Expectation Definition and 689 Threads

Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...

For question 2.2: <Ψ0|p|Ψ0> = ∫Ψ0 -iħ d/dx(Ψ0) =M Using Integration by parts i get: M = -Ψ0 iħ d/dx(Ψ0) (assuming hilbert space) Implying the expectation values for momentum are zero , however i get all the expectation values are zero for x and momentum in both states which makes no sense :(

I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?

Okay so I begin first by mentioning the length of the well to be L, with upper bound, L/2 and lower bound, -L/2 and the conjugate u* = Aexp{-iz} First I begin by writing out the expectation formula: ## \langle p \rangle = \int_{\frac{L}{2}}^{ \frac{L}{2} } Aexp(-iu) -i \hbar \frac{ \partial }{...

In the derivation of the Fermi-Dirac and Bose-Einstein distributions, we compute the Grand Partion Function ##Q##. With ##Q##, we can compute the espection value of the occupation number ##n_{l}##. This is the number of particles in the same energy level ##\varepsilon _{l}##. The book I am...

I'm trying calculate the expected number of steps in one node in a random walk , ##\langle s\rangle=\sum sp^s##. By deduction I have a possible solution (for rational probabilities ##p=n/m,\ n< m##) in ##\bar{s}=\langle s\rangle= nm/(m-n)^2##, which looks pretty weird but I have not found a...

Watching Dr. Susskind show how to find the time evolution of the average of an observable K, he writes: I can not for the life of me figure out he derived it, and he also did something which I found terribly annoying throughout which is set hbar to 1, so after steps you lose where the hbar...

Homework Statement Homework Equations VD= -1/(8m2c2) [pi,[pi,Vc(r)]] VC(r) = -Ze2/r Energy shift Δ = <nlm|VD|nlm> The Attempt at a Solution I can't figure out how to evaluate the expectation values that result from the Δ equation. When I do out the commutator, I get p2V-2pVp+Vp2. This...

E(X) of probability density function f(x) is \int x f(x) dx E(X2) of probability density function f(x) is \int x^2 f(x) dx Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ? I tried to find E(5 + 10X) from pdf. I did two ways: 1. I found E(X) then using...

I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?

I've been studying quantum mechanics, and working problems to get a feel for expectation values and what causes them to be real. I was working the problem of finite 1D wells, when I came across a situation I did not understand. A stationary state solution is made up of a forward and reverse...

Homework Statement If x is a continuous variable which is uniformly distributed over the real line from x=0 to x -> infinity according to the distribution f (x) =exp(-4x) then the expectation value of cos 4x is? Answer is 1/2 Follow· 01 Request Homework Equations the expectation value of any...

The question is as follows: A particle of mass m has the wave function psi(x, t) = A * e^( -a ( ( m*x^2 / hbar) +i*t ) ) where A and a are positive real constants. i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the...

The expectation value of any function ##f(x)## is given by <f(x)>= \int_{-\infty}^{\infty}f(x)\psi^2(x) dx But what is ##f(x)## actually? In a physical sense. For example if ##f(x)=x## or ##f(x)=x^2##, what do these functions represent on a physical level?

Homework Statement Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which $$\langle\vec{\sigma}\rangle =\vec{e},$$ where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle## Homework Equations :[/B] are all given...

I am tackling a technique to determine the parameters of a Moffat Point Spread Function (PSF) defined by: ## \text {PSF} (r, c) = \bigg (1 + \dfrac {r ^ 2 + c ^ 2} {\alpha ^ 2} \bigg) ^ {- \beta} ## with the parameter "(r, c) =" line, column "(not necessarily integers). The observation of a...

Homework Statement I have a general question how I calculate the expectation value of V (potential energy) with Ehrenfest’s theorem. Do I have to integrate d<p>/dt with respect to d<x>. Also if the potential is symmetric (even) would that mean the expectation value of the potential is 0...

Homework Statement How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom. Homework Equations The Attempt at a Solution I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I...

Homework Statement Can someone explain to me what particles (fermions, scalar/vector bosons, gravitons, ...) can have their vacuum expectation values and why? Which components of these fields can have VEV-s? The Attempt at a Solution I am assuming only scalar boson fields have it (like Higgs...

Hi! I want to know under what conditions the operator expectation values of a product of operators can be expressed as a product of their individual expectation values. Specifically, under what conditions does the following relation hold for quantum operators (For my specific purpose, these are...

Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...

Homework Statement [/B] ##-1\leq\alpha\leq 1## ##f(y_1,y_2)=[1-\alpha\{(1-2e^{-y_1})(1-2e^{-y_2})\}]e^{-y_1-y_2}, 0\leq y_1, 0\leq y_2## and ##0## otherwise. Find ##V(Y_1-Y_2)##. Within what limits would you expect ##Y_1-Y_2## to fall? Homework Equations N/A The Attempt at a Solution...

Hi all, I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For...

Suppose I am trying to approximate a function which I do not know, but I can measure. Each measurement takes a lot of effort. Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]## Supose I know the expectation and variance of ##f(x)##. Is there a way to compute the confidence...

I've been trying to form a proof using , using majorly dirac notation.There has been claims that its much better to use in QM. The question i wanted to generally show that the expected value is Zero for all odd energy levels.I believe i have solved the question but I am a bit Iffy about a step...

Hello physics forums. I'm trying to solve an old exam question. Would love your help. Homework Statement A free particle in one dimension is described by: ## H = \frac{p^2}{2m} = \frac{\hbar}{2m}\frac{\partial^2}{\partial x^2}## at ##t = 0## The wavefunction is described by: ## \Psi(x,0) =...

I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result. Homework Statement At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by: ## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...

Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...

Working on a homework at the moment involving spinors. The algebra isn't hard at all, I just want to make sure my understanding is right and I'm not doing this incorrectly. 1. Homework Statement An electron in a one-dimensional infinite well in the region 0≤x≤a is described by the spinor ψ(x)...

Homework Statement Show that, for a general one-dimensional free-particle wave packet $$\psi (x,t) = (2 \pi h)^{-1/2} \int_{-\infty}^{\infty} exp [i (p_x x - p_x^2 t / 2 m)/h] \phi (p_x) dp_x$$ the expectation value <x> of the position coordinate satisfies the equation $$<x> = <x>_{t=t_0}...

Homework Statement (a) If a particle is in the spin state ## χ = 1/5 \begin{pmatrix} i \\ 3 \\ \end{pmatrix} ## , calculate the expectation value <Sy>(b) If you measured the observable Sy on the particle in spin state given in (a), what values might you get and what is the probability of...

Homework Statement A particle is moving in a one-dimensional harmonic oscillator, described by the Hamilton operator: H = \hbar \omega (a_+ a_- + \frac{1}{2}) at t = 0 we have \Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x)+i\psi_1(x)) Find the expectation value and variance of harmonic oscillator...

Homework Statement Suppose that { |ψ1>, |ψ2>,...,|ψn>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψj> = qj|ψj> for all j = 1...n. A particle is in the state |Φ>. Show that for this particle the expectation value of <Q> is ∑j=1nqj |<Φ|...

Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...

1. The problem statement Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by H = \frac{p^2}{2m} + \frac{m \omega ^2 x^2}{2} Knowing that the ground state of the particle at a certain instant is described by the wave...

Homework Statement A particle moving in a periodic potential has one-dimensional dynamics according to a Hamiltonian ## \hat H = \hat p_x^2/2m+V_0(1-cos(\hat x))## a) Express ## \frac{d <\hat x>}{dt}## in terms of ##<\hat p_x>##. b) Express ## \frac{d <\hat p_x>}{dt}## in terms of ##<sin(\hat...

Homework Statement At ##t = 0##, a particle of mass m in the harmonic oscillator potential, ##V(x) = \frac1 2 mw^2x^2## has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$ where A is a constant If we make a measurement of the energy, what possible...

Homework Statement Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected...

If Eψ = Hψ, then why is expected energy ∫ψ*Hψ dx? It makes more sense if I see the ψ on the right side of H as the ψ in ∫Q(ψ*ψ) dx, where Q is some quantity we want to measure the expectation of. But if true, then since H is defined as (h2/2m) (d2/dx) + V, then what does it mean to calculate...

Homework Statement Homework Equations I know that there are two eigenstates of the operator C: |B> = (1 0) as a column vector with eigenvalue 1 |R> = (0 1) also a column vector with eigenvalue -1 The Attempt at a Solution My work is shown here: If anyone could point me in the right...

Imagine a particle in an equally weighted superposition of being located in three distant regions P, Q, and R, and imagine you stand in region P with a measuring device. The probability of finding the particle there is 1/3. Now imagine a large number N of particles prepared in that same state...

I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...

Homework Statement A few questions: Q1) How does 1.29 flow to 1.30 and 1.31? How was the integral-by-parts done? Q2) The author states that <v> = d<x>/dt represents the expectation value of velocity. What does this actually mean? I tried to rationalise that d<x>/dt represented the velocity...

In 1D QM: I understand that if a given potential well, U(x), is symmetric about x = L, then the expectation value for operator [x] would be <x> = L. (I am not even entirely sure why this is, guessing that the region where x<L and x>L are equally probable) Is it possible to draw conclusion...

Hi. I came across the following in the solution to a question I was looking , regarding expectation values of momentum in 3-D < p12p22p32 > = < p12 > < p22 > <p32 > ie. the expectation value has been factorised. I can't figure out why this is true and also why it doesn't apply to the following...

Homework Statement I have a variable ##s_i## with probability distribution ##w(s_i)## ##(\Delta(s_i))^2## denotes the variance ##=<(s-<s>)^2>=<s^2>-<s>^2## I want to show ## \sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0 ## where ## < > ## denote expectation My book has: ## <\Delta...

Homework Statement At t=0, the system is in the state . What is the expectation value of the energy at t=0? I'm not sure if this is straight forward scalar multiplication, surprised if it was, but we didn't cover this in class really, just glossed through it. If someone could walk me through...

Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...

Homework Statement Show that: d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket Homework Equations Liouville theorem The Attempt at a Solution <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ}) So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can...

Homework Statement I am trying to calculate the expectation value of ##\hat{P}^3## for the harmonic oscillator in energy eigenstate ##|n\rangle## Homework EquationsThe Attempt at a Solution [/B] ##\hat{P}^3 = (i \sqrt{\frac{\hbar \omega m}{2}} (\hat{a}^\dagger - \hat{a}))^3 = -i(\frac{\hbar...