Expectation Definition and 689 Threads

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...

Homework Statement Say X has a density f(x) = 3x^(-4) if x > 1, and 0 otherwise. Now say X1,...,X16 are independent with density f. Let Y = (X1X2...X16)^(1/16). Find E(Y) and Var(Y). Homework Equations Var(Z) = E(Z^2) - [E(Z)]^2 E(Z) = Integral from -inf to +inf of z*f(z)dz The...

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> =...

QM Harmonic Oscillator, expectation values Hello. I am working on a problem involving the 1-dimensional quantum harmonic oscillator with energy eigenstates |n>. The idea of the exercise is to use ladder operators to obtain the results. I feel I am getting a reasonably good hang of this, but my...

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...

Homework Statement We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates...

A probability distribution,f(x) ,can be represented as a generating function,G(n) , as \sum_{x} f(x) n^x . The expectation of f(x) can be got from G'(1) . A bivariate generating function, G(m,n) of the joint distribution f(x,y) can be represented as \sum_{x} \sum_{y} f(x,y) n^x m^y ...

I have two random variables X and Y, and I need to calculate E(XY). The expectation of X, E(X) = aZ, and the expectation of Y, E(Y) = bZ, where a and b are known constants and Z is a random variable. So the question is how would I calculate E(XY)? I was thinking that I could do the...

Very basic question which has confused me: if the variance of an expectation value <A> is: uncertainty of A=<(A-<A>)^2>^0.5 how is this equal to: (<A^2>-<A>^2)^0.5 ??

Homework Statement Prove that E(X) > a.P(X>a) Homework Equations E(X) is expectation, a is a positive constant and X is the random variable. (Note, > should be 'greater than or equal to' but I'm not too sure how to do it) The Attempt at a Solution Well I can show it easy enough...

Homework Statement 6) A particle in the infinite square well has the initial wave function Ψ(x,0)= Ax when 0<=x<=a/2 Ψ(x,0)= A(a-x) when a/2<=x<=a a) Sketch Ψ(x,0), and determine the constant A. b) Find Ψ(x,t) c) Compute <x> and <p> as functions of time. Do they oscillate? With what...

Homework Statement I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square? Homework Equations \langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx \langle Q^2 \rangle = \int_{-\infty}^{\infty}...

1. Problem statement This isn't a homework question itself, but is related to one. More specifically, I'm computing the time-derivative of \langle x \rangle using the correspondence principle. One side simplifies to \left\langle \frac{\hat{p}}{m} \right\rangle, but what is the physical meaning...

Homework Statement Calculate the expectation values of x, x^2 for a particle in a one dimensional box in state \Psi_n Homework Equations \Psi_n = \sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a}) The Attempt at a Solution i formed the integral \int_{-\infty}^{+\infty}\Psi ^2 x dx as the...

Homework Statement I need to show that a particle in an infinite potential well in the nth energy level, obeys the uncertainty principle and also show which state comes closest to the limit of the uncertainty principle. This means i have to calculate <x>, <x^2>, <p> and <p^2>Homework...

How much sense does it make to compute expectation value of an observable in a limited interval? i.e. \int_a^b \psi^* \hat Q \psi dx. rather than \int_{-\infty}^{\infty} \psi \hat Q \psi dx Apparently, it shouldn't make any sense for it gives weird results when you compute e.v. of momentum for...

Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do...

I'm given that there is a harmonic oscillator in a state that is a superposition of the ground and first excited stationary states given by \psi = \frac{1}{\sqrt{2}}\abs{\psi_0(x,t) + \psi_1(x,t)}, where \psi_0 = \psi_0(x)e^{\frac{-iE_0t}{\hbar}} and \psi_1 = \psi_1(x)e^{\frac{-iE_1t}{\hbar}}...

When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate? Also as I am...

My question says: "Evaluate the expectation value <1/r> of the 1s state of hydrogen. How does this result compare to the result found using the Bohr theory?" Firstly, I have been told that <1/r> does not mean <1/r> but rather that it means 1/<r>. Having made it this far I now do the 1/<r>...

OK, here is the problem: An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is): \phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}]...

is there a better way to check for hermicity than doing expecation values? for example, what if you had xp (operators) - px (operators), or pxp (operators again); how can you tell if these combos are hermetian or not, without going through the clumsy integration (that doesn't give a solid...