Expectation value Definition and 337 Threads

I am quite new to Quantum Mechanics and I am studying it from the book by Griffiths, as kind of a self-study..no instructor and all... For a gaussian wavefunction \Psi=Aexp(-x^{2}), I calculated <p^{2}> and found it to be equal to ah^{2}/(1-2aiht/m) (By h I mean h-bar..not so good at...

Homework Statement I have a random, uniformly distributed vector with Cartesian components x,y,z. I should calculate the expectation value of the products of the components, e.g. <x\cdot x>, <x\cdot y>, ..., <z\cdot z>. Homework Equations In spherical coordinates the x,y,z components...

In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?

[SOLVED] expectation value of P^2 for particle in 2d box I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral...

Homework Statement If X is a real valued random variable with E[|X|] finite. <-> \sum(P(|X|>n)) finite , with the sum over all natural numbers from 1 to infinity. Homework Equations As a tip I am given that for all integer valued X>0 E(X) = \sum(P(X)>k , where the sum goes over all k =1 to...

I'm confused re a particle of energy E < V inside a square potential of width 'a' centered at x = 0 with depth V. They give the wavefunction for outside the well as \Psi(x) = Ae^{k|x|} for |x| > a/2 and k^2 = -\frac{2ME}{\hbar^2} => k = i\frac{\sqrt{2ME}}{\hbar} ? I need the probability that...

I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...

Homework Statement u(x) = \sqrt{\frac{8}{5}}\left(\frac{3}{4}u_{1}(x)-\frac{1}{4}u_{3}(x)\right) Determine the time-dependent expectation value of position of this wave function (the particle is in an infinite potential well between x = 0 and x = a). The Attempt at a Solution I...

I am trying to show that \frac{d}{dt}<x^2>=\frac{1}{m}(<xp>+<px>)....(1) With the wavefunction \Psi being both normalized to unity and square integrable Here is what I tried... <xp> = \int_{-\infty}^{\infty}{\Psi}^*xp{\Psi}dx <px> =...

Homework Statement Can somebody help me integrate \int{x\cdot p(x)} where p(x) is the Gaussian distribution (from here http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html) The Attempt at a Solution I can't really get anywhere. It's true that \int{e^{x^2}} has no analytical...

I am trying to find <x> for \psi(x,t) = A exp\left(-|x|/L - i*E*t/\hbar\right) I found the normalization factor of 1/L and I took \int_{-\infty}^{\infty}\left( x * exp(|x|/L) \right) in two integrals however I got as a final result: L * -\infty * exp(-\infty/ L) - L *...

Homework Statement Find the expectation value for a hydrogen atom's radius if n=25 and l=0. Homework Equations expectation value = <f|o|f> where f=wavefunction and o=operator The Attempt at a Solution So I know that to find an expectation value, you integrate over all relevant...

Here's a silly question. I'm sure I should know the answer, but alas my most recent QM course was 9 years ago. I sat down to calculate the expectation value of momentum in the H-atom today, because some kid on another forum wanted to know how fast an electron in an atom is. I was going to...

Homework Statement We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates...

Homework Statement I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square? Homework Equations \langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx \langle Q^2 \rangle = \int_{-\infty}^{\infty}...

1. Problem statement This isn't a homework question itself, but is related to one. More specifically, I'm computing the time-derivative of \langle x \rangle using the correspondence principle. One side simplifies to \left\langle \frac{\hat{p}}{m} \right\rangle, but what is the physical meaning...

When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate? Also as I am...

My question says: "Evaluate the expectation value <1/r> of the 1s state of hydrogen. How does this result compare to the result found using the Bohr theory?" Firstly, I have been told that <1/r> does not mean <1/r> but rather that it means 1/<r>. Having made it this far I now do the 1/<r>...

OK, here is the problem: An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is): \phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}]...

is there a better way to check for hermicity than doing expecation values? for example, what if you had xp (operators) - px (operators), or pxp (operators again); how can you tell if these combos are hermetian or not, without going through the clumsy integration (that doesn't give a solid...

I'm still really confused on how to go about calculating this for non eigenstates. I'm trying to do the problem below, and am wondering how to go about it. \Psi (x,0) = A (1-2 \sqrt {\frac{m \omega}{\hbar}} x)^2 e^ {-\frac{m \omega x^2}{2 \hbar}} So I can't calculate the expectation...

I have calculated the expectation value of a particle in a box of width a to be a/2. The wavefunction of the particle is: N Sin(k_n x) Exp[-i \frac{E_n t}{\hbar}] Now, in the first excited state with k_n equal to 2\pi / a the position probability density peaks at a/4 and 3a/4 but is zero...

I need to calculate <x^n> and <p^n> for psi(x)=exp(-ax^2/2) for n even. For <x^n>: <x^n>=integral(exp(-ax^2)*x^n )dx from -inf to +inf then i use integration by parts to get an infinite series and i use a formula to find the finite sum of the series =[exp(-ax^2)*x^(n+1)/((n+1-2a*(n+1)^2)]...

hi all can sombody show me the way I could get the square expectation value http://06.up.c-ar.net/03/fd4f.jpg for a particle in a box where the answer is given to us : http://06.up.c-ar.net/03/87d0.jpg

I have a wavefunction given by: \psi = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} With boundary conditions 0<x<L. When I compute the expectation value for the momentum like this: \overline{p_x} = \int_0^L \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} \left(-i\hbar \frac{\partial}{\partial...

how do you find the expectation value <x> for the 1st 2 states of a harmonic oscillator?

Hi.. i am doing this question for Probability Theory, to find E[x] of a continuous random variable E[x] = the integral from (0 to infinity) of 2x^2 * e^(-x^2) dx So I used integration by parts... u = x^2 du = 2xdx dv = e^(-x^2) <--- ahh... how do you integrate that. (it dosn't look like...

We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator. I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do...

I need to find the momentum expectation value of the function in the attached picture. It is the function of the harmonic oscillator (first excited state). :confused: I know that the expectation value is the value that we measure with the highest probability if we measure the system. But...

Hi, I have to find the expectation values of xp and px for nth energy eigenstate in the 1-d harmonic oscillator. If I know <xp> I can immediately find <px>since [x,p]=ih. I use the ladder operators a_{\pm}=\tfrac1{\sqrt{2\hslash m\omega}}(\mp ip+m\omega x) to find <xp>, but I get a complex...

Hi, could anyone tell me how one would show that the expectation value of a anti-Hermitian operator is a pure imaginary number? Thanks.

This is the problem: Calculate: \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \frac {d \mean{p}}_{dt} Here's a few more points to keep in mind... (A) The assumption is that <p> is defined as: \newcommand{\mean}[1]{{<\!\!{#1}\!\!>}} \mean{p} = -i \hbar \int \left( \psi^* \frac...

I want to find the expectation value \langle x^2 \rangle in some problem. To do this I make a change-of-variable, \xi = \sqrt{\frac{m\omega}{\hslash}}x, and compute the expectation value \langle \xi^2 \rangle like this: \langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi...

When I'm in a dimension higher than 1, do I need to integrate over all space (V) or only the x axis? Thanks in advance.

I had thought that the expectation value would be the same...whether you did it in momentum space or position space. Could someone explain what is going on in this particular problem? \psi (x) = \sqrt{b} e^{-b |x| + i p_0 x / \hbar } Taking the Fourier transform, I can get this...

If the expectation value <x> of a particle trapped in a box L wide is L/2, which means its average position in the middle of the box. Find the expectation value <x squared>. How do I go about doing this? I am really confused.

Expectation value problem pleasezzz help ASAP Hi Everyone, I have a problem on one of my problems in the quantum course. I need tofind the expectation values <x>,<x^2>, <p> & <p^2> for the function e^(-(x-xo)^2/2k^2) please email me if you need theformulaes.. i have them but i...