Expectation value Definition and 337 Threads

Homework Statement The expectation value of the time derivative of an arbitrary quantum operator \hat{O} is given by the expression: d\langle\hat{O}\rangle/dt\equiv\langled\hat{O}/dt\rangle=\langle∂\hat{O}/∂t\rangle+i/hbar\langle[\hat{H},\hat{O}]\rangle Obtain an expression for...

We have for two random variables X and Y (one sum runs over j and one over k): E(X+Y) = ƩƩ(xj+yk)P(X=xk,Y=yk) = ƩƩxjP(X=xk,Y=yk) + ƩƩykP(X=xk,Y=yk) Now this can be simplified to obtain E(X+Y)=E(X)+E(Y) if we use that: P(X=xk,Y=yk) = P(X=xk)P(Y=yk), because then (and same goes the other...

why we find expectation value? why do we take expectation value of sime entity? and how come we know that our expected value is correct

Homework Statement In a QM problem I must calculate the expectation value of the kinetic energy, namely ##\langle \Psi _c ,\frac{ \hat { \vec p } ^2}{2m} \Psi _c \rangle##. Where ##\Psi _c=ce^{-\alpha x^2}##. Homework Equations ##\int _{-\infty}^\infty \exp \{ -a x^2 +bx \} dx =...

I realize that at ground state of a harmonic oscillator the energy will be at zero. I'm assuming that the expectation value will also be at zero. Could someone confirm this & possibly explain just a little more. Thank you

Homework Statement The kinetic energy is given by \left\langle E_{kin} \right\rangle = \frac{\left\langle \widehat{p}^2 \right\rangle}{2m} In Dirac notation we have \left\langle E_{kin} \right\rangle = \frac{1}{2m} \left\langle \widehat{p}\Psi | \widehat{p}\Psi \right\rangle Homework...

we have variables X,Y with f(m,n)=P(X=m,Y=n) with f(0.1)=0.1 f(1.0)=0.1 f(1.1)=0.344 find the expectation value E(X+Y) i need help because i don't how to start to solve this , if i begin with the definition of the expected value i can't do anything any ideas?

Homework Statement Say you have a large column of gas with insulating walls standing on the Earth's surface, which is L high and at room temperature (25degC) at the interface of the surface and column. Assuming the potential energy on the gas is only duel to gravity, U = mgx where x is the...

Hi! I've seen it stated that because of Lorenz and translational invariance \langle 0| \phi(x) |0 \rangle has to be a constant and I wondered how to formally verify this?

Homework Statement The expectation value of motion of a particle over a time interval t-to is C(t,to) = <0|x(t)x(to)|0> (product of position operators in Heisenberg representation for ground state harmonic oscillator) Homework Equations Schrodinger picture: <ψ(t)|Ω|ψ(t)> =...

Homework Statement This may be incredibly obvious, but I just need to check. Of course we all know that physical observables must yield real expectation values. What if you tried to calculate, say, <xV(d/dx)>, where x is the position, d/dx is a first derivative, and V is the potential? This...

Homework Statement Why does the extra phase factor cancel out? Is it because you are multiplying the wave-function with the extra phase factor by its conjugate and if so, why should it matter that the extra phase factor is independent of x? All relevant information, the solution and equation...

I have a question regarding an exercise I am doing. It is an electron confined to move on a cylinder and I am asked to: "Find the expectation value of Ly and Lz" in the unperturbed basis. I am just not sure what is meant by the expectation value in a basis? I know what the expectation value is...

Homework Statement I don't know how the writer of the book took integral of the first statement and got the second statement? Can anybody clarify on this? Homework Equations Given in the photoThe Attempt at a Solution When I took the integral I just ended up with the exact same statement but...

Homework Statement Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the Fourier transform show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k 2. The attempt at a solution Let n = 1 and substitute the expression for the momentum operator. Transform the...

Homework Statement A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values hω/2 or 3hω/2, each with a probability of one-half. The average values of the momentum <p> at time t = 0 is √mωh/2. This information...

Hey, I'm having trouble interpreting a question, as the solutions say something different... Anyways the question part d) below: So we want to determine the expectation value of the y-component of the electron spin on the eigenstate of Sx, now I would of thought this was given by...

Homework Statement This problem comes from the second edition of Griffiths's, Introduction to Quantum Mechanics. Given the Gaussian Distribution: p(x) = Aec(x-a)2 find <x>, that is, the expectation (or mean) value of x. Clearly, to do this you evaluate the following integral: ∫xp(x)dx on...

Homework Statement Basically I need to produce a state for a spin-1/2 particle such that the expectation value of <Jz> = 0 where <Jz> is for a spin-1 particle. Homework Equations Jz = (1 0 0, 0 0 0, 0 0 -1) <--[3x3] matrix The Attempt at a Solution I don't quite understand how to do this...

Homework Statement What is the expectation value of <p*x> aka the momentum times the position operator, for a particle in a box. Homework Equations Psi(x) = root(2/l) sin (n∏x/l) P= -ih(bar)d/dx X=x The Attempt at a Solution All integrals are from 0 to L I'm typing this on a playbook so I...

Homework Statement A particle in an infinite box is in the first excited state (n=2). Obtain the expectation value 1/2<xp+px> 2. The attempt at a solution Honestly, I don't even know where to begin. I assumed V<0, V>L is V=∞ and 0<V<L is V=0 I tried setting up the expectation...

It seems that the energy expectation value is independent of time. I did it for an infinite square well. And when you time evolve your wave function the time evolution cancels when you complex conjugate it and then do the integral. <E>=<ψ|E|ψ> it seem like this might always...

Homework Statement Homework Equations The Attempt at a Solution I am just trying to figure out how to start the problem. Any help would be greatly appreciated.

Going from the abstract state vector lψ> and the mean-value of an observable x (operator) given by: <x> = <ψlxlψ> I want to show how that is done in the position basis: So I take: <x> = <ψlxlψ> And insert completeness in front of the state vector to get the expansion involving the...

Good Evening Fellows, I have the following question, So far I have learned that the expectation value of momentum is equal the time derivative of the expectation value of position. If the potential only depends upon position and not on time. Then, if we use the time independent schrodinger...

I have to find the expectation value of the z component of the angular momentum for a particle on a ring and the expectation value of the z component of the angular momentum squared for a particle on a ring. The wavefunction is e^((± imx)) I've determined that the expectation value for the...

Homework Statement Show that the expectation value for r for an electron in the groundstate of a one-electron-atom is: <r>=(3/2)a_{0}/Z Homework Equations Expectationvalue: <f(x)>=∫\psi*f(x)\psidx, -∞<x>∞ \psi_{100}=C_{100} exp(-Zr/a_{0}), a_{o}\ =\ 0.5291\ \times\ 10^{-10}m , h\...

Homework Statement Assume that a particle travels with a certain known (average) velocity ##v = \left\langle\hat{p}/m\right\rangle##. You know it's position with an uncertainty ##Δx##. Use the uncertainty principle to determine the least possible value for the article's kinetic energy...

I am asked a problem where I'm supposed to integrate the expectation value of a dynamic variable (operator) to solve a differential equation. OK, is the expectation value supposed to be a variable? But it seems to me like its a definite integral over allspace and thus is a number. So...

[b]1. consider this wave function ψ(x)=(√(30/L^5))(L-x) if 0≤x≤L and 0 else [b]2. Compute the expectation value of the momentum. Compute the expectation value of the kinetic energy. Compute Δ p⋅Δ x...

Hello, I was just curious about expectation values. One of the postulates of quantum mechanics state: The only possible results of a measurement is an eigenvalue of the operator. Now, is the expectation value considered a measurement, thus considered an eigenvalue? Thanks!

difference between eigenvalue and an expectation value of an observable. in what circumstances may they be the same? from what i understand, an expectation value is the average value of a repeated value, it might be the same as eigen value, when the system is a pure eigenstate.. am i right?

Consider two Hermitian operator A, B; Define [A,B]=iC, then operator C is also Hermitian. we calculate the expectation value with respect to |a>, one eigenstate of A with the eigenvalue a. From the left side, we have: <a|[A,B]|a>=<a|(AB-BA)|a>=(a-a)<a|B|a>=0, while on the right side...

Homework Statement the first two energy eigenstates of a 1 nm wide finite well of barrier height 8vo have energy eigen values of 0.66ε and 2.6ε. calculate the expectation value of a linear superposition of these states? Homework Equations airy equations The Attempt at a Solution...

expectation value for a particle in a 1-D well how do i calculate the expectation value for the particles energy in a 1-D well. i have attached a word file, with my working out, just not quite sure if I am on the right track... i appreciate any help...thanks a mill

Dear all, I'm wondering, how one could justify mathematically the equality \int O(E(\vec{x}_1,...\vec{x}_N)) exp(-\beta E(\vec{x}_1,...,\vec{x}_N)) d\vec{x}_1...d\vec{x}_N = \int g(E) O(E) exp(-\beta E) dE where O(E(x)) is an observable and g(E) the density of states. Is there a...

Homework Statement Hi I have read a paper, where they want to find the average number of photons in a cavity. They have an expression for \langle{\hat a}\rangle, and then they use \langle{\hat a}\rangle^* = \langle{\hat a^\dagger}\rangle to find \langle{\hat a^\dagger \hat a}\rangle. I agree...

Homework Statement Hi My book uses the following in a calculation \left\langle a \right\rangle \left\langle {b^\dagger } \right\rangle + \left\langle {a^\dagger } \right\rangle \left\langle b \right\rangle = 2\operatorname{Re} \left[ {\left\langle a \right\rangle \left\langle {b^\dagger...

Homework Statement given \mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle ) where \mid1\rangle, \mid2\rangle are orthonormal calculate i)density operator ii) \langle A \rangle where A is an observable Homework Equations The Attempt at a Solution i) \rho = \frac{1}{2}...

Hi, (Sorry for the slight misnomer in the title... I can't edit it!) I'm doing several problems to compute the expectation value and variances of sub-samples & operations on the normal distribution; and I am having trouble getting results that agree with numerical simulations. I have several...

I was told that given a probability distribution p(x) dx, the expected value for x is given by: <x> = Ʃ xi P(xi) = ∫ x P(x) dx This part makes sense to me. It was justified to me through the use of weighted averages. However, my teacher then made a hand-wavy move to generalize the above...

Dear All: I have a quite mysterious and cumbersome question concerning with the expectation values for a system of identical particles. For example, suppose I have a system of N identical bosons given by the wavefunction ψ(x1,x2,...xN), which is of course symmetrized. My concern is: 1...

Hi everyone, What is the difference between an expectation value and an average. I may have this wrong, but is it something along these lines: You perform a series of measurements on a given observable, such as momentum, and the average value of all these measurements is your expectation...

How can we compute the expectation value, <\widehat{x}\widehat{p}> where ψ(x) is a normalized wavefunction? (The result is i\hbar/2)

Homework Statement At time t=0 a particle is described by a one dimensional wavefunction (capital)ψ(x,0)= (2a/)^(1/4) e^(-ikx)e^(-ax^2) (three lines)=(2a/)^(1/4) e^(-ikx-ax^2)--------equation 1 k and a are real positive constants Homework Equations I think this is the one <p subscript(x)> =...

Hi, as I am new in Matlab, so I need your help. I want to replace the following inverse matrix (X'*X)^-1 with its expectation value: E{(X'*X)^-1} = E{|1/Xk|^2}I X'*X and (X'*X)^-1 is a diagonal matrix. Could anyone give me an Idea how to write it in MATLAB this expectation value...

Hi, In Birrel and Davies ch4 they write: \langle \psi|:T_{ab}:|\psi \rangle =\langle \psi|T_{ab}|\psi \rangle -\langle 0|T_{ab}|0 \rangle this is for the usual Mink field modes and vac state. Why does normal ordering reduce to this expression, could anybody point me the way to...

Homework Statement In my homework assignment I have a wavefunction defined as \Psi(x)=N\exp(-|x|/a) and I am asked to find the expectation value of momentum squared in configuration space. Homework Equations \int\Psi*(x)\hat{p^2}\Psi(x)dx The Attempt at a Solution N is 1/\sqrt{a}...

I know that <Aψ|Aψ> = <A2>. Can anyone tell please tell me how you get one line to the another?

Homework Statement Consider an observable A associated to an operator A with eigenvalues an. Using the formula <A> = ∫ψ*Aψ compute the expectation value of A for the following wave function: \Psi=\frac{1}{\sqrt{3}}\phi_{1}+\frac{1}{\sqrt{6}}\phi_{2}+\frac{1}{\sqrt{2}}\phi_{3} where...