Expectation Definition and 654 Threads

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 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 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...

Homework Statement Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix). Homework Equations I think the only one relevant is the expectation value: <a> = integral[psi*(a)psi] / integral[psi*psi] where psi* is...

Hi, I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable...

Homework Statement Psi(x) = Ax -a<x<a I am trying to find the probability that my measured momentum is between h/a and 2h/a Homework Equations I have normalized A= sqrt(3/(2a^3)) I know that if I was finding the expected momentum I would use \int\Psi * p \Psi dx The...

Homework Statement I am familiar with the following kind of conditional expectation expression: \mathbb{E}[Y|X=x], where X and Y are random variables. I am wondering what the following conditional expectation stands for: \mathbb{E}[Y|X] How these two are related? How the second...

In QFT expressions such as these hold: real scalar: \Delta_F(x-x')\propto\langle 0| T\phi(x)\phi(x')|0\rangle 4-spinor S_F(x-x')]\propto\langle 0| T\psi(x)\bar{\psi}(x')|0\rangle where T is the time-ordering operation and the proportionality depends on the choice of normalization...

Hi, I have the following problem: Suppose you have a coin that has chance p of landing heads. Suppose you flip the coin n times and let X denote the number of 'head runs' in n flips. A 'head run' is defined as any sequence of heads. For example the sequence HHTHHHHHTTTTHHTHT contains 4 head...

Homework Statement Suppose X ~ uniform (0,1) and the conditional distribution of Y given X = x is binomial (n, p=x), i.e. P(Y=y|X=x) = nCy x^{y} (1-x)^{n-y} for y = 0, 1,..., n. Homework Equations FInd E(y) and the distribution of Y.The Attempt at a Solution f(x) = \frac{1}{b-a} = \frac{1}{1-0}...

Show that the expectation value of Lz is -2h for the radial wavefunction Y2,-2. ? Can someone do this?

Homework Statement Show that the expectation value of the Coulomb potential v(\vec{r_1},\vec{r_2})=\frac{e^2}{|\vec{r_1}-\vec{r_2}|}, between two electrons depends on the relative orientation of spin of the two electrons. Assume each electron is in the product state form...

Is there any way of proving <p> = 0 for a discrete (bound) state given it's wave function? I've seen proofs using the hermitian properties of p but I'm interested in proving that the integral of Psi*(x) Psi'(x) is identically zero regardless of Psi(x) as long as it's a solution of Schroedinger's...

hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...

Ok I am not sure if I should put this question in the homework category of here but it’s a problem from schaums outline and I know the solution to it but I don’t understand the solution 100% so maybe someone can explain this to me. Let X and Y be defined by: \begin{array}{l} X = \cos \theta...

If I have H=p^2/2m+V(x), |a'> are energy eigenkets with eigenvalue E_{a'}, isn't the expectation value of [H,x] wrt |a'> not always 0? Don't I have that <a'|[H,x]|a'> = <a'|(Hx-xH)|a'> = <a'|Hx|a'> - <a'|xH|a'> = 0 ? But if I calculate the commutator, I get: <a'|[H,x]|a'> = <a'|-i p \hbar /...

I've been struggling with this problem for more than 4 days now: Let A, B and C be exponential distributed random variables with parameters lambda_A, lambda_B and lambda_C, respectively. Calculate E [ B | A < B < C ] in terms of the lambda's. I always seem get an integral which is...

Let A be an observable (opeator), and we're assuming that for a given state psi(x), the value of A is given by A acting on psi(x), namely - A|psi>. Also we assume that - P(x) = |psi(x)|^2 So, I'de expect the Expectation value of A to be defined like so: <A> = Integral[-Inf:+Inf]{ P(x) A...

Homework Statement Calculate the expectation value of \hat{H}' in the state \psi(x,t=0). \hat{H}'=k(\hat{x}\hat{p}+\hat{p}\hat{x}) \psi(x,t=0)=A(\sqrt{3}i\varphi_{1}(x)+\varphi_{3}(x)), where A=\frac{1}{2} Homework Equations The Attempt at a Solution I know it's found by...

I have small question computing vacuum expectation values here http://www.cns.gatech.edu/FieldTheory/extras/SrednickiQFT03.pdf" from Mark Srednicki. My problem is with equation 210 on the pdf page 69. In the second line of 210, where does the second term come from? Z(J) and W(J) are defined...

Homework Statement I have E(a) = 0, E(b) = x but E(|a+b|)=?? where E is the expectations operator and x is a known constant which is greater than zero. Homework Equations Any one know how I would go about determining E(|a+b|)?

Homework Statement Say, it is known that E_X[f(X)] = E_X[g(X)] = a where f(X) and g(X) are two functions of the same random variable X. What is the relationship between f(X) and g(X)? Homework Equations The Attempt at a Solution My answer is f(X) = g(X) + h(X) where E_X[h(X)] =...

Homework Statement Hello, have a stats question I am hoping you guys can help with. The expectation of a function g of a random variable X is: E[g(X)] = \int^{\infty}_{-\infty} g(x)fx(x)dx where fx is the pdf of X. For example, the particular expectation I am considering right now...

In the book "The mathematic of Gambling", the author considers a fair coin with 50% getting head and 50% for tail. The expectation of such coin, of course, will be zero. Here is what I read from the text, it reads Consider the fair game example mentioned earlier in the chapter (fair coin)...