{r} (position operator) as a function of Z? Assuming origin at nucleus. Homework Equations for Z=1 < ψ |...
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