Expectation value Definition and 347 Threads

In studying the Aharonov-Bohm effect, a model of an electron confined in a box is used, for example, on page 353 of Modern Quantum Mechanics by Sakurai et al. The box makes one turn along a closed loop surrounding a magnetic flux line. In the derivation, there will be an integration involving...

So first I derived the expressions for the dynamics of the spin operators and got: $$ \frac{d\hat{S}_y}{dt} = w\hat{S}_x^H $$ $$ \frac{d\hat{S}_x}{dt} = w\hat{S}_y^H $$ $$ \frac{d\hat{S}_z}{dt} = 0 $$ Now I want to calculate the time-dependence of the expectation values of the spin operators...

Hi all, I am not familiar with stochastic processes, but I would like to know how to evaluate the following expectation value: $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$ where ##\mathbb{E}[V_{i}(t)] = 0,\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')## for some...

Hello! I am confused about the derivation in the screenshot below. This is in the context of a diatomic molecular potential, but the question is quite general. Say that the potential describing the interaction between 2 masses, as a function of the radius between them is given by the anharmonic...

hi all how do I prove that $$ <A^{n}>=<A>^{n} $$ It seems intuitive but how do I rigorously prove it, My attempt was like , the LHS can be written as: $$ \bra{\Psi}\hat{A}.\hat{A}.\hat{A}...\ket{\Psi}=\lambda^{n} \bra{\Psi}\ket{\Psi}=\lambda^{n}\delta_{ii}=\lambda^{n} $$ and the RHS equal: $$...

##\begin{align} \langle E \rangle &= \int_{\mathbb R^3} \int_{\mathbb R^3} \int_{\mathbb R^3} \int_{\mathbb R^3} g^\dagger (\tilde K) g(K) |\psi_0(x)|^2 \left(E_0 +\frac{\hbar^2 |K|^2}{2m}\right) e^{i(K-\tilde K)\cdot X -\frac{i}{\hbar} \left(\frac{\hbar^2 |K|^2}{2m}-\frac{\hbar^2...

Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then ##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0## will require that the momentum expectation value remains...

The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...

Apart from the usual integral method, are there any other ways to find expectation value of momentum? I know one way is by using ehrenfest theorem, relating it time derivative of expectation value of position operator. Even using the uncertainty principle, we might get it if we know the...

so from Fourier transform we know that Ψ(r)=1/2πℏ∫φ(p)exp(ipr/ℏ)dp I proved that <p>= ∫φ(p)*pφ(p)dp from <p>=∫Ψ(r)*pΨ(r)dr so will the same hold any operator??

When the expectation value of spin in the z direction for one particle is zero and I make measurements for an even number of particles in the same state, do I get exactly half to be spin up and half to be spin down along the z direction? More generally, what does spin expectation value for one...

I have found an answer to all of them (a-e) but I don’t know how to plot the function. Thanks!

Hi A theorem states that if V(x , t) ≥ V0 then <E> is real and <E> ≥V0 for any normalizable state. The proof contains the following line <E> = (ħ2/2m)∫∇ψ*∇ψ d3x + ∫ Vψ*ψ d3x ≥ ∫ V0ψ*ψ Can anybody explain why that inequality is true ? Thanks

So, I have a hamiltonian for screening effect, written like: $$ H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k} $$ And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##. I...

Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...

In non relativistic quantum mechanics, the expectation value of an operator ##\hat{O}## in state ##\psi## is defined as $$<\psi |\hat{O}|\psi>=\int\psi^* \hat{O} \psi dx$$. Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial...

In This wikipedia article is said: "If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a...

Hi all, I found this notation of expectation values in a NMR text. In class, I learned that expectation values are given by $$<\hat{X}>=\int_{-\infty}^\infty\psi^*x\psi dx$$ why does this textbook divide by the integral of probability density ##\int \psi^*\psi dx##?

I know that the eigenstates of momentum operator are given by exp(ikx) To construct a real-valued and normalized wavefunction out of these eigenstates, I have, psi(x) = [exp(ikx) + exp(-ikx)]/ sqrt(2) But my trouble is, how do I find the expectation value of momentum operator <p> using this...

The expectation value of the kinetic energy operator in the ground state ##\psi_0## is given by $$<\psi_0|\frac{\hat{p^2}}{2m}|\psi_0>$$ $$=<\psi_0|\frac{1}{2m}\Big(-i\sqrt{\frac{\hbar mw}{2}}(\hat{a}-\hat{a^{\dagger}})\Big)^2|\psi_0>$$ $$=\frac{-\hbar...

A recent thread by @coolcantalope was accidentally deleted by a Mentor (I won't say which one...), so to restore it we had to use the cached version from Yahoo.com. Below are the posts and replies from that thread. The cached 2-page thread can be found by searching on the thread title, and is...

I may have misunderstood the expectation value, but if not then with the Copenhagen Interpretation it is easy to understand the expectation value for a wave function. It is just based on the probability of each event. If there were 4 possible events, and the probability of the event having a...

In the 3rd edition of the Introduction to Quantum Mechanics textbook by Griffiths, he normally does the notation of the expectation value as <x> for example. But, in Chapter 3 when he derives the uncertainity principle, he keeps the operator notation in the expectation value. See the pasted...

Apologies if this isn't the right forum for this. In my stats homework we have to prove that the expected value of aX and bY is aE[X]+bE[Y] where X and Y are random variables and a and b are constants. I have come across this proof but I'm a little rusty with summations. How is the jump from the...

I can not solve this problem: However, I have a similar problem with proper solution: Can you please guide me to solve my question? I am not being able to relate Y R (from first question) and U (from second question), and solve the question at the top above...

Hi, I have a question which asks me to use the generalised Ehrenfest Theorem to find expressions for ##\frac {d<Sx>} {dt}## and ##\frac {d<Sy>} {dt}## - I have worked out <Sx> and <Sy> earlier in the question. Since the generalised Ehrenfest Theorem takes the form...

Summary:: Linear Quantum harmonic oscillator and expectation value of the potential energy (time dependent) Hello, I have attached a picture of the full question, but I am stuck on part b). I have found the expectation value of the <momentum> and the <total energy> However I am struggling with...

Given that operator ##S_M##, which consists entirely of ##Y## and ##Z## Pauli operators, is a stabilizer of some graph state ##G## i.e. the eigenvalue equation is given as ##S_MG = G## (eigenvalue ##1##). In the paper 'Graph States as a Resource for Quantum Metrology' (page 3) it states that...

I am struggling to figure out how to calculate the expectation value because I am finding it hard to do something with the exponential. I tried using Euler's formula and some commutator relations, but I am always left with some term like ##\exp(L_z)## that I am not sure how to get rid of.

I can show that ##\frac{d}{dt} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{1}{m} \langle \psi (t) \vert PX+XP \vert \psi (t) \rangle##. Taking another derivative with respect to time of this, I get ##\frac{d^2}{dt^2} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{i}{m...

⟨Lx⟩=⟨l,m|Lx|l,m⟩=−iℏ⟨l,m|[Ly,Lz]|l,m⟩

For question 2.2: <Ψ0|p|Ψ0> = ∫Ψ0 -iħ d/dx(Ψ0) =M Using Integration by parts i get: M = -Ψ0 iħ d/dx(Ψ0) (assuming hilbert space) Implying the expectation values for momentum are zero , however i get all the expectation values are zero for x and momentum in both states which makes no sense :(

I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?

Okay so I begin first by mentioning the length of the well to be L, with upper bound, L/2 and lower bound, -L/2 and the conjugate u* = Aexp{-iz} First I begin by writing out the expectation formula: ## \langle p \rangle = \int_{\frac{L}{2}}^{ \frac{L}{2} } Aexp(-iu) -i \hbar \frac{ \partial }{...

I am currently studying this paper on quantum synchronization. The first page gives an introduction to synchronization and the basic setup of the ensembles in the cavity. My query is on the second page where the following statements are made. Can anyone see why the implication is that all...

In the derivation of the Fermi-Dirac and Bose-Einstein distributions, we compute the Grand Partion Function ##Q##. With ##Q##, we can compute the espection value of the occupation number ##n_{l}##. This is the number of particles in the same energy level ##\varepsilon _{l}##. The book I am...

Watching Dr. Susskind show how to find the time evolution of the average of an observable K, he writes: I can not for the life of me figure out he derived it, and he also did something which I found terribly annoying throughout which is set hbar to 1, so after steps you lose where the hbar...

Homework Statement Hello today I am solving a problem where an electron is trapped in a potential well. I have a solved Schrodinger's Equation. I am having problems in figuring out what the wave function should be. When I solved the equation I got a complex exponential. I know I cannot use the...

Homework Statement Homework Equations VD= -1/(8m2c2) [pi,[pi,Vc(r)]] VC(r) = -Ze2/r Energy shift Δ = <nlm|VD|nlm> The Attempt at a Solution I can't figure out how to evaluate the expectation values that result from the Δ equation. When I do out the commutator, I get p2V-2pVp+Vp2. This...

I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?

Homework Statement If x is a continuous variable which is uniformly distributed over the real line from x=0 to x -> infinity according to the distribution f (x) =exp(-4x) then the expectation value of cos 4x is? Answer is 1/2 Follow· 01 Request Homework Equations the expectation value of any...

The question is as follows: A particle of mass m has the wave function psi(x, t) = A * e^( -a ( ( m*x^2 / hbar) +i*t ) ) where A and a are positive real constants. i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the...

Homework Statement Let ##U_t = e^{-iHt/\hbar}## be the evolution operator associated with the Hamiltonian ##H##, and let ##P=\vert\phi\rangle\langle \phi\vert## be the projector on some normalized state vector ##\vert \phi\rangle##. Show that $$\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n\text{...

Homework Statement Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which $$\langle\vec{\sigma}\rangle =\vec{e},$$ where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle## Homework Equations :[/B] are all given...

Homework Statement I have a general question how I calculate the expectation value of V (potential energy) with Ehrenfest’s theorem. Do I have to integrate d<p>/dt with respect to d<x>. Also if the potential is symmetric (even) would that mean the expectation value of the potential is 0...

Homework Statement How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom. Homework Equations The Attempt at a Solution I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I...

Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...

Hi all, I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For...

I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result. Homework Statement At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by: ## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...

Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...