Expectation Definition and 689 Threads

1. What is the expectation value, <x>, for the given distribution over the interval from – to + infinity of the function: f(x)=e^(-.5(x-mu)^2(sigma^-2)) 2. This is a statistics problem i think. I just need to know how this type of problem is worked out because it is relevant to my...

Homework Statement If x is the position of a particle then the expectation value of x, <x> is : (I got lazy and just copied an image from Wiki, just pretend <x> is on the lhs of the eqn) When Griffith derives an expression for d<x> / dt, he uses the fact that dx/dt is zero, since "the wave...

Homework Statement Calculate the expectation values of p and p2 for a particle in state n=2 in a square well potential. Homework Equations \Psi(x,y) = (2/L)*sin(n1\pix/L)*sin(n2\piy/L) p= -i\hbar\partial/\partialx The Attempt at a Solution \int\Psip\Psidxdy...

Homework Statement Find the expectation value of position as a function of time. Homework Equations This is in the latter half of a multi-part question, previously we were given that: Eqn 1: Ψ(x, t) = A(ψ1(x)e−iE1t/h¯ + iψ2(x)e−iE2t/h¯) and in an even earlier part: Eqn 2: ψn(x) =...

Hi everyone, I have a feeling the following property is true but I can't find it stated in any textbook/online reference. Maybe it's not true... Can someone verify/disprove this equation? E(A+B|C) = E(A|C) + E(B|C)

Can anyone explain to me why the only time that the expectation of L^2 operator and the expectation value of L_3^2 are equal only when there is no angular dependence? And what does this mean? Does this have something to do with being restricted to the z-axis which is what L_3 is associated...

Homework Statement I'm wondering how I go about calculating the expectation of a random variable? Is it a different process for a discrete and a continuous? Can you show me an example? Say Poisson and expoential? Also, in the formula Var(X) = E[X^2] - (E[X])^2 how does one...

Hi. First post. I'm trying to understand if electronic energy levels have fixed values, or merely fixed expectation values (in the latter case, orbital electrons could have any energy and it's only the average that would be fixed). Here's my argument for the latter. If it's incorrect, could...

Given some state \left|\psi\right\rangle, and two operators \hat{A} and \hat{B}, how do you prove that if \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle then \hat{A} = \hat{B} ?

Given a stationary state H \psi = E \psi \Rightarrow \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right)\psi = E\psi Firstly is it true that \left<p\right> = \frac{\hbar}{i}\int\psi^* \frac{\partial \psi}{\partial x} dx= 0 ?? If it is, how do we prove it?

Does the following make sense: E(e^{-X}) = 0 \Rightarrow X = \infty\quad a.s. ? (Intuitively yes, but mathematically?) Thank you in advance for your help! :-) /O

Homework Statement Kindly take a look at the attachment for the problem statement. Homework Equations Schrodinger Equation: H psi = E psi The Attempt at a Solution For Part A: H psi = E psi S= 1/2 So energy of ground state is -1/2.K.H.hcross Could you tell me if I am on the...

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value from searching in internet I found that E{X|X>a}=a+E{x} but I can not prove it Help please

Homework Statement For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value Homework Equations The Attempt at a Solution

Suppose X is a uniformly distributed random variable on an interval [-a,a] for some real a. Let Y=X^2. Then what could you say about this distribution of Y? I have no idea how to think about this distribution. Also how could we compute the expectation of Y? I know that E[X]=0 but what could I...

Hi everyone, I was searching an answer for E(XY), where X and Y are two dependent random variables, number of observations n=21 and Sum(x*y)= 1060.84. Can somebody help me? It's not mentioned, but I think that each x and y of the distributions have the same probability to occur. Thank you.

Homework Statement how do you delete a post?

Hello, I want to find <xftf|x(t)|xiti> in harmonic oscillator. I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff. Thank you

I haven't posted any of my working for this as I only want to check my answer. Q. For a hydrogen atom with n=2, l=1, m=0 calculate <r^2> My answer = 0.75 * a^2 where a is the bhor radius. Am I right?

Homework Statement given a wavefuntion \Psi = (1/sqrt50) (3\mu1 + 5\mu2 - 4\mu3) what is the expectation value of the total energy? My thoughts were to calculate <\Psi|\hat{}H|\Psi> but the previous part to the question asks for the probability of each outcome(which I know how to...

Please teach me this: Why the minima of potential of classical Lagrangian is called the ''vacuum expectation value of Phi(field function)''.Is it really a vacuum expectation value of field operator at the vacuum states(at this state,the potential part of classical Lagrangian equals zero)...

Homework Statement I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0) \left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0 and \left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0...

Homework Statement Hi, I'm stuck on the very last part of 5.b https://www.maths.ox.ac.uk/system/files/attachments/PaperC2003.pdf Homework Equations The Attempt at a Solution I can't prove the inequality, would it be right to say the expected premium would be...

Hi, I've found the expectation value of Sz, which is hbar/2 (|\psiup|2 - |\psidown|2) by using the formula: <Si> = <\psi|Si\psi> where i can bex, y or z and \psi is the 'spinor' vector. I tried to find Sx using the same formula, however, I could only get as far as: hbar/2 ((\psiup)*\psidown...

Concerning expectation values... Also, the derivation in terms of bra-ket rather than wage function would be appreciated. Where \psi is the system state Knowing that <A>\psi=<\psi|A|\psi> And A is comprised of a complete eigenvector set \phij w/ corresponding eigenvalues aj How do you...

I have a problem understanding why I need the expectation of two variable which are dependent. What is the physical meaning of this E[xy]. I know that E[X] is the likelihood f finidng say a particle from a experiment repreated N time atthe same place. What Kind of physical meaning exist of two...

Homework Statement Hi Say I have an operator O, and I find its expectation value <O>. Now, if I wish to find the expectation value of O† († denoting Hermitian conjugate), then will this just equal the complex conjugate of <O>? Niles.

Homework Statement A particle of mass m that is confined to a harmonic oscillator potential V(x) = \frac{1}{2} m \omega^2 x^2 is described by a wave packet having the probability density, |\Psi (x,t) |^2 = \left(\frac{m\omega}{\pi\hbar} \right )^{1/2}\textrm{exp}\left[-\frac{mw}{\hbar}(x -...

I want to calculate E[x] of the following continuous distribution having density: f(x)=e^-2|x| for x in the reals (x e R) I did the calculation with integral bounds infinity and minus infinity, are these the right bounds to use since we are only told x e R? I got 0 as the answer, can someone...

Homework Statement Discrete random variables X and Y , whose values are positive integers, have the joint probability mass function , (, ) = 2−−. Determine the marginal probability mass functions () and (). Are X and Y independent? Determine [], [ ], and [ ]. The Attempt at a Solution...

Homework Statement Compute the complex conjugate of <p> using eq 1.35 (<p>=∫ψ*(h/i)∂/∂x ψ dx) and prove that <p> is real (<p>=<p>*) Homework Equations equation 1.35 is given above The Attempt at a Solution to take the c.c. don't i just add a minus to the i and switch the stars like...

Homework Statement (Question is #6 on p.171 in An Introduction to Probability and Statistics by Ruhatgi & Saleh) Let X have PMF Pλ{X=x} = λxe-λ/x!, x=0,1,2... and suppose that λ is a realization of a RV Λ with PDF f(λ)=e-λ, λ>0. Find E(e-Λ|X=1) The Attempt at a Solution The...

It is defined that the population variance is S^{2}= \frac{1}{N-1}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2} or \sigma^{2}= \frac{1}{N}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2}. Also that the V\left[\bar{y}_{n}\right] = \frac{N-n}{N}\frac{S^{2}}{n} = \left(\frac{1}{n} -...

Homework Statement Hey forum, I copied the problem from a pdf file and uploaded the image: http://img232.imageshack.us/img232/6345/problem4.png What is the probability that the measurement of L^{2} will yield 2\hbar^{2} Homework Equations \left\langle L^{2} \right\rangle = \left\langle \Psi...

Homework Statement Suppose that $Y$ is a random variable, $\mathcal{G}$ a $\sigma$-algebra, $E|Y| < \infty$. Show that $Y = E(Y|\mathcal{G})$ a.s. (a.s. = almost surely). Homework Equations We're given $Y$ integrable. The Attempt at a Solution It's recommended as a hint to prove...

Homework Statement A particle is represented(at t=0) by the wavefunction u(x,0) = A(a^2 - x^2) if -a<x<a = 0 otherwise Determine <x> & <p>. It is given in the book that in this case <p> \neq m*d/dt<x>. Could someone please tell me the reason...

Homework Statement Prove that for a particle in a potential V(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque: d/dt <L> = <N> Where N = r x(-del V) N, r, and L are vectors. Homework Equations...

Hi, I want to proof what the distribution will be when I apply a normal distributed x to a linear function y = a*x + b. What will be the mean and the variance of y ? The expectations can be calculated than with this formula ( probably with this formula what i want can be proofed with...

Homework Statement Show: <Jx>=<Jy>=0 Homework Equations Jx=1/2(J++J-) Jy=1/2(J+-J-) The Attempt at a Solution <jm l Jx l jm> = < jm l 1/2 J+ l jm> + < jm l j- l jm > = < jm l h/2 sqrt [(j-m)(j+m+1)] + h/2sqrt[(j+m)(j+m+1) l jm > i am not sure how to apply the next step

Homework Statement Hi Say I have the following number: \left\langle {\psi _i |A|\psi _j } \right\rangle 1) First of all, am I correct when saying that \left\langle {\psi _i |A|\psi _j } \right\rangle = \left\langle {\psi _j |A^\dag |\psi _i } \right\rangle ^* where...

Homework Statement The expectation value of the sum of two random variables is given as: \langle x + y \rangle = \langle x \rangle + \langel y \rangle My textbook provides the following derivation of this relationship. Suppose that we have two random variables, x and y. Let p_{ij}...

Hello, in relation to Markov chains, could you please clarify the following equations: In particular, could you please expand on why the first line is equal. Surely from , along with the first equation, this implies that: I just don't see why they are all equal. Please could you...

Homework Statement [PLAIN]http://img222.imageshack.us/img222/2781/statsqk.jpg Homework Equations f_{X} (x) = \int^{\infty}_{-\infty} f_{X,Y} (x,y)\;dy f_{Y} (y) = \int^{\infty}_{-\infty} f_{X,Y} (x,y)\;dx f_{X|Y} (x|y) = \frac{f_{X,Y} (x,y)}{f_Y (y)} f_{Y|X} (y|x) =...

Homework Statement (a) Let Q be an operator which is not a function of time, abd Let H be the Hamiltonian operator. Show that: i(hbar)(\delta<q> / dt =<[Q,H]> Here <q> is the expectation value of Q for any arbirtary time-dependent wave function Psi, which is not necessarily an...

Homework Statement Find E[X^Y], where X and Y are independent random variables which are uniform on [0,1]. Homework Equations The Attempt at a Solution I know that to get E[f(x)] for a function of one continuous random variable X, you integrate xf(x) between minus and plus infinity...

My understanding was that the expectation value of an observable H for a state |a> is just <a|H|a>. But in a homework problem, my prof. used <H> = <a|H|a>/<a|a>. I'm a little confused by the discrepancy, why the discrepancy?

In quantum mechanics, when is this true \langle\psi|AB|\psi\rangle=\langle\psi | A|\psi\rangle\langle\psi |B|\psi\rangle ? In probability theory, when the two variables are independent, the mean value of the product is the product of the mean values. What about QM?

Homework Statement the origial question given is: Show that the difference in energies between 2s and 2p radial wavefunctions is equal to the energy of the angular part of the 2p wavefunction, and thus that they have the same overall energy. hints given:a)use virial theorem to determine...

Homework Statement Evaluate <x^2> for the wave function \psi(x)=\int_{-\infty}^{\infty}dk exp(-|k|/k_0)exp(ikx) My calculation yields a negative answer and I can't find my error Homework Equations...

Homework Statement |O> = k |R1> + 1/9 |R2> a) Find k if |O> has already been normalized, and b) then the expectation value. The Attempt at a Solution a) To Normalise: |(|O>)|2 = (1/9 |R2> + k |R1>).(1/9 |R2> - k|R1>) = 1/81|R2>2 - k2|R1>2 = 1 I just assumed that |k| = (1-(1/81))0.5, but...