Expectation Definition and 689 Threads

Homework Statement Homework EquationsThe Attempt at a Solution I tried to solve (a), but i don't know which approach is right ((1) or (2)) and how to solve (b).[/B]

Homework Statement Homework Equations The Attempt at a Solution When I take the second formula, multiply by it's conjugate and then by x and do the integral of the first formula, I get 0, and not L/2, for <x>. Am I missing a formula ? The complex conjugate of the exponential part...

Admins: Please excuse my E and brackets in the title, and correct if possible. My questions are these, please: 1. What is the physical significance of (\hat{a}\cdot\boldsymbol{\sigma}_{1}) in \left\langle...

I have to calculate the Expectation Value of an Energy Eigenstate : < En > The integral is ∫ ψ* En ψ dx I have : A ) ψ = √L/2 sin nπx/L , a single standing wave of the wave function B ) ψ = BsinBcosD , the wave function of the particle C ) ψ = ΣCn ψn = C , sum of all the...

Hello, I'm a beginner at quantum mechanics. I'm working through problems of the textbook A Modern Approach to Quantum Mechanics without a professor since I am not going to college right now, so I need a brief bit of help on problem 1.10. Everything else I have gotten right so far, but I am...

Q The amount of time (in minutes) that an executive of a certain firm talks on the telephone is a random variable having the probability density: $$f(x) = \begin{cases} \dfrac{x}{4}&\text{for $0 < x \le 2$}\\ \dfrac{4}{x^3}&\text{for $x > 2$}\\...

Homework Statement Write down a spinor that represents the spin state of the particle at any time t > 0. Use the expression to find the expectation values of ##S_x## and ##S_y## Homework Equations The particle is a spin-##\frac 1 2## particle, the gyromagnetic ratio is ##\gamma_s \lt 0##, and...

Hello Forum, I understand that in order to calculate the average of a certain operator (observable), whatever that observable may be that we are interested in, we need to prepare many many many identical copies of the same state and apply the operator of interest to those identical state. By...

Hello all. I am working on proving some theorems about Monte Carlo simulation and have proven a theorem that, in a certain formula, it is valid to replace a random variable in the denominator of a fraction by its expected value. I have been wondering whether this result can be generalised to...

Homework Statement Let w(1) = event of a random walk with right drift (p > q, p+q = 1) starting at 1 returns to 0 Let p(w(1)) = probability of w(1) Let S=min{t>=0:wt(1)=0} be the minimum number of steps t a walk starting from 1 hits 0. What is E[S|w(1)]? Homework Equations I know E[S|w(0)] = 0...

Homework Statement See Image, Sorry Its easier for me to attach images than writing all equation on the forum's keyboard! I only need to check if I'm working it out correctly up to the position expectation value because I don't want to dive in the rest on wrong basis ! Homework Equations...

Homework Statement You are playing a game with two bells. Bell A rings according to a homogeneous poisson process at a rate r per hour and Bell B rings once at a time T that is uniformly distributed from 0 to 1 hr (inclusive). You get $1 each time A rings and can quit anytime but if B rings...

Homework Statement Show the mean position and momentum of a particle in a QHO in the state ψγ to be: <x> = sqrt(2ħ/mω) Re(γ) <p> = sqrt (2ħmω) Im(γ) Homework Equations ##\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})##The Attempt at a Solution I put ψγ into...

Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?

Homework Statement Given ##\psi = AR_{21}[BY_1^1 + BY_1^{-1} + CY_1^0]##, find ##\left<L_z\right>## and ##\left<L^2\right>##. (This is not the beginning of the homework problem, but I know my work is correct up to here. I am not looking for a solution, only an answer as to whether or not my...

Homework Statement In a coherent state ##|\alpha\rangle##, letting ##P(n)## denote the probability of finding ##n^{\text{th}}## harmomic oscillator state. Show that $$\displaystyle{\langle\hat{n}\rangle \equiv \sum\limits_{n}n\ P(n)=|\alpha|^{2}}$$ Homework Equations The Attempt at a...

Hi, Initially X and Y are exponential random variables with rate respectively $$\mu \lambda$$, and I am aware that E[X|X>Y] is obtained using joint distribution but I can not build up the integral structure, I intuitively think the result is just 1/mu, but I can not prove it to myself could you...

Hi, If E[wwH]=T, where w is a zero-mean row-vector and H is the Hermitian transpose then assuming that H is another random matrix, it holds that E[H w (H w)H] = T H HH or T E[H HH] ?? In other words, the expectation operation still holds as in the latter expression or vanishes as in the...

hello :-) here is my problem...: 1. Homework Statement For a linear harmonic oscillator, \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2x^2 a) show that the expectation values for position, \bar{x}, and momentum \bar{p} oscillate around zero with angular frequency \omega. Hint...

Hey all, I have been doing some math lately where I need to find the conditional expectation of a function of random variables. I also at some point need to find a derivative with respect to the variable that has been conditioned. I am not sure of my work and would appreciate it if you guys can...

Homework Statement Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, ##A##, is given (in the same basis) by \begin{bmatrix} 0 &a \\ a & 0 \end{bmatrix} where ##a\in\mathbb{R}^+##. The initial state of the system...

I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...

It would be really appreciated if somebody could clarify something for me: I know that stationary states are states of definite energy. But are all states of definite energy also stationary state? This question occurred to me when I considered the free particle(plane wave, not a Gaussian...

Hello! Could somebody please tell me how i can compute the expectation value of the momentum in the case of a free particle(monochromatic wave)? When i take the integral, i get infinity, but i have seen somewhere that we know how much the particle's velocity is, so i thought that we can get it...

Homework Statement [/B] For a general operator ## \hat{O}##, let ##\hat{O}_{mn}(t)## be defined as: $$ \hat{O_{mn}}(t) = \int u^{*}_{m}(x,t) \hat{O} u_{n}(x,t) $$ and $$ \hat{O_{mn}} = \int u^{*}_{m}(x) \hat{O} u_{n}(x) $$ ##u_{m}## and ##u_{n}## are energy eigenstates with corresponding...

It says in Susskind's TM: ##\langle L \rangle = Tr \; \rho L = \sum_{a,a'}L_{a',a} \rho_{a,a'}## with ##a## the index of a basisvector, ##L## an observable and ##\rho## a density matrix. Is this correct? What about the trace in the third part of this equation?

Homework Statement √[/B] A particle in an infinite square well has the initial wave function: Ψ(x, 0) = A x ( a - x ) a) Normalize Ψ(x, 0) b) Compute <x>, <p>, and <H> at t = 0. (Note: you cannot get <p> by differentiating <x> because you only know <x> at one instance of time)Homework...

Homework Statement (a) Suppose we flip a fair coin until two Tails in a row come up. What is the expected number, NTT, of flips we perform? Hint: Let D be the tree diagram for this process. Explain why D = H · D + T · (H · D + T). Use the Law of Total Expectation (b) Suppose we flip a fair...

There is another topic for this but I didn't quite see it and I don't know how I've gone so far through my course not asking this simple question. So what's the difference? My thought process for hydrogen. I know it can have quantised values of energy, the energy values are the Eigen values of...

I'm having a hard time following the arguments of how the Higgs gives mass in the Standard Model. In particular, the textbook by Srednicki gives the Higgs potential as: $$V(\phi)=\frac{\lambda}{4}(\phi^\dagger \phi-\frac{1}{2}\nu^2)^2 $$ and states that because of this, $$\langle 0 | \phi(x)...

Homework Statement [/B] Particle in one dimensional box, with potential ##V(x) = 0 , 0 \leq x \leq L## and infinity outside. ##\psi (x,t) = \frac{1}{\sqrt{8}} (\sqrt{5} \psi_1 (x,t) + i \sqrt{3} \psi_3 (x,t))## Calculate the expectation value of the Hamilton operator ##\hat{H}## . Compare it...

So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...

Good morning. Can you help me to solve this exercise. The correct answer should be the 2, but how is it calculated? Thanks. Let $l_{+}$ be the set of nonnegative simple rv’s. Pick $X=7\cdot I _{\left \{ X\leqslant 7 \right \}}+7\varepsilon \cdot I_{\left \{ X> 7 \right \}}\epsilon l_{+}$ , for...

Homework Statement The Hamiltonian of an electron in solids is given by H. We know that H is an Hermitian operator, it satisfies the following eigenvalue equation: H|Φn> = εn|Φn> Let us define the following operators in terms of H as: U = e^[(iHt)/ħ] , S = sin[(Ht)/ħ] , G = (ε -...

Homework Statement ## H ## is the Hamiltonian of an electron and is a Hermitian operator. It satisfies the following equation: ##H |\phi_n\rangle = E_n |\phi_n\rangle ## Let ## U = e^{\frac {iHt}{\hbar}} ##. Find the expectation value of U in state ##|\phi_n\rangle## Homework Equations ##...

An animation of the CHSH experiment to generate correlated photons is at: http://www.animatedphysics.com/games/photon_longdistance_chsh.htm @georgir has a program to show the calculations using the formula for photon detection return Math.random() < (Math.cos(r(p-a)*2)+1)/2; yields the...

I am following Griffiths' intro to quantum mechanics and struggling(already) on page 16. When a particle is in state ##\Psi##, $$\frac{d<x>}{dt} = \frac{i\hbar}{2m}\int_{-\infty}^{\infty} x\frac{\partial}{\partial t}\bigg (\Psi^*\frac{\partial \Psi}{\partial x}-\frac{\partial \Psi^*}{\partial...

It's my first post so big thanks in advance :) 1. Homework Statement So the question states "By interpreting <pxΨ|pxΨ> in terms of an integral over x, express <Ekin> in terms of an integral involving |∂Ψ/∂x|. Confirm explicitly that your answer cannot be negative in value." ##The 'px's should...

I am not sure why a factor of (½) appears in front of the summation over orbitals, i, j to N, of the Coulomb and exchange integrals in the HF energy expectation value.

Srednicki page 65 it says "Let us compute the vacuum expectation value of the field $$\phi(x)$$ which is given by $$\langle 0| \phi (x)|0 \rangle = \frac{\delta}{\delta J(x)} Z_{1}(J) |_{J=0}$$ This expression is then the sum of all diagrams that have a single source, with the source removed."...

Homework Statement Consider a particle, with mass m, charge q, moving in a uniform e-field with magnitude E and direction X_1. The Hamiltonian is (where X, P, and X_1 are operators): The initial expectation of position and momentum are <X(0)> = 0 and <P(0)>=0 Calculate the expectation...

Homework Statement Show that for a two spin 1/2 particle system, the expectation value is \langle S_{z1} S_{n2} \rangle = -\frac{\hbar^2}{4}\cos \theta when the system is prepared to be in the singlet state...

Homework Statement The number of customers visiting a store during a day is a random variable with mean EX=100and variance Var(X)=225. Using Chebyshev's inequality, find an upper bound for having more than 120 or less than 80customers in a day. That is, find an upper bound on P(X≤80 or X≥120)...

Homework Statement Let X∼Geometric(p). Using Markov's inequality find an upper bound for P(X≥a), for a positive integer a. Compare the upper bound with the real value of P(X≥a). Then, using Chebyshev's inequality, find an upper bound for P(|X - EX| ≥ b). Homework Equations P(X≥a) ≤ Ex / a...

Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework EquationsThe Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and...

How can position^2 expectation be greater than the Length of "box"? I mean <x^2> = L^2 / 3. Say L=100m then we have <x^2> = 333m. How is this possible?

Homework Statement Consider the bipartite observable O_AB = (sigma_A · n) ⊗ (sigma_B · m) Where n and m are three vectors and sigma_i = (sigma_1_i, sigma_2_i, sigma_3_i) with i = [A,B] are the Pauli vectors. Compute using abstract and matrix representation the expectation value of O_AB...

Hi, I have trouble with the following problem: Gaussian random variable is defined as follows \phi(t) = P(G \leq t)= 1/\sqrt{2\pi} \int^{t}_{-\infty} exp(-x^2/2)dx. Calculate the expected value E(exp(G^2\lambda/2)). Hint: Because \phi is a cumulative distribution function, \phi(+\infty) =...

The Question Let X and Y be two independent Geometric(p) random variables. Find E[(X^2+Y^2)/XY]. Formulas Px(k) = py(k) = pq^(k-1) E(x) = Σx(p(x)) My attempt at a solution I am really struggling with this question because I want to apply the LOTUS equation but am unsure how to do it for...

I am stuck on this Self-test 1.6 in molecular quantum mechanics by atkins and friedman. Probably making use of the completeness relation the question is the following: Show that if <Ωf>*=-Ωf*, then <Ω>=0 for any real function f. Anyone got a clue?