Expectation Definition and 689 Threads

Hello, I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following, c = \frac{(a + ib)}{\sqrt{2}}...

Homework Statement Let Bt be a standard Brownian motion. Let s<t: a) Compute P(\sigma B_{t}+\mu t|B_{s}=c) b) Compute E(B_{t}-t|B_{s}=c) Homework Equations Defition of brownian motion: B(t) is a (one-dim) brownian motion with variance \sigma^{2}if it satisfies the following conditions: (a)...

Every quantum mechanical operator has an observable in classical mechanics <x> - position ... <x^2> - ? <p^2> - ? What is the meaning on these expectation values? v^2 = <x^2> - <x>^2 What is the meaning of this? edit: It looks to me like uncertainty in position. Is it the average...

fx,y = 6(x-y)dydx, if 0<y<x<1 how do you find E(XY), i know the formula...g(x,y)fxy(x,y)dydx but i don't know what 'g(x,y)' represents and the limits to use??

Homework Statement The variance of an observable Qhat in a state with wavefunction psi is, (delta Qhat)2=<(Qhat-<Qhat>)2> Show that this can be written as, (delta Qhat)2=<Qhat2>-<Qhat>2 Homework Equations As above. The Attempt at a Solution (delta...

Homework Statement calculate <x>, when \Psi(x,t)=A*exp(-(\sqrt{}Cm/2h)x^{}2 Homework Equations <x>=\int\Psi^{}*x\Psidx over all space.. \intexp(-\alphax^{}2)=\sqrt{}\pi/\alpha The Attempt at a Solution ok know how to do this but how do i do the intergral... my maths isn't so good...

http://img23.imageshack.us/img23/1649/93412460.th.png For question 2 in the above link, I calculated the expectation of the energy by E=<\hat{H}>=\int_0^a \psi^* \hat{H} \psi dx where \psi=\psi^*=x(a-x) this gave E=0. this answer confused me for two reasons: (i) is it ok for the...

Homework Statement http://209.85.48.12/3560/8/upload/p2791776.jpg Homework Equations The most relevant identity to the part that I'm confused about is the following identity: for any cumulative distribution function F, with the inverse function F-1, if U has uniform (0,1) distribution...

Hi all. I have a question which arose from the answer of a homework problem. A particle is in the state given by \left| \psi \right\rangle = \frac{1}{{\sqrt 3 }}\left[ {\left| \psi \right\rangle _1 + \left| \psi \right\rangle _2 + \left| \psi \right\rangle _3 } \right], where {\left|...

Hi quick question: Suppose you have a function of random variables given in the following way Z=X if condition A Z=Y if condition B where both X and Y are random variables, and conditions A & B are disjoint. Then would the expectation of Z be E[Z]=E[X]*Pr(A)+E[Y]*Pr(B)? Thanks in advance.

Homework Statement Let X and Y be contnious random variables with joint probability density function - f(x,y) = 10x^2y if 0<x<y<1 0 othewise a) Determine P( Y < \frac{X}{2}) b) Determine P(x \leq 1/2 | Y < X^2) c) Determine the marginal density functions of X and Y, respectively...

This result isn't in our book, but it is in my notes and I want to make sure it's correct. Please verify if you can. Homework Statement I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z. E(Y \...

If W=g(X) is a function of continuous random variable X, then E(W)=E[g(X)]= ∞ ∫g(x) [fX(x)] dx -∞ ============================ Even though X is continuous, g(X) might not be continuous. If W happens to be a discrete random variable, does the above still hold? Do we still integrate ∫...

Is the expectation value of momentum/position/energy the value that we're most likely to measure? So suppose we measure 100 particles with the same wavefunction, would we expect most of them to have momentum/position/energy that's equal to the expectation value? And I was wondering, how do we...

Homework Statement prove or disprove that E[X^2] = E(X)^2 Homework Equations E[X] = \sumxi*pr(xi) The Attempt at a Solution I really don't know where to start, I believe that it is not true, so I should try to disprove it, and the easiest way to do that would be by...

Hi, anyone help please. Let X and Y are independent uniform random variables over the interval [0,1] E[|X-Y|a]=? where, a>0

Thanks for all the help on the first question but now I have to solve for <T>. I have no idea how to do this, and I could use some help for a kick start. thanks!

first post! but for bad reasons lol Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2 i solved for x: x=sqrt(h/2mw)((a+)+(a-)) so <x> integral of si x ((a+)+(a-)) x si. therefor the integral of si(n+1) x si + si(n-1) x si. si(n+1)...

Homework Statement Consider a wave function \psi (x,t) = R(x,t) exp(i S(x,t)) what is the expectation value of momentum?Homework Equations <f(x)> = \int^{\infty}_{-\infty} \psi^* f(x) \psi dx \hat{p} = -i \hbar \frac{\partial}{\partial x} The Attempt at a Solution This is for an intro to...

Homework Statement The expectation value <r> of the electron-nucleus separation distance 'r' is: <r> = ʃ r |ψ|² dV. (a) Determine <r> for the 1, 0, 0 state of hydrogen. The Attempt at a Solution Right, I've obtained the value for ψ = (1/πa³)^1/2 exp(-r/a) I then...

Homework Statement I have to prove the following: \hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle Edit: L = Angular Momentum & N = Torque Homework Equations I used Ehrenfest's theorem, and I've got the equation in the following form: \frac{1}{i} \left(\left[L,H\right]\right) +...

Homework Statement I need to find the expectation value for E but I don't know how b acts on the vacuum state. Homework Equations b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t) | \psi(t)\rangle = b^\dagger| 0\rangle The Attempt at a Solution \langle \psi(t) |...

Homework Statement I we know the eigenstates of the system be |\psi_1\rangle and |\psi_2\rangle. Current state of the system is |\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle Try to find the expectation value of electric dipole moment \mu (assume it is real) and write it in...

Homework Statement Obtain an expression for the expectation value <Pxn>n N=1,2... of a particle in an infinite box ( V=\infty for x<0 and x>L ; V=0 for 0<X<L) which is in an eigenstate of the energy. Homework Equations Pn =+- \sqrt{2*m*En } = +- (n*pi*Hbar) / L The Attempt at a...

Homework Statement Hi all, i have a problem: i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). Ψ(x,t) = (1/sqrt2)[Ψ0(x).e-[i(E0)t/h] + Ψ1(x)e-[i(E1)t/h]], where Ψ0,1(x) are the ground and...

Homework Statement For a given wave function Psi(x,t)=Aexp^-(x/a)^2*exp^-iwt*sin(kx) find the expectation value of the linear momentum. Homework Equations <p>=integral(-inf,inf) psi* p^ psi dx p^=-ih(bar) d/dx sin x = (exp ix - exp -ix)/2i cos x = (exp ix + exp -ix)/2 The Attempt...

Quantum homework - Average Expectation Values?? Hi people, I'm struggling with my quantum mechanics homework - I've included links to photographs of my attempts at solutions, but i know they are wrong because I am given what the answers are supposed to be. Can somebody help me spot where I am...

If \Psi (x,t) = \psi (x) g(t), should I then use \Psi or \psi when calculating <p> and <p ^2>?

Homework Statement Calculate \Delta x = \sqrt{\left\langle(x - \left\langle x \right\rangle )^2 \right\rangle} if \left\langle x \right\rangle = 0 and \left\langle x^2 \right\rangle = a^2(\frac{\pi - 6}{12 \pi^2}) 2. The attempt at a solution \left\langle(x - \left\langle x...

I seem to be having a rather difficult time understand all the details of the notation used in this quantum material. If I'm given the eigenvalue equations for L^{2} and L_z and L_{\stackrel{+}{-}} for the state |\ell,m>, how do I compute <L_{x}> using bra-ket formalism? I know that L_x =...

Homework Statement A particle is in a infinite square poteltian well between x=0 and x=a. Find <p> of a particle whose wave function is \psi(x) = \sqrt{\frac{2}{a}}sin\frac{n \pi x}{a} (the ground state). 2. The attempt at a solution <p> = \frac{2 \hbar k}{\pi} \int^{a}_{0}sin^{2}...

Hi, You know famous equation, \frac{d<A>}{dt} = <\frac{i}{\hbar}[\hat{H},\hat{A}] + \frac{\partial\hat{H}}{\partial t} > But liboff said if \frac{\partial \hat{A} }{\partial t} = 0 then, \frac{d<\hat{A}>}{dt} = 0 this is the proof \frac{d<A>}{dt} =...

the number of hairs Nsub.1 on a certain rare species can only be the number 2sup.l(l=0,1,2...) The probability of finding such an animal with 2sup.l hairs is exp-1/l ! what is the expectation,<N>? what is deltaN?

Let X and Y be two random variables. Say, for example, they have the following joint probability mass function x -1 0 1 -1 0 1/4 0 y 0 1/4 0 1/4 1 0 1/4 0 What is the proper way of computing E(XY)? Can I let Z=XY and find E(Z)=∑...

How does this follow from the defintion of the expectation value of x

Homework Statement Here is another True or False question from the same practice test. Since the expectation of the momentum operator <p>=<n|pn> is zero for an energy eigen state of the harmonic oscillator, a measurement of the momentum will give zero every time (True or False)...

I need help about conditional expectation for my research. I get stucked on this point. Could anyone show me how to prove that: "Let E[|Y|]<∞. By checking that Definition is satisfied, show that if Y is measurable F0, then E[Y|F0]=Y." Def: Let Y be a random variable defined on an underlying...

Homework Statement Hi all. The expectation value for S_x (spin in x-direction) is: \left\langle {S_x } \right\rangle = \left\langle {\phi |S_x \phi } \right\rangle = \phi ^\dag S_x \phi where \phi is the state and \phi^"sword" is the hermite conjugate. My question is: I...

Homework Statement A cell diverges into X new cells. Each of them reproduces in the same manner. X is a geometric random variable with success parameter of 0.25. What is the expectation of the number of great-grandsons a cell have? 2. The attempt at a solution I thought about using the...

I am having some great difficulty getting intuition out of the standard quantization of the Klein-Gordon Lagrangian. consider the H operator. In QM, the expectation values for H in any eigenstates |n> is just <n|H|n> now, in QFT, suppose I have a state |p> in the universe, what do I get if I...

Homework Statement I need to find the expectation value of x of an aharmonic oscillator of a given potential: V_{(x)} = c x^2 - g x^3 - f x^4 Homework Equations Two relevant equations: First: I'm using the partition function to find the expectation value <x>= \frac { \int x Z...

Homework Statement First off, this is my first time posting here so please excuse any editing mistakes or guidelines I may have overlooked. This is problem 1.17(c) from Griffiths, Introduction to Quantum Mechanics 2nd edition. It reads: \Psi(x, 0) = A(a^2 - x^2), -a\leqx\leqa. \Psi(x, 0)...

Homework Statement Hi all. Let's say that i have a wave function \Psi (x,t) = A \cdot \exp ( - \lambda \cdot \left| x \right|) \cdot \exp ( - i\omega t) I want to find the expectation value for x. For this I use \left\langle x \right\rangle = \int_{ - \infty }^\infty x \left| \Psi...

Problem Consider an operator \hat{A} whose commutator with the Hamiltonian \hat{H} is the constant c... ie [\hat{H}, \hat{A}] = c. Find \langle A \rangle at t > 0, given that the system is in a normalized eigenstate of \hat{A} at t=0, corresponding to the eigenvalue a. Attempt Solution We...

On pages 16-17 of Griffith's Intro to QM, he writes the following: \frac{d\left\langle x \right\rangle}{dt}= \int x \frac{\partial}{\partial t}|\Psi|^{2} dx = \frac{i\hbar}{2m}\int x \frac{\partial}{\partial x} \left( \Psi^{*}\frac{\partial\Psi}{\partial x}- \frac{\partial\Psi^{*}}{\partial...

The state \Psi = \frac{1}{\sqrt{6}}\Psi-1 + \frac{1}{\sqrt{2}}\Psi1 + \frac{1}{\sqrt{3}}\Psi2 is a linear combination of three orthonormal eigenstates of the operator Ô corresponding to eigenvalues -1, 1, and 2. What is the expectation value of Ô for this state? (A) 2/3 (B)...

I need some help with "law of total expectation". Sorry for my English, I don't know the right English expressions. The Problem is: People come (show in) with with Poisson rate of 10 people per hour. There is a 0.2 chance that a person will give money to a beggar sitting in the corner. The...

We all know the concept of expectation value,it is the average of all possible outcomes of an experiment. Mathematically average of x is written as (Σnkxk / Σnk ). Quantum-mechanically nk is represented by probability density(P), where P = ∫Ψ*Ψ d3r, then <r> = ∫ r P(r) d3r -----------(1)...

I am quite new to Quantum Mechanics and I am studying it from the book by Griffiths, as kind of a self-study..no instructor and all... For a gaussian wavefunction \Psi=Aexp(-x^{2}), I calculated <p^{2}> and found it to be equal to ah^{2}/(1-2aiht/m) (By h I mean h-bar..not so good at...

i'm just not sure on this little detail, and its keeping me from finishing this problem. take the arbitrary operator \tilde{p}^{n}\tilde{y}^{m} where p is the momentum operator , and x is the x position operator the expectation value is then <\tilde{p}^{n}\tilde{y}^{m} > is this the same...