Expectation Definition and 654 Threads

Homework Statement A wavefunction of angular momentum states is given: \psi = \frac{1}{\sqrt{7}}|1,-1\rangle + \frac{\sqrt{35}}{7}|1,0\rangle+\sqrt{\frac{1}{7}}|1,1\rangle Calculate \langle \psi| L_{\pm} |\psi \rangle and \langle 1,1|L_+^2|\psi\rangle3. Attempt at a solution. If the...

Quantum Mechanics "Expectation" Homework Statement 1. Calculate the expectation value <p_{x}> of the momentum of a particle trapped in a one-dimensional box. 2. Find the expectation value <x> of the position of a particle trapped in a box L wide.Homework Equations \psi...

hello! can any1 please help me with the following proofs? thanks let X and Y be random variables. prove the following: (a) if X = 1, then E(X) = 1 (b) If X ≥ 0, then E(X) ≥ 0 (c) If Y ≤ X, then E(Y) ≤ E(X) (d) |E(X)|≤ E(|X|) (e) E(X)= \sumP(X≥n)

Homework Statement I am trying to derive for myself the velocity of the expectation value from the information given, specifically that <x> = \int_{-\infty}^{\infty}x|\Psi (x,t)|^2 dt (1) Eq (1) can be transformed into, \frac{d<x>}{dt} =...

Homework Statement What are the possible results and their probabilities for a system with l=1 in the angular momentum state u = \frac{1}{\sqrt{2}}(1 1 0)? What is the expectation value? ((1 1 0) is a vertical matrix but I can't see how to format that) Homework Equations The...

How to compute E[X|Y1,Y2]? Assume all random variables are discrete. I tried E[X|Y1,Y2] = \sum_x{x p(x|y1,y2) but I'm not sure how to compute p(x|y1,y2] = \frac{p(x \cap y1 \cap y2)}{p(y1 \cap y2)} If it is correct, how can I simplify the expression if Y1 and Y2 are iid?

Homework Statement Find the expectation value of x (Find <x>) given the wave function: \psi(x)=[sqrt(m*alpha)/h_bar]e^[(-m*alpha*|x|)/(h_bar)^2] This wave function represents the single bound state for the delta-function potential. It's the solution to the shrodinger equation given the...

Does anybody help me how to find the average (expectation) of terms involving double summation? Here is the equation which I'm trying solve. [\tex]E\Big[2\sum_{k=0}^{N-2}\sum_{j=k+1}^{N-1}f(k,j)\cos[2\pi(j-k)t-\theta_{k,j}]\Big][\tex] where f(k,j) and [\tex]\theta_{k,j}[\tex] are some...

Homework Statement Consider a hydrogen atom whose wave function at time t=0 is the following superposition of normalised energy eigenfunctions: Ψ(r,t=0)=1/3 [2ϕ100(r) -2ϕ321(r) -ϕ430(r) ] What is the expectation value of the angular momentum squared? Homework Equations I know...

Homework Statement If an electron is in an eigen state with eigen vector : [1] [0] what are the expectation values of the operators S_{x}, and S_{z} Interpret answer in terms of the Stern-Gerlach experiment. The Attempt at a Solution Im not too sure how to calculate the...

1. If X is uniform distributed in (0,pi), what is E(X|sinX)? 2. Suppose X and Y are Gaussian random variables N(0,sigma_x) and N(0,sigma_y). what is the distribution of E(X-Y|2X-Y) Can anyone help? thanks

Do we have to use normalised wavefunction to calculate the expectation and probability of finding the particle? If yes, why?

I'm trying to calculate the expectation value of the momentum squared (p^2) of the harmonic oscillator ground state. The integral involves the second derivative of a Gaussian (exponential of a negative squared term) Then the integral involves, after working it out, an x^2 term times...

In a paper in Physical Review A, the author discusses a wave function for one particle, Ψ(r,t), where r is the position vector. He writes "The probability distribution for one-particle detection at a point r is given by [SIZE="4"]|<r|Ψ >|[SIZE="3"]2 ". Is that correct? The above...

Hi I'm going through some presentation material and i can't understand how the following has been derived \sum^{n}_{j=1} \mathbb{E}[ ln(1 +K_{j})] = n \mathbb{E}[ln(1+K_{1})] Could someone point me in the right direction on why this makes sense ? Thanks

Homework Statement 8. Suppose that X and Y are independent continuous random variables, and each is uniformly distributed on the interval [0,1] (thus the pdfs for X and Y are zero outside of this interval and equal to one on [0,1]). (a) Find the mean and variance for X+Y. (b) Calculate...

Homework Statement I need to find <x>, <x2>, <p>, and <p2> for a particle in the first state of a harmonic oscillator. Homework Equations The harmonic oscillator in the first state is described by \psi(x)=A\alpha1/2*x*e-\alpha*x2/2. I'm using the definition <Q>=(\int\psi1*Q*\psi)dx...

I'm looking at a question... The last part is this: find the expectation values of energy at t=0 The function that describes the particle of mass m is A.SUM[(1/sqrt2)^n].\varphi_n where I've found A to be 1/sqrt2. The energy eigenstates are \varphi_n with eigenvalue E_n=(n + 1/2)hw...

ok. this is an easy enough thing to prove in one dimension but my question seems to be 3 dimensional and it's causing me some hassle: show the expectation value of the kinetic energy in a bound state described by the spherically symmetric wavefunction \psi_T(r) may be written \langle...

I have two random variables Y and X and Y is dependent of X, though X is not the only source of variability of Y. With fixed X=x, Y(x) follows gaussian law. X also follows gaussian law. In what cases can I move from E[ Y(X) ] to E[ Y(E(X))] someone has any idea? is there a text...

Hello, this is just a general question, how is <x^2> evaluated, if <x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket) Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ? I'm only wondering how the squared works...

If the expectation of some observable is constant then can it be measured at Lab.

I've never seen an expectation value taken and would greatly appreciate seeing a step by step of how it is done. Feel free to use any wavefunction, this is the one I've been trying to do: In the case of \Psi=c1\Psi1 + c2\Psi2 + ... + cn\Psin And the operator A(hat) => A(hat)\Psi1 =...

I have been stuck at this calculation. There are two exponential distributions X and Y with mean 6 and 3 respectively. We need to find E[y-x|y>x] I keep getting it negative, which is clearly wrong. Anybody wants to try it?

Find the time dependence of the expectation value <x> in a quantum harmonic oscillator, where the potential is given by V=\frac{1}{2}kx^2 I'm assuming some wavefunction of the form \Psi(x,t)=\psi(x) e^{-iEt/\hbar} When I apply the position operator, I get: <x>=\int_{-\infty}^\infty...

Many thanks in advance Suppose x is normal variable x~N(a,b) and y=160*x^2 I need calculate E(y)=∫yf(y)d(y) f(y) is the density function of y how can I write it as an integral of x since we know x's distribution, I mean use the density function of x to substitute the original integral...

Hi: If we want to work out the expectation of: <0|T(φ1φ2)|0> ie. <0|<0|T(φ1φ2)|0>|0> apparently it is acceptable to pull out the <0|T(φ1φ2)|0>: So <0|<0|T(φ1φ2)|0>|0>=<0|T(φ1φ2)|0><0|I|0> I do realize this is a really stupid question, but I want to be 100% sure. Is this simply...

If I have x1,x2 iid normal with N(0,1) and I want to find E(x1*x2 | x1 + x2 = x) Can I simply say: x1 = x - x2 and thus E(x1*x2 | x1 + x2 = x) = E[ (x - x2)*x2) = E[ (x * x2) - ((x2)^2) ] <=> x*E[x2] - E[x2^2] = 0 - 1 = -1?

Hi had this question on my last "Statistical Inference" exam. And I still have some doubts about it. I determined that the maximum likelihood estimator of an Uniform distribution U(0,k) is equal to the maximum value observed in the sample. That is correct. So say my textbooks. After that the...

Homework Statement i. Confirming the wavefunction is normalised ii. Calculating the expectation values: <\hat{x}> , <\hat{x^{2}}> , <\hat{p}> , <\hat{p^{2}}> as a function of \sigma iii. Interpreting the results in regards to Heisenberg's uncertainty relation. Homework Equations...

Hello, I am trying to learn about some basic quantum mechanics. http://farside.ph.utexas.edu/teaching/qmech/lectures/node35.html this website shows that the time derivative of the momentum expectation d<p>/dt = -<dV/dx> The part that i am not getting is how the writer goes from the...

Homework Statement Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0). 1. Find E(Z) 2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1) Homework Equations E(Z) = EX (E(X|Z))...

Homework Statement A particle of mass m is in the state Psi(x,t) = Ae^(-a[(mx^2)+it]) where A and a are positive real constants. a) Find A b) For what potential energy function V(x) does Psi satisfy the Shrodinger equation? c) Calculate the expectation values of x, x^2, p, and...

Homework Statement I'm told that of n couples, each of whom have at least one child, with couples procreating independently and no limits on family size, births single and independent, and for the ith couple the probability of a boy is p_i and of a girl is q_i with p_i + q_i = 1. 1. Show...

Homework Statement Let X1,...,Xn denote a random sample from a N(\mu , \sigma) distribution. Let Y = \Sigma \frac{(X_i - \overline{X})^2}{n} Homework Equations The Attempt at a Solution How would I find E(Y)? Any help would be greately appreciated.

Hello, Can someone explain to me how the expectation values are calculated in the following picture: I mean , What did they do after the brackets? What did they multiply with what? thanks

I am aware of the expectation value \left\langle\ r \right\rangle. But I was wondering what is physically meant by the expectation value: \left\langle\frac{1}{r}\right\rangle The reason I am asking is because calculating this (reciprocal) expectation value for the 1s state of hydrogen, one...

Homework Statement This has been driving me CRAZY: Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle and \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle Homework Equations Raising/lowering eigenvalue equations: a |n...

Let's consider eigenstates |nlm\rangle of hamiltonian of hydrogen atom. Can anyone prove that \langle r \rangle = \langle nlm|r|nlm\rangle = \frac{a}{2}(3 n^2-l(l+1)). Where a - bohr radious. I've been trying to prove it using some property of Laguerre polynomials (which are radial part...

Given X follows an exponential distribution \theta how could i show something like \operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta ? i have get the idea of using Memorylessness property here, but how can i combine the probabilty with the expectation? thanks. casper

How can I do this? Let X,Y r.v., \mathbb{E}(X|Y)=Y and \mathbb{E}(Y|X)=X. Proove that X=Y a.s.

Homework Statement The ground state (lowest energy) radial wave function for an electron bound to a proton to form a hydrogen atom is given by the 1s (n=1, l=0) wave function: R10 = (2 / a3/2) exp(-r / a) where r is the distance of the electron from the proton and a is a constant. a)...

" The law of total expectation is: E(Y) = E[E(Y|X)]. It can be generalized to the vector case: E(Y) = E[E(Y|X1,X2)]. Further extension: (i) E(Y|X1) = E[E(Y|X1,X2)|X1] (ii) E(Y|X1,X2) = E[E(Y|X1,X2,X3)|X1,X2] " ==================== I understand the law of total expectation itself, but...

Homework Statement Using the fact that ,\left\langle \hat{L}_{x}^{2} \right\rangle = \left\langle \hat{L}_{y}^{2} \right\rangle show that \left\langle \hat{L}_{x}^{2} \right\rangle = 1/2 \hbar^{2}(l(l+1)-m^{2}. The Attempt at a Solution L^{2} \left|l,m\right\rangle = \hbar^{2}l(l+1)...

Homework Statement Express Lx in terms of the commutator of Ly and Lz and, using this result, show that <Lx>=0 for this particle. The Attempt at a Solution [Ly,Lz]=i(hbar)Lx <Lx>=< l,m l Lx l l,m> then what?

Homework Statement In my textbook, the formula for the expectation value is written as: <x> = \int \Psi^{*}\Psi dx Shouldn't there be an x next to |\Psi|^{2} ? Thanks. Homework Equations The Attempt at a Solution

Homework Statement Calculate the expectation value of the operator _{}Sz for a spin-half particle known to be in an eigenstate of the operator _{}Sz Homework Equations The Attempt at a Solution I know the eigenvalues for the _{}Sz but how can I find the expectation values...

Hi I have a question. Let X1 & X2 be stochastic variables and X1<=X2, then can we say E[X1]<=E[X2] or SD[X1]<=SD[X2]? why or why not? Looking forward to some reply Thanks!

Expectation value of operator A is given by following formula in Dirac notation. <A> = <x|A|x> where A : Operator <A> : Expectation value of A |x> : State Somehow I am unable to convince myself that this formula is true. Would someone please explain it to me? Thanks

Homework Statement Evaluate the expectation value of p and p² using the momentum-space wave function Homework Equations Momentum-space wave function: \sqrt{\frac{d}{\hbar\sqrt{\pi}}}e^{\frac{-\left(p'-\hbar k\right)^2d^2}{2\hbar^2}} The Attempt at a Solution I can get \langle...