Expectation Definition and 654 Threads

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[/color])? Can I let Z=XY and find...

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 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 (Σn[SIZE="1"]kx[SIZE="1"]k / Σn[SIZE="1"]k ). Quantum-mechanically n[SIZE="1"]k is represented by probability density(P), where P = ∫Ψ*Ψ...

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...

can anyone give me any ideas on how to evaluate this: <z>=<\Phi1|z|\Phi2> (for say hydrogen wavefunctions). Similarly <x+iy>=<\Phi1|x+iy|\Phi2> FYI, I'm trying to understand how radiation is polarised (an external B field polarises radiation, so we must consider the dipole...

Given x,y and z are standard normal distributions with mean 0 and standard deviation 1. x,y and z are also statistically independent. Find E{x|x+y+z=1}.

I am re-writing up some lecture notes and one of the proofs that E[X] for the negative binomial is r/p where r is the number of trials...The problem is there are a number of books that say r(1-p)/p is the correct expectation whilst others agree with 1/p Which one is correct...for what its...

Hi I am reading Hoare's original paper where he derives the complexity of quicksort. I am trying to figure how he derives the expectation for the number of exchanges (sorry if this is a very CS-specific question): \frac{(N-r-1)(r-1)}{N} \frac{N}{6}+\frac{5}{6N} I can't see...

Homework Statement A particle moves in a sequence of steps of length L. The polar angle \theta for each step is taken from the (normalized) probability density p(\theta). The azimuthal angle is uniformly distributed. Suppose the particle makes N steps. My question is how do I find the...

I'm trying to evaluate the expectation of position and momentum of \exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle} where \hat{a},\hat{a}^\dag are the anihilation/creation operators respectively. Recall \hat{x}...

Homework Statement I have a random, uniformly distributed vector with Cartesian components x,y,z. I should calculate the expectation value of the products of the components, e.g. <x\cdot x>, <x\cdot y>, ..., <z\cdot z>. Homework Equations In spherical coordinates the x,y,z components...

Let X, Y be independent exponential random variables with means 1 and 2 respectively. Let Z = 1, if X < Y Z = 0, otherwise Find E(X|Z) and V(X|Z). We should first find E(X|Z=z) E(X|Z=z) = integral (from 0 to inf) of xf(x|z). However, how do we find f(x|z) ?

Help me in conditional expectation Hi all.. I read one article couple days ago, yet, there is some equations that I could not understand. let assume that y = u + v where u is normally distributed with mean = 0 and variance = s -> u ~ N (0, s) and v is normally distributed with mean =...

[SOLVED] Conditional Expectation I'm trying to understand the following proof I saw in a book. It says that: E[Xg(Y)|Y] = g(Y)E[X|Y] where X and Y are discrete random variables and g(Y) is a function of the random variable Y. Now they give the following proof: E[Xg(Y)|Y] = \sum_{x}x g(Y)...

In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?

I have a random variable Y that represents the size of a population. I know that the expectation E(Y) = a. Now suppose, I have another random variable X that represents the number of people in that population that have a certain disease. The expectation is that on average half the population...

[SOLVED] expectation value of P^2 for particle in 2d box I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral...

[SOLVED] Expectation Values of Spin Operators Homework Statement b) Find the expectation values of S_{x}, S_{y}, and S_{z} Homework Equations From part a) X = A \begin{pmatrix}3i \\ 4 \end{pmatrix} Which was found to be: A = \frac{1}{5} S_{x} = \begin{pmatrix}0 & 1 \\ 1 & 0...

Let Y = a + bZ + cZ2 where Z (0,1) is a standard normal random variable. (i) Compute E[Y], E[Z], E[YZ], E[Y^2] and E[Z^2]. HINT: You will need to determine E[Z^r], r = 1, 2, 3, 4. When r = 1, 2 you should use known results. Integration by parts will help when r = 3, 4. I am struggling with the...

Hello, This time my question is not about Catalan numbers but something much more interesting (to me at least;)) I was wondering how the maximum of a multinormal random vector is distributed, for example let X \approx N(\mu_1,\sigma_1^2) Y \approx N(\mu_2,\sigma_2^2) be normally...

For a particle in the state Y(l=3, m=+2), how do I find <Lx^2> + <Ly^2> ? I'm lost. THanks!

Homework Statement If X is a real valued random variable with E[|X|] finite. <-> \sum(P(|X|>n)) finite , with the sum over all natural numbers from 1 to infinity. Homework Equations As a tip I am given that for all integer valued X>0 E(X) = \sum(P(X)>k , where the sum goes over all k =1 to...

I'm confused re a particle of energy E < V inside a square potential of width 'a' centered at x = 0 with depth V. They give the wavefunction for outside the well as \Psi(x) = Ae^{k|x|} for |x| > a/2 and k^2 = -\frac{2ME}{\hbar^2} => k = i\frac{\sqrt{2ME}}{\hbar} ? I need the probability that...

I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...

Homework Statement WTS \langle \hat{A} \rangle = \langle \hat{A} \rangle^\ast The Attempt at a Solution \langle \hat{A} \rangle^\ast = \left(\int \phi_l^\ast \hat{A} \phi_m dx\right)^\ast=\left(\int (\hat{A}\phi_l)^\ast \phi_m dx\right)^\ast= \int \phi_m^\ast \hat{A}\phi_l dx. So...

Homework Statement u(x) = \sqrt{\frac{8}{5}}\left(\frac{3}{4}u_{1}(x)-\frac{1}{4}u_{3}(x)\right) Determine the time-dependent expectation value of position of this wave function (the particle is in an infinite potential well between x = 0 and x = a). The Attempt at a Solution I...

Homework Statement How do I get the expectation value of operator \sigma using density matrix \rho in a trace: Tr\left(\sigma\rho\right) I have \sigma and \rho in matrix form but how do I get a number out of the trace?

I am trying to show that \frac{d}{dt}<x^2>=\frac{1}{m}(<xp>+<px>)....(1) With the wavefunction \Psi being both normalized to unity and square integrable Here is what I tried... <xp> = \int_{-\infty}^{\infty}{\Psi}^*xp{\Psi}dx <px> =...

Homework Statement Can somebody help me integrate \int{x\cdot p(x)} where p(x) is the Gaussian distribution (from here http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html) The Attempt at a Solution I can't really get anywhere. It's true that \int{e^{x^2}} has no analytical...

Hi More of a general integration question, but I just saw the following proof for the derivation of the expectation of a normal variable: E[X] = \frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^{\infty}{x exp\left( -\frac{1}{2\sigma^2}(x-\mu)^2 \right) dx} Set z=(x-mu): E[X] =...

I am just starting an introduction to quantum mechanics this semester, and it's hard for me to do some of my homework and follow some of the lectures because I can't grasp the actual 'physical' meaning of some of the concepts. What do they mean by the expectation values? For example...

I am trying to find <x> for \psi(x,t) = A exp\left(-|x|/L - i*E*t/\hbar\right) I found the normalization factor of 1/L and I took \int_{-\infty}^{\infty}\left( x * exp(|x|/L) \right) in two integrals however I got as a final result: L * -\infty * exp(-\infty/ L) - L *...

Homework Statement Find the expectation value for a hydrogen atom's radius if n=25 and l=0. Homework Equations expectation value = <f|o|f> where f=wavefunction and o=operator The Attempt at a Solution So I know that to find an expectation value, you integrate over all relevant...

Here's a silly question. I'm sure I should know the answer, but alas my most recent QM course was 9 years ago. I sat down to calculate the expectation value of momentum in the H-atom today, because some kid on another forum wanted to know how fast an electron in an atom is. I was going to...

Hi, Let x,z continuous random vectors and n discrete random vector: n=[n1,n2,...]. I'm trying to find for instance, E_z|n3{ E_n|z(x)} = ?. Thanks...