Expectation Definition and 689 Threads

Homework Statement If X1 has mean -3 and variance 2 while X2 has mean 5 and variance 4 and the two are independent find a) E(X1 - X2) b) Var(X1 - X2)The Attempt at a Solution I am not very clear on what I am supposed to be doing for this problem. I don't fully understand this expectation value...

Homework Statement X ~ Uniform (0,1) Y = e-X Find FY (y) - or the CDF Find fY(y) - or the PDF Find E[Y] 2. Homework Equations E[Y] = E[e-X] = ∫0 , 1 e-xfx(x)dx FY(y) = P(Y < y) fY(y) = F'Y(y) The Attempt at a Solution FX(x) = { 0 for x<0 x for 0<x<1 1 for 1<x } fX(x) = { 1 for...

Homework Statement Under what conditions is \left\langle{{\mathbf{x} \cdot \mathbf{p}}}\right\rangle a constant. A proof of the quantum virial theorem starts with the computation of the commutator of \left[{\mathbf{x} \cdot \mathbf{p}},{H}\right] . Using that one can show for Heisenberg...

I'm given an operator $\mathcal{L}$ is Hermitian, and asked to show $<\mathcal{L}^2>$ is $\ge 0$ I believe $<\mathcal{L}>$ is the expectation value, $=\int_{}^{}\Psi^* \mathcal{L} \Psi \,d\tau $ (Side issue: I am not sure what $d\tau $ is, perhaps a small region of space? And the interval?) I...

Hi all, Let X be a random EDIT variable with (infinite) sample space S. Are there some results dealing with how to maximize E(X|s ) (conditional expectation of X given s ) for s in S ? Thanks.

While reading a paper, i came across the following Expectations: Given that the ##E\left\{e^2_{n-i-1}e^2_{n-j-1}\right\}=E\left\{e^2_{n-i-1}\right\}E\left\{e^2_{n-j-1}\right\}## for ##i\neq j##.\\ Then as ##n\rightarrow\infty## ##E\left\{\left(\sum\limits_{i=0}^{n-2}\alpha^i...

Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...

Hey, I'm stuck halfway through the solution it seems. I could use some tips on how to continue. 1. Homework Statement I have to determine a linear combination of the states |0\rangle, |1\rangle, of a one dimensional harmonic oscillator, so that the expectation value \langle x \rangle is a...

Homework Statement Consider the following inital states of the symmetric 2D harmonic oscillator ket (phi 1) = 1/sqrt(2) (ket(0)_x ket(1)_y + ket (1)_x ket (0)_y) ket (phi 2) = 1/sqrt(2) (ket(0)_x ket(0)_y + ket (1)_x ket (0)_y) Calculate the <p_x (t)> for each state Homework EquationsThe...

Homework Statement Homework Equations $$ \psi_{100} = \frac {1}{\sqrt{\pi a^{3}}} e^{-r/a} $$ The Attempt at a Solution a) $$\langle r \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{3} e^{-2r/a} dr$$ This comes out to be ##\frac {3}{2}a##...

Homework Statement Demonstrate that the expectation value of momentum (p) for the wave function: ψ(x)∝e^(-γx) when x>0, ψ(x)=0 when x<0. Hint: Pay special attention to the discontinuity at x=0.[/B] Homework Equations <p>=<ψ|p|ψ>=∫dxψ*(x)[-iħ∂/∂x]ψ(x) from -∞ to ∞. [/B]The Attempt at a...

Hello, I've been trying to find <p'|φ(x)|p> for a free scalar field. and integral of <p'|φ(x)φ(x)|p> over 3d in doing the space In writing φ(x) as In doing the first, I get the creation and annihilation operators acting on |p> giving |p+1> and |p-1> which are different from the bra state |p>...

Hi, I am really really stuck on my life-decision (look at my previous posts). I do REALLY LOVE PHYSICS A LOT. I want to be able to work in the next big thing - Quantum Computation, Theory Of Everything and so on. I am passionate about physics it grabs my interest straight away. I am only 15, I...

Why does the expectation values of some operators, such as 'number' operator ##a^\dagger a## and atomic population operator ##\sigma^\dagger\sigma##, are always nonnegative? Can we prove this from a mathematical point? For example, are these operators positive semidefinite?

I have an inner product ## \langle \alpha|f| \beta \rangle## where ##f## is an operator that is a function of position ##x## operator (1D). According to the book I read (and I'm sure in any other book as well), that inner product can be written in position representation as ## \int...

If I have the following expectation value for a general operator A < psi | cA | psi > where c is a complex constant and I want to take c outside the bracket does it go as c or its complex conjugate ?

Homework Statement I will denote operators by capital letters. The question is calculate <p | XXPP | x> / <p | x > Homework Equations X |x> = x |x> P |p> = p |p> P |x> = -i(hbar)d/dx X |p> = i(hbar)d/dp The Attempt at a Solution If I start on the RHS and take PP out I get...

Hi, I'll start by sketching the specific model I'm looking at, a quantum spin chain. This is defined as N spins (2 basis states) on a 1D lattice i.e. the sites are a subset of ##\mathbb{Z}##. Then we defined the state-space as \{ \psi\in\mathcal{H}_N \text{ with } ||\psi|| = 1\}\cong S^d...

Hello, I am very confused how this is true? Where does this come from?? $$<f| \hat{Q}f> = (\sum_{n}a_{n}^{*} |\psi_{n}>)(\sum_{m}a_{m} \hat{Q} | \psi_{m}>)$$ thanks

Homework Statement The expectation value of <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx For the Guassian wave-packet ψ(x)=(1/(π^1/4)(√d))e^-((x^2)/(2d^2)) Limits on all integrals are ∞ to -∞. Homework Equations <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx ψ(x)=(1/(π^1/4)(√d))e^(iKx)-((x^2)/(2d^2)) The Attempt at a Solution Ok...

Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ. Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying...

Can I ask a basic question. This was a question in a test, I could not solve this. When is it true that the result of a single measurement for a dynamical variable is equal to the expectation value of the operator corresponding to that dynamical variable? Thank you for your help. Sincerely...

Where one can find a proof of the expectation value of operator expression. <A> = < Ψ | A | Ψ > or <A> = integral( Ψ* A Ψ dx ) Thanks.

Homework Statement The problem asks me to find the expectation value of W. Homework Equations The given ψ[x,t] is Asin(πx/a) e^((-i Eot)/ħ). By QM postulate 2 the QM operator of W is: iħ δ/δt or equivalently -ħ/i δ/δt. The Attempt at a Solution <w>=∫ψ*iħδ/δtψ= iħδ/δt 1/(2e^(-iEot/)ħ)...

If you have some wave function of some particle, say... |¥> And you calculate the expectation value of momentum, say... <¥|p|¥> What ensures that that spatial integral is real valued? Separately, all the components of the integral are complex valued

Homework Statement I'm curious in proving that expectation value of momentum for any bound state is zero. So the problem is how to prove this.Homework Equations $$ \langle \mathbf{p_n} \rangle \propto \int \psi^*(\mathbf{r_1}, \dots ,\mathbf{r_N}) \nabla_n \psi(\mathbf{r_1}, \dots...

If (X, Y) has the normal distribution in two dimensions with zero means and unit variances and correlation coefficient \rho, then to prove that the expectation of the greater of X and Y is \sqrt{(1-\rho)\pi}. How to proceed with it? Help please.

I have come across a bit of conflict in wording of some physics and chemistry textbooks about the probability of finding particles in certain places. To be more specific, I have come across 3 different statements: 1. $$\int_a^b {| \psi(x) |^2 dx}$$ The above integral is said to give the...

Homework Statement It's an old assignment for exam, but the solution manual gives little help: Describing a particle of mass m moving in one dimension (x) the wave function at time t=0 is: ## \Psi(x,t=0) = A \frac{1}{\sqrt{(x-x_0)^4 + l^4}} ## ##x_0## and ##l## are positive constants...

The problem is:Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...

Homework Statement The position-space representation of the radial component of the momentum operator is given by ## p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right ) ## Show that for its expectation value to be real:## \left \langle \psi|p_r|\psi \right...

Homework Statement Consider a one-dimensional particle subject to the Hamiltonian H with wavefunction \Psi(r,t) =\sum_{n=1}^{2} a_{n}\Psi _{n}(x)e^{\frac{-iE_{n}t}{\hbar}} where H\Psi _{n}(x)=E_{n}\Psi _{n}(x) and where a_{1} = a_{2} = \frac{1}{\sqrt{2}}. Calculate the expectation value of the...

Homework Statement Hello, I'm a bit confused about the calculation of the expectation values. Normally, when I have a wave function of sort and I want to calculate the expectation value of some operator, I just insert it into the braket <ψ|A|ψ>, where ψ for example is a wave function composed...

Homework Statement Here's the problem with the solution provided: Homework Equations Fundamental Theorem of Calculus (FToC) The Attempt at a Solution So I understand everything up to where I need to take the derivative of the integral(s). Couple of things I know is that the derivative of...

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

When calculating the expectation value of momentum of a real wavefunction is it always zero ? The momentum operator introduces an i into the integral and with real wavefunctions there is no other i to cancel and all Hermitian operators have real expectation values.

Homework Statement A particle at time zero has a wave function Psi(x,t=0) = A*[phi_1(x)-i*sin(x)], where phi_1 and phi_2 are orthonormal stationary states for a Schrodinger equation with some potential V(x) and energy eigenvalues E1, E2, respectively. a) Compute the normalization constant A. b)...

we have a wavefunction \psi (x) the question asks for \psi (p) and says to use this to calculate the expectation value of momentum. The problem is the expectation value of momentum is integrated over dx so after transforming how do you get the integral to be over dp? thanks for any help with...

Homework Statement Let X and Y be independent Bernoulli RV's with parameter p. Find, \mathbb{E}[X\vert 1_{\{X+Y=0\}}] and \mathbb{E}[Y\vert 1_{\{X+Y=0\}}] Homework EquationsThe Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...

Homework Statement System of 2 particles with spin 1/2. Let \vert + \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \\ \vert - \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} singlet state \vert \Phi \rangle = \frac{1}{\sqrt{2}} \Big( \vert + \rangle \otimes \vert - \rangle - \vert - \rangle...

CAn someone help me calculate expectation value of energy of electron in conduction band for semi conductor

I know the difference between the expectation value and probability density, but how do you calculate the probability density of an observable other than position? For position, the probability of the particle being in a particular spot is given by |\Psi|^2, which is the probability density, and...

Homework Statement Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y). Homework EquationsThe Attempt at a Solution I'm pretty sure I have it, just want to make sure. Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so...

1. I have a problem that I cannot figure out how to solve. I want to find the following: E(X|X<Y) where X follows exp(a) and Y follows exp(b) (exp is for exponential distribution). Any ideas on how to solve it? [b]I got E(X|X<Y) = \int_{-∞}^{∞} E(X|X<y)f_{y}(y)dy = \int_{-∞}^{∞}...

Imagine a million different names are in a hat, yours among them. Some number N of names will be drawn, decided by people that you know too little about to decide a meaningful expectation on N. The drawing is done in secret, and the newspaper reports one winner each day, in no particular...

Homework Statement The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction ##ψ=e^(-αr) ## show that Homework Equations ##<ψ|H|ψ>/<ψ|ψ> = (\hbarα)^2/2m + 3mω^2/2α^2## The following...

Homework Statement Given the following normalised time-independent wave function the question asks for the expectation value of the energy of the particle. The well has V(x)=0 for 0<x<a Homework Equations ψ( x ) = √(1/a) ( 1+2cos(∏x/a) )sin(∏x/a) The Attempt at a Solution I...

Homework Statement I've been reading Leonard Susskind's Theoretical Minimum volume on QM, and enjoying it quite a bit - the book doesn't include exercise solutions at the end though, and if they exist online for this volume I haven't been able to find them. (Perhaps if such solutions...

Homework Statement Hello, I need to calculate the expectation value for position and momentum for a wavefunction that fulfills the following relation: ψ0(-x)=ψ0(x)=ψ*0(x) The wave function is normalised. Homework Equations There is also a second wave equation that is orthogonal...

Homework Statement A particle is under a central potential. Initially its wave function is an eigenfunction ##\psi## such that ##\hat {\vec L ^2} \psi = 2 \hbar ^2## , ##\hat L_3 \psi =0##. Calculate the expectation value of ##\hat {\vec L}## for all times. Homework Equations...