Expectation Definition and 689 Threads

I'm a bit confused about the nature of probability conservation and expectation values. According to probability conservation, \frac{∂P(r,t)}{∂t}=0. Does that mean that expectation values e.g. <x>, <p> and <E> depend only on the position of the particle and not on time? Thanks

I am reading the fine structure article from Wikipedia at http://en.wikipedia.org/wiki/Fine_structure. Under the heading 'Kinetic energy relativistic correction', we have the following: For the hydrogen atom, V = e2/r. This implies that the expectation of V = -e2/a0n2. Now, I know that...

I know the E[X] = Integral between [-inf,inf] of X*f(x) dx Where X is normally distributed and f(x) is the PDF How do I find the expectation of X4? Bare with me because I'm useless in Latex So far what I've done is written the integral as Integral between [-inf,inf] of X4*f(x) dx...

I just have a simple question to get me started. If I am given an initial value wavefunction ψ(x,0) and I am asked to find <P> at t = 0 can I use this: <P> = -ih∫ψ*(x,0)\frac{∂}{∂x}ψ(x,0)dx or do I need to find ψ(x,t) before I find <P>?

Homework Statement Suppose that X and Y have a continuous joint distribution with joint pdf given by f (x, y) = { x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0 otherwise. Suppose that a person can pay a cost c for the opportunity of observing the value of X before...

I know that the formula for the expectation value is: <Q(x,p)> = ∫ψ*Q(x,(h/i)d/dx)ψ dx For instance, the expectation value for momentum is. -ih∫ψ*(dψ/dx)dx But, why? How is it derived?

Homework Statement An electron starts in a spin state |\psi(t=0)\rangle = |z \uparrow \rangle and evolves in a magnetic field B_0(\hat{x} + \hat{z}). The Hamiltonian of the system is \hat{H} = \alpha \vec{B}\cdot\vec{S}. Evaluate \langle \psi (t_{1/2}) | S_x | \psi(t_{1/2}) \rangle...

"A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. Let a person choose, at random and without replacement, 3 chips from this bowl. If a person is to receive the sum of the resulting amounts, find his expectation." Here is my attempt: The possible...

Homework Statement Given an observable quantity A, when will it happen that the same value for A will be measured every time? What is the relationship between the operator \hat{A} and \Psi for this case? and What is the relationship between \widehat{A} and \widehat{H}, the...

Homework Statement Let Ω = [0,1] with the σ-field of Borel sets and let P be the Lebesgue measure on [0,1]. Find E(X|Y) if: Homework Equations X(w)=5w^2 Y(w)= \left\{ \begin{array}{ll} 4 & \mbox{if $w \in [0,\frac{1}{4}]$} \\ 2 & \mbox{if $w \in (\frac{1}{4},1]$} \\ \end{array}...

Is it possible to express ANY observable A(X,P) in terms of the ladder operators? I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be...

So I'm a little confused on the notation when working with wave functions constructed as a linear combination of an orthornormal basis set. Like on the form: \Phi=Ʃn cnψn If I want to find the expectation value represented by the operator O for the state described by \Phi, I would...

So for example, if I have a random variable X, take it to be normally distributed. How do you find the expectation and variance of the random variable e^X in terms of μ and σ? Integrating the entire normal function with the f(x) is it?

Homework Statement A wave function ψ is A(eix+e-ix) in the region -π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x=0 and x=π/8, and (b) between x=0 and x=π/4. Homework Equations The Attempt at a Solution So to...

Homework Statement I am given ψ(x), want to calculate <x^{2}>. Homework Equations \psi(x) = a\exp(ibx-(c/2)(x-d)^2) <x^2> = \int\limits_{-∞}^∞ \psi^*x^2\psi \mathrm{d}x The Attempt at a Solution Well, I normalized the wave function and found a = (\frac{c}{\pi})^{1/4}. So, the...

In a linear superposition, what is the relationship between expectation value of, say, energy and the amplitude coefficients of the eigenfunctions?

Hello everybody, I'm looking for a proof of the following equation: <x6> = <x>6+15s2<x>4 where the brackets denote an expectationvalue and s is the standard deviation. Thanks in advance!

Homework Statement {\Psi (x,t)} = \frac {m \omega}{\pi h_{bar}}^{1/4}e^{- \frac {m \omega}{2h_{bar}}(x^{2}+ \frac {a^2}{2}(1+e^{-2i \omega t}+\frac {ih_{bar}t}{m}-2axe^{-i \omega t})} Problems: Find |ψ(x,t)|2 Compute <x> and <p> Homework Equations {x = \int^\infty_{-\infty} x | \Psi...

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 1. For the ground-state of the 2D rigid rotor what is the expectation value of the angular momentum? And what is the corresponding uncertainty in this value? 2. Describe in words what the uncertainty in position is. 3. Explain why the rigid rotor can have vanishing...

Homework Statement Show that for the expectation values the following relations hold: d \langle x \rangle /dt =\langle p \rangle /m and d \langle p \rangle /dt = - \langle d V/dx \rangle. Homework Equations \langle x \rangle = \int _{- \infty}^{\infty} \Psi ^* x \Psi dx. The Attempt...

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...