# 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. ### The time-dependence of the expectation values of spin operators

So first I derived the expressions for the dynamics of the spin operators and got: $$\frac{d\hat{S}_y}{dt} = w\hat{S}_x^H$$ $$\frac{d\hat{S}_x}{dt} = w\hat{S}_y^H$$ $$\frac{d\hat{S}_z}{dt} = 0$$ Now I want to calculate the time-dependence of the expectation values of the spin operators...
2. ### I Evaluating a Stochastic Average

Hi all, I am not familiar with stochastic processes, but I would like to know how to evaluate the following expectation value: $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$ where ##\mathbb{E}[V_{i}(t)] = 0,\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')## for some...
3. ### I Help with a derivation from a paper (diatomic molecular potential)

Hello! I am confused about the derivation in the screenshot below. This is in the context of a diatomic molecular potential, but the question is quite general. Say that the potential describing the interaction between 2 masses, as a function of the radius between them is given by the anharmonic...

16. ### A QFT with vanishing vacuum expectation value and perturbation theory

In This wikipedia article is said: "If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a...

Hi all, I found this notation of expectation values in a NMR text. In class, I learned that expectation values are given by $$<\hat{X}>=\int_{-\infty}^\infty\psi^*x\psi dx$$ why does this textbook divide by the integral of probability density ##\int \psi^*\psi dx##?
18. ### Expectation value of momentum operator

I know that the eigenstates of momentum operator are given by exp(ikx) To construct a real-valued and normalized wavefunction out of these eigenstates, I have, psi(x) = [exp(ikx) + exp(-ikx)]/ sqrt(2) But my trouble is, how do I find the expectation value of momentum operator <p> using this...

43. ### Find the spinor-state for a given expectation value

Homework Statement Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which $$\langle\vec{\sigma}\rangle =\vec{e},$$ where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle## Homework Equations :[/B] are all given...
44. ### Calculate the expectation value of V from Ehrenfest's theorem

Homework Statement I have a general question how I calculate the expectation value of V (potential energy) with Ehrenfest’s theorem. Do I have to integrate d<p>/dt with respect to d<x>. Also if the potential is symmetric (even) would that mean the expectation value of the potential is 0...
45. ### Expectation value <p> of the ground state of hydrogen

Homework Statement How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom. Homework Equations The Attempt at a Solution I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I...
46. ### Time Derivative of Expectation Value of Position

Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...
47. ### I What is the correct expectation value for this game with redraw?

Hi all, I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For...
48. ### Expectation value of raising and lower operator

I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result. Homework Statement At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by: ## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...
49. ### Expectation Value of a Stochastic Quantity

Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...
50. ### Expectation value of position

Homework Statement Show that, for a general one-dimensional free-particle wave packet $$\psi (x,t) = (2 \pi h)^{-1/2} \int_{-\infty}^{\infty} exp [i (p_x x - p_x^2 t / 2 m)/h] \phi (p_x) dp_x$$ the expectation value <x> of the position coordinate satisfies the equation <x> = <x>_{t=t_0}...