Expectation Definition and 654 Threads

Homework Statement Determine for the hydrogen atom states 1s and 2p the expectation value of the radius r and the associated mean square error Δr. Homework Equations Wave Functions for 1s and 2p from Demtroeder's Experimental Physics Volume 3 (it says "The normalized complete...

Homework Statement I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...

I've managed to get myself confused on a seemingly simple point of mathematics. When I calculate the expectation value of momentum in quantum mechanics <p>=\int{\psi* \frac{\hbar}{i} \frac{d}{dx} \psi dx} To what should I be applying the derivative? \psi?

Homework Statement given a certain state |ψ> that is an eigenstate of L^2 and Lz Calculate <Lx^2> and <Lx> Homework Equations L^2|ψ> = l(l+1)h^2 Lz|ψ> = mh|ψ> Lx = YPz - ZPy The Attempt at a Solution <Lx^2> = (1/2)(h^2)[l(l+1)-(m1)^2] for Lx i got <Lx> = ∫ψ(YPz-ZPy)ψ dx

I understand that the the Higgs field has a vacuum expectation value of 246 GeV. I think that means that the expectation of the Hamiltonian energy operator applied to the vacuum state is 246 GeV. What does this imply for the energy density of the Higgs field in the vacuum (i.e. in Joules /...

Hi all! Homework Statement If we consider the hydrogen atom as a spinless particle. Let this system in the state \Psi ( \vec{r} )= \frac{1}{6} [4 \Psi_{100} ( \vec{r} )+ 3 \Psi_{211}- \Psi_{210} ( \vec{r} ) + \sqrt{10}\Psi_{21-1} ( \vec{r} )] Calculate: 1) Expectation value of...

Homework Statement Show that the expectation value of angular momentum <Lx> is zero Homework Equations L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩ L± = Lx ± iLy The Attempt at a Solution I'm supposed to use ladder operators here to show <Lx> is zero. I start with...

Just to check something: If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ? Or can you not split up the operators like <a|A (B|a>) ? Thanks.

Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...

When i have stuff like E(E(X)) or E(E(XY|X)) etc how do i evaluate a double expectation? and for random vectors stuff like E(E(X)'Y) or E(X'E(Y)) etc.

Homework Statement The ground state wave function for a particle of mass m moving with energy E in a one-dimensional harmonic oscillator potential with classical frequency omega is: u(subscript 0) (x)= N(subscript 0) exp((-alpha^2)(x^2)/2) and alpha=sqrt (m *omega/h-bar) where...

Ψ(x)=(2/a)^(1/2) [csin(nπx/a)] The Expectation value of momentum <P>=∫Ψ＊(x)[-ih d/dx ] Ψ(x) dx = 0 the average momentum is zero.It means the particle is moving equally in the +x and -x. And if Ψ(x)=(2/a)^(1/2) {csin(πx/a)+dsin(2πx/a)} I calculate the average momentum is also...

Homework Statement What is the expectation valueof the Sχ for a system in the time-dependent state |Ψ> = 2e-2iωt |z+> -ieiωt |z-> Homework Equations maybe the state must be normalised first i.e 1/√5 times the initial ψ The Attempt at a Solution And then say<ψ|Sχ|ψ> where ψ...

Find the expectation value of (px)2, keeping in mind that ψ0(x) = A0e−ax2 where A0 = (2mω0/h)^1/4, and <x2> = ∫x2|ψ|2dx = h_bar / 2mω0 <ψ(x)|px2|ψ(x)> = ∫ψ(x)(pop2)ψ(x) dx pop = [hbar / i] (\delta/\deltax) I'm not going to attempt to type out me solving the integral because it...

"Expectation value", a few questions I've read that in quantum mechanics we use the term "expectation value" for example for the energy of a system. Despite its name, the expectation value of the energy of for instance the quantum harmonic oscillator is not the most probable measured energy of...

Homework Statement Suppose X1 , X2 , . . . , Xn are independent random variables, with common expectation μ and variance σ^2 . Let Sn = X1 + X2 + · · · + Xn . Find the variance of Sn. The attempt at a solution Expected value: E[S_n] = n E[X_i] = n\mu \hspace{10 cm} (1)...

Homework Statement I'm re-hashing a problem from my notes; given the wave function \psi(x)=Ne^{-(x-x_0)/2k^2} Find the expectation value <x>. Homework Equations The normalization constant N for this is in my notes as N^2=1/\sqrt{2 \pi k^2} N=1/(2\pi k^2)^{(1/4)} It should be...

Homework Statement Homework Equations The Attempt at a Solution The issue I'm having here is that the problem should be able to be done rather quickly. I can see how to solve for <H> using the operator, but there's a quick way that I'm not picking up on. I thought about solving <H> = <p^2> /...

Homework Statement 1. What is <x^{2}>, in terms of position and expectation values. 2. How can I use the correspondence principal to explain the quantum vs classical results (below). My textbook (Serway, Modern Physics) uses <x> as the expectation value, meaning the average position of a...

Homework Statement Show that, if [H,A] = 0 and dA/dt = 0, then <&Delta;A> is constant in time. Homework Equations d<A>/dt = <i/ℏ[H,A] + dA/dt> The Attempt at a Solution I am trying to use the above equation to show that d<&Delta;A>/dt is 0, and I can get to d&Delta;A/dt = 0, but I...

Homework Statement For the Harmonic Oscillator, the state |ψ> = (|0> + |1>) / √(2) Find \overline{x} = <ψ|x|ψ> \overline{p} = <ψ|p|ψ> \overline{x^2} = <ψ|x^{2}|ψ> and \overline{p^2} = <ψ|p^{2}|ψ> and <ψ| (x - \overline{x})^2 |ψ><ψ| (p - \overline{p})^2 |ψ> [b]2. Homework Equations...

My book tries to illustrate the conditional expectation for a random variable X(\omega) on a probability space (\Omega,\mathscr F,P) by asking me to consider the sigma-algebra \mathscr G = \{ \emptyset, \Omega \}, \mathscr G \subset \mathscr F. It then argues that E[X|\mathscr G] = E[X] (I'm...

Homework Statement Calculate <x> for the Gaussian wave packet \psi(x)=Ne-(x-x0)/2k2 Homework Equations \left\langle x \right\rangle = \int dx x|\psi(x)|2 The Attempt at a Solution So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm...

The wavefunction of hydrogen is given by \psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta, \phi) If I am only given the radial part, and asked to find the expectation value of the radial part I integrate the square of the wavefunction multiplied by r cubed allowing r to range from 0 to...

How to calculate E(x^2) given that x are i.i.d random variables distributed as a standard normal i.e. N(0,1) ? Thank you.

Suppose you know E[X] = 0 for a given (continuous) random variable. Does that mean E[|X|] < \infty? This is what my professor told me today, though it doesn't really make much sense...

Homework Statement The expectation value of the position observable x is <x> = ∫ψ*xψdx. The expectation value of the expectation value, <<x>>=<x>, is still the expectation value...why? Homework Equations The Attempt at a Solution All I can think of is that the expectation value...

Homework Statement Homework Equations I have that p(\theta)d\theta = \frac{1}{\pi}d\theta, this is definitely correct. Also y=r*sin(theta) so dy=r*cos(theta)*d(theta). Substituting d(theta) in above and simplifying, I have: p(y)dy = \frac{1}{pi} \frac{dy}{\sqrt{r^2-y^2}} The problem is...

I have a question about expectation values in quantum mechanics. When calculating <x>=\int\Psi*x\Psi does x always make this functions odd? If \Psi is odd then \Psi* I would assume is odd as well and then <x> would be odd*odd*odd, if \Psi is even then I again assume it would be even*odd*even...

I have a question about expectation values in quantum mechanics. Since calculating <x>=\int\Psi*x\Psi does x always make this functions odd? If \Psi is odd then \Psi* I would assume is odd as well and then <x> would be odd*odd*odd, if \Psi is even then I again assume it would be...

1. What is the expectation value, <x>, for the given distribution over the interval from – to + infinity of the function: f(x)=e^(-.5(x-mu)^2(sigma^-2)) 2. This is a statistics problem i think. I just need to know how this type of problem is worked out because it is relevant to my...

Homework Statement If x is the position of a particle then the expectation value of x, <x> is : (I got lazy and just copied an image from Wiki, just pretend <x> is on the lhs of the eqn) When Griffith derives an expression for d<x> / dt, he uses the fact that dx/dt is zero, since "the wave...

Homework Statement Calculate the expectation values of p and p2 for a particle in state n=2 in a square well potential. Homework Equations \Psi(x,y) = (2/L)*sin(n1\pix/L)*sin(n2\piy/L) p= -i\hbar\partial/\partialx The Attempt at a Solution \int\Psip\Psidxdy...

Homework Statement Find the expectation value of position as a function of time. Homework Equations This is in the latter half of a multi-part question, previously we were given that: Eqn 1: Ψ(x, t) = A(ψ1(x)e−iE1t/h¯ + iψ2(x)e−iE2t/h¯) and in an even earlier part: Eqn 2: ψn(x) =...

Hi everyone, I have a feeling the following property is true but I can't find it stated in any textbook/online reference. Maybe it's not true... Can someone verify/disprove this equation? E(A+B|C) = E(A|C) + E(B|C)

Can anyone explain to me why the only time that the expectation of L^2 operator and the expectation value of L_3^2 are equal only when there is no angular dependence? And what does this mean? Does this have something to do with being restricted to the z-axis which is what L_3 is associated...

Homework Statement I'm wondering how I go about calculating the expectation of a random variable? Is it a different process for a discrete and a continuous? Can you show me an example? Say Poisson and expoential? Also, in the formula Var(X) = E[X^2] - (E[X])^2 how does one...

Hi. First post. I'm trying to understand if electronic energy levels have fixed values, or merely fixed expectation values (in the latter case, orbital electrons could have any energy and it's only the average that would be fixed). Here's my argument for the latter. If it's incorrect, could...

Given some state \left|\psi\right\rangle, and two operators \hat{A} and \hat{B}, how do you prove that if \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle then \hat{A} = \hat{B} ?

Given a stationary state H \psi = E \psi \Rightarrow \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right)\psi = E\psi Firstly is it true that \left<p\right> = \frac{\hbar}{i}\int\psi^* \frac{\partial \psi}{\partial x} dx= 0 ?? If it is, how do we prove it?

Does the following make sense: E(e^{-X}) = 0 \Rightarrow X = \infty\quad a.s. ? (Intuitively yes, but mathematically?) Thank you in advance for your help! :-) /O

Homework Statement Kindly take a look at the attachment for the problem statement. Homework Equations Schrodinger Equation: H psi = E psi The Attempt at a Solution For Part A: H psi = E psi S= 1/2 So energy of ground state is -1/2.K.H.hcross Could you tell me if I am on the...

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value from searching in internet I found that E{X|X>a}=a+E{x} but I can not prove it Help please

Homework Statement For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value Homework Equations The Attempt at a Solution

Suppose X is a uniformly distributed random variable on an interval [-a,a] for some real a. Let Y=X^2. Then what could you say about this distribution of Y? I have no idea how to think about this distribution. Also how could we compute the expectation of Y? I know that E[X]=0 but what could I...

Hi everyone, I was searching an answer for E(XY), where X and Y are two dependent random variables, number of observations n=21 and Sum(x*y)= 1060.84. Can somebody help me? It's not mentioned, but I think that each x and y of the distributions have the same probability to occur. Thank you.

Homework Statement how do you delete a post?

Hello, I want to find <xftf|x(t)|xiti> in harmonic oscillator. I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff. Thank you

I haven't posted any of my working for this as I only want to check my answer. Q. For a hydrogen atom with n=2, l=1, m=0 calculate <r^2> My answer = 0.75 * a^2 where a is the bhor radius. Am I right?

Homework Statement given a wavefuntion \Psi = (1/sqrt50) (3\mu1 + 5\mu2 - 4\mu3) what is the expectation value of the total energy? My thoughts were to calculate <\Psi|\hat{}H|\Psi> but the previous part to the question asks for the probability of each outcome(which I know how to...