Expectation Definition and 654 Threads

1. Let the joint pdf be f(x,y) = 2 ; 0<x<y<1 ; 0<y<1 Find E(Y|x) and E(X|y)Homework Equations E(Y|x) = \int Y*f(y|x)dy f(y|x) = f(x,y) / f(x) The Attempt at a Solution f(x) = \int 2dy from 0 to y = 2y f(y|x) = f(x,y)/f(x) = 1/2y E(Y|x) = \int Y/2Y dy from x to 1 = \int 1/2 dy from x to 1 =...

[b]1. consider this wave function ψ(x)=(√(30/L^5))(L-x) if 0≤x≤L and 0 else [b]2. Compute the expectation value of the momentum. Compute the expectation value of the kinetic energy. Compute Δ p⋅Δ x...

Hello, I was just curious about expectation values. One of the postulates of quantum mechanics state: The only possible results of a measurement is an eigenvalue of the operator. Now, is the expectation value considered a measurement, thus considered an eigenvalue? Thanks!

Homework Statement Show that < l,m | Lx2 - Ly2 | l,m > = 0 Homework Equations L2 = Lx2 + Ly2 + Lz2 [ Lx, Ly ] = i h Lz [ L, Lz ] = i h Lx [ Lz, Lx ] = i h Ly The Attempt at a Solution I tried substituting different commutation values in place of Lx and Ly, but I'm...

Hi guys, assume we have an equality involving 2 random variables U and X such that E(U|X) = E(U)=0, now I was told that this assumption implies that E(U^2|X) = E(U^2). However I'm not sure on how to prove this, if anyone could show me that'd be great!

difference between eigenvalue and an expectation value of an observable. in what circumstances may they be the same? from what i understand, an expectation value is the average value of a repeated value, it might be the same as eigen value, when the system is a pure eigenstate.. am i right?

Consider two Hermitian operator A, B; Define [A,B]=iC, then operator C is also Hermitian. we calculate the expectation value with respect to |a>, one eigenstate of A with the eigenvalue a. From the left side, we have: <a|[A,B]|a>=<a|(AB-BA)|a>=(a-a)<a|B|a>=0, while on the right side...

Homework Statement the first two energy eigenstates of a 1 nm wide finite well of barrier height 8vo have energy eigen values of 0.66ε and 2.6ε. calculate the expectation value of a linear superposition of these states? Homework Equations airy equations The Attempt at a Solution...

expectation value for a particle in a 1-D well how do i calculate the expectation value for the particles energy in a 1-D well. i have attached a word file, with my working out, just not quite sure if I am on the right track... i appreciate any help...thanks a mill

Homework Statement This question comes from calculating the Einstein A and B coefficients. I am supposed to find the average value of cos(x)^2 over the solid angle of a sphere which is 1/3. And I need to show this. A similar course in a different uni just says that For...

Homework Statement The probablity density function of the n-state of an electron is proportional to fn=(\frac{rz}{a_{0}})^{2n}e^ \frac{-2Zr}{\large na_{0}} show that the expectation value of the potential energy of the electron in the n-th quantum state of the hydrogen atoms is...

Dear all, I'm wondering, how one could justify mathematically the equality \int O(E(\vec{x}_1,...\vec{x}_N)) exp(-\beta E(\vec{x}_1,...,\vec{x}_N)) d\vec{x}_1...d\vec{x}_N = \int g(E) O(E) exp(-\beta E) dE where O(E(x)) is an observable and g(E) the density of states. Is there a...

Homework Statement Hi I have read a paper, where they want to find the average number of photons in a cavity. They have an expression for \langle{\hat a}\rangle, and then they use \langle{\hat a}\rangle^* = \langle{\hat a^\dagger}\rangle to find \langle{\hat a^\dagger \hat a}\rangle. I agree...

Homework Statement Hi My book uses the following in a calculation \left\langle a \right\rangle \left\langle {b^\dagger } \right\rangle + \left\langle {a^\dagger } \right\rangle \left\langle b \right\rangle = 2\operatorname{Re} \left[ {\left\langle a \right\rangle \left\langle {b^\dagger...

What does the expectation and deviation of an operator mean?? The way I understood it was every observable has a operator to it and the expectation of the observable uses the operator to calculate the deviation ... for ex :: <p>=integral( (si)* momentum operator (si) ) dx ... so what does...

A while back I posted a thread about the probability that someone who is alive at 12 years of age will be alive at age 82. This person could only die if he were murdered or if he died when he was greater than 82 years old. Now, let's assume this person will live forever if he's not murdered...

Homework Statement given \mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle ) where \mid1\rangle, \mid2\rangle are orthonormal calculate i)density operator ii) \langle A \rangle where A is an observable Homework Equations The Attempt at a Solution i) \rho = \frac{1}{2}...

Homework Statement Here's a link to an image of the exam question. It appears in the exam every couple of years, and it's due in my exam this coming week. I've looked in both the textbook and the course notes, and they simply *state* the conclusion, so I don't have a way of proving it, and...

Hi, (Sorry for the slight misnomer in the title... I can't edit it!) I'm doing several problems to compute the expectation value and variances of sub-samples & operations on the normal distribution; and I am having trouble getting results that agree with numerical simulations. I have several...

I was told that given a probability distribution p(x) dx, the expected value for x is given by: <x> = Ʃ xi P(xi) = ∫ x P(x) dx This part makes sense to me. It was justified to me through the use of weighted averages. However, my teacher then made a hand-wavy move to generalize the above...

Let Bt1, Bt2 and Bt3 be standard Brownian motions with ~N(0,1). Then what is E[Bt1.Bt2.Bt3] ? Any help would be much appreciated.

Dear All: I have a quite mysterious and cumbersome question concerning with the expectation values for a system of identical particles. For example, suppose I have a system of N identical bosons given by the wavefunction ψ(x1,x2,...xN), which is of course symmetrized. My concern is: 1...

Hi everyone, What is the difference between an expectation value and an average. I may have this wrong, but is it something along these lines: You perform a series of measurements on a given observable, such as momentum, and the average value of all these measurements is your expectation...

Homework Statement if x1 and x2 are dependent, and y1 and y2 are dependent, but all the x are independent of all the y. Then how can one simplify E(x1y1x2y2)? the textbook says E(x1x2)E(y1y2) So is the rule that you can not just separate two independent variables which they are...

My professor explained this concept absolutely horribly and I have no idea how to do these problems. Let A and B be independent Poisson random variables with parameters α and β, respectively. Find the conditional expectation of A given A + B = c. (Hint: For discrete random variables, there...

How can we compute the expectation value, <\widehat{x}\widehat{p}> where ψ(x) is a normalized wavefunction? (The result is i\hbar/2)

Homework Statement At time t=0 a particle is described by a one dimensional wavefunction (capital)ψ(x,0)= (2a/)^(1/4) e^(-ikx)e^(-ax^2) (three lines)=(2a/)^(1/4) e^(-ikx-ax^2)--------equation 1 k and a are real positive constants Homework Equations I think this is the one <p subscript(x)> =...

Hi, as I am new in Matlab, so I need your help. I want to replace the following inverse matrix (X'*X)^-1 with its expectation value: E{(X'*X)^-1} = E{|1/Xk|^2}I X'*X and (X'*X)^-1 is a diagonal matrix. Could anyone give me an Idea how to write it in MATLAB this expectation value...

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...

Homework Statement Find the expected value of cos(A+B) where A is a constant and B is a random variable with a pdf f(b). Present the answer in terms of f(b). The Attempt at a Solution I don't know how far I can go with the answer -- I have tried for a bit now to remove an integral with no...

If I am given the CDF of a piecewise mixed distribution density starting from a and ending at b, would the expected value just be a + integral(all the pieces) ?

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}...

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

How does on calculate the expectation of the position operator x in a 2D infinite potential well (in the xy plane)? Do we only work with the Psi to the Hamiltonian in that particular coordinate when finding <Psi|x|Psi>?

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...

I'm a bit confused about the nature of probability conservation and expectation values. According to probability conservation, \frac{∂P(r,t)}{∂t}=0. Does that mean that expectation values e.g. <x>, <p> and <E> depend only on the position of the particle and not on time? Thanks

I am reading the fine structure article from Wikipedia at http://en.wikipedia.org/wiki/Fine_structure. Under the heading 'Kinetic energy relativistic correction', we have the following: For the hydrogen atom, V = e2/r. This implies that the expectation of V = -e2/a0n2. Now, I know that...

I know the E[X] = Integral between [-inf,inf] of X*f(x) dx Where X is normally distributed and f(x) is the PDF How do I find the expectation of X4? Bare with me because I'm useless in Latex So far what I've done is written the integral as Integral between [-inf,inf] of X4*f(x) dx...

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>?

I know that the formula for the expectation value is: <Q(x,p)> = ∫ψ*Q(x,(h/i)d/dx)ψ dx For instance, the expectation value for momentum is. -ih∫ψ*(dψ/dx)dx But, why? How is it derived?

"A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. Let a person choose, at random and without replacement, 3 chips from this bowl. If a person is to receive the sum of the resulting amounts, find his expectation." Here is my attempt: The possible...

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...

Homework Statement Let Ω = [0,1] with the σ-field of Borel sets and let P be the Lebesgue measure on [0,1]. Find E(X|Y) if: Homework Equations X(w)=5w^2 Y(w)= \left\{ \begin{array}{ll} 4 & \mbox{if $w \in [0,\frac{1}{4}]$} \\ 2 & \mbox{if $w \in (\frac{1}{4},1]$} \\ \end{array}...

Is it possible to express ANY observable A(X,P) in terms of the ladder operators? I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be...

So I'm a little confused on the notation when working with wave functions constructed as a linear combination of an orthornormal basis set. Like on the form: \Phi=Ʃn cnψn If I want to find the expectation value represented by the operator O for the state described by \Phi, I would...

So for example, if I have a random variable X, take it to be normally distributed. How do you find the expectation and variance of the random variable e^X in terms of μ and σ? Integrating the entire normal function with the f(x) is it?

Homework Statement A wave function ψ is A(eix+e-ix) in the region -π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x=0 and x=π/8, and (b) between x=0 and x=π/4. Homework Equations The Attempt at a Solution So to...

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...

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

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!