Expectation Definition and 654 Threads

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...

A probability distribution,f(x) ,can be represented as a generating function,G(n) , as \sum_{x} f(x) n^x . The expectation of f(x) can be got from G'(1) . A bivariate generating function, G(m,n) of the joint distribution f(x,y) can be represented as \sum_{x} \sum_{y} f(x,y) n^x m^y ...

I have two random variables X and Y, and I need to calculate E(XY). The expectation of X, E(X) = aZ, and the expectation of Y, E(Y) = bZ, where a and b are known constants and Z is a random variable. So the question is how would I calculate E(XY)? I was thinking that I could do the...

Very basic question which has confused me: if the variance of an expectation value <A> is: uncertainty of A=<(A-<A>)^2>^0.5 how is this equal to: (<A^2>-<A>^2)^0.5 ??

Homework Statement Prove that E(X) > a.P(X>a) Homework Equations E(X) is expectation, a is a positive constant and X is the random variable. (Note, > should be 'greater than or equal to' but I'm not too sure how to do it) The Attempt at a Solution Well I can show it easy enough...

Homework Statement 6) A particle in the infinite square well has the initial wave function Ψ(x,0)= Ax when 0<=x<=a/2 Ψ(x,0)= A(a-x) when a/2<=x<=a a) Sketch Ψ(x,0), and determine the constant A. b) Find Ψ(x,t) c) Compute <x> and <p> as functions of time. Do they oscillate? With what...

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}...

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...

Homework Statement Calculate the expectation values of x, x^2 for a particle in a one dimensional box in state \Psi_n Homework Equations \Psi_n = \sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a}) The Attempt at a Solution i formed the integral \int_{-\infty}^{+\infty}\Psi ^2 x dx as the...

Homework Statement I need to show that a particle in an infinite potential well in the nth energy level, obeys the uncertainty principle and also show which state comes closest to the limit of the uncertainty principle. This means i have to calculate <x>, <x^2>, <p> and <p^2>Homework...

How much sense does it make to compute expectation value of an observable in a limited interval? i.e. \int_a^b \psi^* \hat Q \psi dx. rather than \int_{-\infty}^{\infty} \psi \hat Q \psi dx Apparently, it shouldn't make any sense for it gives weird results when you compute e.v. of momentum for...

Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do...

I'm given that there is a harmonic oscillator in a state that is a superposition of the ground and first excited stationary states given by \psi = \frac{1}{\sqrt{2}}\abs{\psi_0(x,t) + \psi_1(x,t)}, where \psi_0 = \psi_0(x)e^{\frac{-iE_0t}{\hbar}} and \psi_1 = \psi_1(x)e^{\frac{-iE_1t}{\hbar}}...

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...

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>...

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}]...

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...

hi there. currently looking at the two conditions that must be met for a process to be wide sense stationary. The first constion is: E[X(t)] = constant what exactly does this mean??isn't is obvious that any random variable (with fixed time) will always yield a constant expextation. I...

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...

Problem 1.17 in griffiths gives, at time t = 0, the state psi =A(a^2-x^2) for -a to a, and 0 otherwise. It asks then to find the expected value of momentum p at 0 and also the uncertainty in p. How do I do this? The only way momentum is defined is md<x>/dt, and since the state is only for time...

I need help in solving the following problem: Let X be uniformly distributed over [0,1]. And for some c in (0,1), define Y = 1 if X>= c and Y = 0 if X < c. Find E[X|Y]. My main problem is that I am having difficulty solving for f(X|Y) since X is continuous (uniform continuous over [0,1])...

Any hints on how to solve for E(Y|X) given the ff: Suppose U and V are independent with exponential distributions f(t) = \lambda \exp^{-\lambda t}, \mbox{ for } t\geq 0 Where X = U + V and Y = UV. I am having difficulty finding f(Y|X)... Also, solving for f(X,Y), I am also having difficulty...

Is it possible to solve for E(Y) and var (Y) when I am only given the distribution f(Y|X)? I can solve for E(Y|X). But is it possible to find E(Y) and var(Y) given only this info?

Is it true that all expectation values must be real? So if I get an imaginary value, does it mean I made a mistake? Or it doesn't matter and I can just take the absolute value of the expectation? The momentum operator has an 'i' in it. But after doing, Psi*[P]Psi, I have an expression with 'i'...

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...

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)]...

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

I need help getting started on this problem: A free particle moving in one dimension is in the initial state \Psi(x,0) . Prove that <\hat{p}> is constant in time by direct calculation (i.e. without recourse to the commutator theorem regarding constants of the motion). Our professor...

Hey, I've got this problem from Peskin & Schroeder (chapter 15). I'm not particularly confident with functional integration, as I'm pretty new to it, and working through such a book by myself is pretty tricky in places. Well here goes The Wilson Loop for QED is defined as U_p(z, z)=\exp...

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...

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

Isn't consistently being inconsistent... consistency? Everything within it's own nature is consistent, and the human expectation and/or desire cannot change that to fit his scenario. Definitions of words in this world are lost in misconceptions. I often get made fun of for making these...

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...

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...

Hello everyone, This really has me stumped! The Washington state lottery has a new game called Zip Bingo. Every ticket costs $2 and consists of 2 regular Bingo cards with 35 call numbers. The prizes are as follows: Regular bingo on card 1: $2 Regular bingo on card 2: $3 Regular...

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...

I am new here and grateful to have found this site! I have a problem: The question is: Roll a fair die 3 times. Find the math expectation of the numerical sum of the outcomes of the rolls. I have no clue. I have looked over all notes, and I just don't "get it". Can someone help...

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...

What is the relationship between Expectations, Moments, and Autocorrelation. Can somone please please give me some examples? thanks

Hello, I have difficulty understanding the vacuum expectation: consider <0|A_{mu}|0>, we can understand it as the possibility ampitude of a photon turn into vacuum(although 0 in common), but in the spontaneous of gauge symmetry, we should understand <0|A_{mu}|0> as the strength of a...

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

X~N(0,1), Y=X^2~\chi^2(1), find E(XY). My thoughts are in the following: To calculate E(XY), I need to know f(x,y), since E(XY)=\int{xyf(x,y)dxdy}. To calculate f(x,y), I need to know F(x,y), since f(x,y)=d(F(x,y)/dxdy. F(x,y)=P(X\leq x, Y\leq y) \\ =P(X\leq x, X^{2} \leq...

Two quick ones :) Hi, two questions: 1) How can I find the expectation value of the x-component of the angular momentum, \langle L_x \rangle, when I know \langle L^2 \rangle and \langle L_z \rangle? 2) Say, I have a state |\Psi \rangle and two operators A and B represented as matrices...

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...

Given the wave function: \psi (x,t) = Ae^ {-\lambda \mid x\mid}e^ {-(i ) \omega t} where A, \lambda , and \omega are positive real constants I'm asked to find the expectation values of x and x^2. I know that the values are given by <x> = \int_{-\infty}^{+\infty} x(A^2)e^...

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...

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.

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...

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.

Here is a question that I have found in my quantum text which I have been thinking about for a few days and am unable to make sense of. If there is an operator A whose commutator with the Hamiltonian H is the constant c. [H,A]=c Find <A> at t>0, given that the system is in a normalized...