Expectation Definition and 689 Threads

Homework Statement Homework Equations The Attempt at a Solution I am just trying to figure out how to start the problem. Any help would be greatly appreciated.

Going from the abstract state vector lψ> and the mean-value of an observable x (operator) given by: <x> = <ψlxlψ> I want to show how that is done in the position basis: So I take: <x> = <ψlxlψ> And insert completeness in front of the state vector to get the expansion involving the...

My professor made a rather concise statement in class, which sums to this: E(Y|X=xi) = constant. E(Y|X )= variable. Could anyone help me understand how the expectation is calculated for the second case? I understand that for different values of xi, we'll have different values for the...

1. Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2. A) Calculate the characteristics function Φ_T(u,v,w) of T. B) Calculate the density of T. 2. Let X and Y be N(0,1) (standard normals), not necessarily...

So, this has been bothering me for a while. Lets say we have the wavefunction of a harmonic oscillator as a general superposition of energy eigenstates: \Psi = \sum c_{n} \psi _{n} exp(i(E_{n}-E_{m})t/h) Is it true in this case that <V> =(1/2) <E> . I tried calculating this but i...

Good Evening Fellows, I have the following question, So far I have learned that the expectation value of momentum is equal the time derivative of the expectation value of position. If the potential only depends upon position and not on time. Then, if we use the time independent schrodinger...

Homework Statement I am just trying to figure out how to calculate the expectation of something. The context is for a random sample from a normal distribution with known mean μ and unknown variance σ2. Homework Equations 3. The solution So for the purposes of this question we set θ = σ2 I...

Suppose that α and β are independently distributed random variables, with means; μ_α, μ_b and variances; δ_α^2, δ_β^2, respectively. Further, let c=αβ+e, where e is independently distributed from α and β with mean 0 and variance δ_e^2. Does it hold that E(αβ | c) = E(α|c)...

A robot arm solders a component on a motherboard. The arm has small tiny errors when locating the correct place on the board. This exercise tries to determine the magnitude of the error so that we know the physical limitations for the size of the component connections. Let us say that the...

Hi, Please refer to this book (in google archive), and go to section 7.7 (page 85)...

I have to find the expectation value of the z component of the angular momentum for a particle on a ring and the expectation value of the z component of the angular momentum squared for a particle on a ring. The wavefunction is e^((± imx)) I've determined that the expectation value for the...

Homework Statement Show that the expectation value for r for an electron in the groundstate of a one-electron-atom is: <r>=(3/2)a_{0}/Z Homework Equations Expectationvalue: <f(x)>=∫\psi*f(x)\psidx, -∞<x>∞ \psi_{100}=C_{100} exp(-Zr/a_{0}), a_{o}\ =\ 0.5291\ \times\ 10^{-10}m , h\...

Homework Statement Assume that a particle travels with a certain known (average) velocity ##v = \left\langle\hat{p}/m\right\rangle##. You know it's position with an uncertainty ##Δx##. Use the uncertainty principle to determine the least possible value for the article's kinetic energy...

I am asked a problem where I'm supposed to integrate the expectation value of a dynamic variable (operator) to solve a differential equation. OK, is the expectation value supposed to be a variable? But it seems to me like its a definite integral over allspace and thus is a number. So...

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...

Homework Statement What is the expected number of flips of a biased coin with probability of heads 'p', until two consecutive flips are heads?Homework Equations The Attempt at a Solution Let T_1 = first flip is tails, H_1 = first flip is heads. and T_2, H_2 for second flip. \mathbb{E}[X] =...

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...