Expectation Definition and 654 Threads

Homework Statement An electron is in the spin state in the Sz representation |ψ> = A (1-2i 2)T <- this is a 2 X 1 matrix If Sx is measured, what values and probabilities do you get? What is the expectation value of Sx? Homework Equations The Attempt at a Solution...

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Homework Statement Consider a particle in an infinite one-dimensional box that has a length L and is centered at the origin. (Use h for Planck's constant, n, and L, as necessary.) Evaluate <x^2> for <x^2> at n=1. Homework Equations <x^2>= (2/L)[(L^3/24)-(L^3/4n^2*pi^2)cos(n*pi)] The...

I'm trying to check that the expectation value <E> is E for the wavefunction sqrt(2/L) sin(2pix / L) I know the shortcut way of doing it by saying that the hamiltonian multiplied by the function is just the eigenvalue E multiplied by the function, and since the function is normalized the...

My teacher said that angular momentum doesn't have orientation in space - but how can that be? Isn't cos(theta) = L_z / |L vector| ? Also (an unrelated question) could somebody give an example of how the integration process goes when you are trying to get an expectation value for something...

Why should symmetries require a field that acquires vacuum expectation value to have the same quantum numbers as the vacuum? Please give me a reference also..if possible...

Homework Statement The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0. Homework Equations The Attempt at a Solution The only thing I'm having a problem...

I want to find <x> and ,<x^2>, <p>, and <p^2> of a particle in an infinite well where: V(x)=0, \frac{-a}{4}<x<\frac{3a}{4} Using the usual method, I found the wavefunction to be: \psi(x)=\sqrt{\frac{2}{a}}sin[\frac{n\pi}{a}(x+\frac{a}{4})] I also found...

Homework Statement I am trying to find the variance of p for a wave function \Psi(x,0)=A(a^2-x^2) I'm confused about how to set up the integral. it should be something like -i^2h^2\int_{-a}^a A(a^2-x^2) (\frac{\partial\Psi}{\partial x})^2 dx I'm confused about the partial...

Homework Statement Using the Feynman-Hellman theorem, determine the expectation values of 1/r and 1/r^2 for the hydrogen atom. Homework Equations Hamiltonian: H=-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} energy...

Hi, I am not sure if I understand well : is there a difference between avarage case complexity to expectation of running time? thank you Naftali

If i know that Pr(X_t>b)=0, where X_t>0 and b is positive finite, then should the expectation of X_t be finite? Is there any case where it is infinite?

Okay, so I'm now reviewing ladder operators (no, not homework). While reviewing a quantum problem involving the L_z operator at this website (http://quantummechanics.ucsd.edu/ph130a/130_notes/node219.html#example:expectLz"), I found myself confused. Okay, here's my question: don't we need to...

Homework Statement Let X denote a random variable with the following probability mass function: P(j)= 2^(-j), j=1,2,3,... (a) Compute the moment generating function of X. (b) Use your answer to part (a) to compute the expectation of X. Homework Equations m.g.f of X is M (t) =...

Good Evening: I'm given this problem: A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of...

Hello everybody, I have two questions on conditional expectation w.r.t (Polynomial) OLS: Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how...

The distribution function of a random variable X is given by: F(x) = 0 if x <-3 3/8 if -3 <= x < 0 1/2 if 0 <= x < 3 3/4 if 3 <= x <4 1 if x => 4 Calculate E(X) and E(X2 - 2|X|) Well I'm at a loss of E(X) although once I know this the other should be fairly simple.. Ive got...

Homework Statement Find the expectation value <x> if: from 0 <= x <= a, psi = A x/a from a <= x <= b, psi = A(b-x)/(b-a) Normalizing gives me that A = sqrt(3/b) (verified correct)Homework Equations The Attempt at a Solution <x> = \int_0^b x \psi^2 dx = \int_0^a \frac{A^2 x^3}{a^2}dx + \int_a^b...

Homework Statement In quantum mechanics, how to show that the expectation value is always positive? Homework Equations The Attempt at a Solution

Just a quick question. I finished an expectation value sum and noticed that the given solution had me stumped. Ive attached a quick picture of the simplifying process which was given as the solution. The only thing i don't understand is how to get the value iCm/(pi*hbar)^1/2. I don't know...

Hello, I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following, c = \frac{(a + ib)}{\sqrt{2}}...

Homework Statement Let Bt be a standard Brownian motion. Let s<t: a) Compute P(\sigma B_{t}+\mu t|B_{s}=c) b) Compute E(B_{t}-t|B_{s}=c) Homework Equations Defition of brownian motion: B(t) is a (one-dim) brownian motion with variance \sigma^{2}if it satisfies the following conditions: (a)...

Every quantum mechanical operator has an observable in classical mechanics <x> - position ... <x^2> - ? <p^2> - ? What is the meaning on these expectation values? v^2 = <x^2> - <x>^2 What is the meaning of this? edit: It looks to me like uncertainty in position. Is it the average...

fx,y = 6(x-y)dydx, if 0<y<x<1 how do you find E(XY), i know the formula...g(x,y)fxy(x,y)dydx but i don't know what 'g(x,y)' represents and the limits to use??

Homework Statement The variance of an observable Qhat in a state with wavefunction psi is, (delta Qhat)2=<(Qhat-<Qhat>)2> Show that this can be written as, (delta Qhat)2=<Qhat2>-<Qhat>2 Homework Equations As above. The Attempt at a Solution (delta...

Homework Statement calculate <x>, when \Psi(x,t)=A*exp(-(\sqrt{}Cm/2h)x^{}2 Homework Equations <x>=\int\Psi^{}*x\Psidx over all space.. \intexp(-\alphax^{}2)=\sqrt{}\pi/\alpha The Attempt at a Solution ok know how to do this but how do i do the intergral... my maths isn't so good...

http://img23.imageshack.us/img23/1649/93412460.th.png For question 2 in the above link, I calculated the expectation of the energy by E=<\hat{H}>=\int_0^a \psi^* \hat{H} \psi dx where \psi=\psi^*=x(a-x) this gave E=0. this answer confused me for two reasons: (i) is it ok for the...

Homework Statement http://209.85.48.12/3560/8/upload/p2791776.jpg Homework Equations The most relevant identity to the part that I'm confused about is the following identity: for any cumulative distribution function F, with the inverse function F-1, if U has uniform (0,1) distribution...

Hi all. I have a question which arose from the answer of a homework problem. A particle is in the state given by \left| \psi \right\rangle = \frac{1}{{\sqrt 3 }}\left[ {\left| \psi \right\rangle _1 + \left| \psi \right\rangle _2 + \left| \psi \right\rangle _3 } \right], where {\left|...

Hi quick question: Suppose you have a function of random variables given in the following way Z=X if condition A Z=Y if condition B where both X and Y are random variables, and conditions A & B are disjoint. Then would the expectation of Z be E[Z]=E[X]*Pr(A)+E[Y]*Pr(B)? Thanks in advance.

Homework Statement Let X and Y be contnious random variables with joint probability density function - f(x,y) = 10x^2y if 0<x<y<1 0 othewise a) Determine P( Y < \frac{X}{2}) b) Determine P(x \leq 1/2 | Y < X^2) c) Determine the marginal density functions of X and Y, respectively...

This result isn't in our book, but it is in my notes and I want to make sure it's correct. Please verify if you can. Homework Statement I have two I.I.D random variables. I want the conditional expectation of Y given Y is less than some other independent random variable Z. E(Y \...

If W=g(X) is a function of continuous random variable X, then E(W)=E[g(X)]= ∞ ∫g(x) [fX(x)] dx -∞ ============================ Even though X is continuous, g(X) might not be continuous. If W happens to be a discrete random variable, does the above still hold? Do we still integrate ∫...

Is the expectation value of momentum/position/energy the value that we're most likely to measure? So suppose we measure 100 particles with the same wavefunction, would we expect most of them to have momentum/position/energy that's equal to the expectation value? And I was wondering, how do we...

Homework Statement prove or disprove that E[X^2] = E(X)^2 Homework Equations E[X] = \sumxi*pr(xi) The Attempt at a Solution I really don't know where to start, I believe that it is not true, so I should try to disprove it, and the easiest way to do that would be by...

Hi, anyone help please. Let X and Y are independent uniform random variables over the interval [0,1] E[|X-Y|a]=? where, a>0

Thanks for all the help on the first question but now I have to solve for <T>. I have no idea how to do this, and I could use some help for a kick start. thanks!

first post! but for bad reasons lol Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2 i solved for x: x=sqrt(h/2mw)((a+)+(a-)) so <x> integral of si x ((a+)+(a-)) x si. therefor the integral of si(n+1) x si + si(n-1) x si. si(n+1)...

Homework Statement Consider a wave function \psi (x,t) = R(x,t) exp(i S(x,t)) what is the expectation value of momentum?Homework Equations <f(x)> = \int^{\infty}_{-\infty} \psi^* f(x) \psi dx \hat{p} = -i \hbar \frac{\partial}{\partial x} The Attempt at a Solution This is for an intro to...

Homework Statement The expectation value <r> of the electron-nucleus separation distance 'r' is: <r> = ʃ r |ψ|² dV. (a) Determine <r> for the 1, 0, 0 state of hydrogen. The Attempt at a Solution Right, I've obtained the value for ψ = (1/πa³)^1/2 exp(-r/a) I then...

Homework Statement I have to prove the following: \hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle Edit: L = Angular Momentum & N = Torque Homework Equations I used Ehrenfest's theorem, and I've got the equation in the following form: \frac{1}{i} \left(\left[L,H\right]\right) +...

Homework Statement I need to find the expectation value for E but I don't know how b acts on the vacuum state. Homework Equations b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t) | \psi(t)\rangle = b^\dagger| 0\rangle The Attempt at a Solution \langle \psi(t) |...

Homework Statement I we know the eigenstates of the system be |\psi_1\rangle and |\psi_2\rangle. Current state of the system is |\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle Try to find the expectation value of electric dipole moment \mu (assume it is real) and write it in...

Homework Statement Obtain an expression for the expectation value <Pxn>n N=1,2... of a particle in an infinite box ( V=\infty for x<0 and x>L ; V=0 for 0<X<L) which is in an eigenstate of the energy. Homework Equations Pn =+- \sqrt{2*m*En } = +- (n*pi*Hbar) / L The Attempt at a...

Homework Statement Hi all, i have a problem: i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). Ψ(x,t) = (1/sqrt2)[Ψ0(x).e-[i(E0)t/h] + Ψ1(x)e-[i(E1)t/h]], where Ψ0,1(x) are the ground and...

Homework Statement For a given wave function Psi(x,t)=Aexp^-(x/a)^2*exp^-iwt*sin(kx) find the expectation value of the linear momentum. Homework Equations <p>=integral(-inf,inf) psi* p^ psi dx p^=-ih(bar) d/dx sin x = (exp ix - exp -ix)/2i cos x = (exp ix + exp -ix)/2 The Attempt...

Quantum homework - Average Expectation Values?? Hi people, I'm struggling with my quantum mechanics homework - I've included links to photographs of my attempts at solutions, but i know they are wrong because I am given what the answers are supposed to be. Can somebody help me spot where I am...

If \Psi (x,t) = \psi (x) g(t), should I then use \Psi or \psi when calculating <p> and <p ^2>?

Homework Statement Calculate \Delta x = \sqrt{\left\langle(x - \left\langle x \right\rangle )^2 \right\rangle} if \left\langle x \right\rangle = 0 and \left\langle x^2 \right\rangle = a^2(\frac{\pi - 6}{12 \pi^2}) 2. The attempt at a solution \left\langle(x - \left\langle x...

I seem to be having a rather difficult time understand all the details of the notation used in this quantum material. If I'm given the eigenvalue equations for L^{2} and L_z and L_{\stackrel{+}{-}} for the state |\ell,m>, how do I compute <L_{x}> using bra-ket formalism? I know that L_x =...