Expectation value Definition and 347 Threads

Homework Statement Show that, for a general one-dimensional free-particle wave packet $$\psi (x,t) = (2 \pi h)^{-1/2} \int_{-\infty}^{\infty} exp [i (p_x x - p_x^2 t / 2 m)/h] \phi (p_x) dp_x$$ the expectation value <x> of the position coordinate satisfies the equation $$<x> = <x>_{t=t_0}...

Homework Statement (a) If a particle is in the spin state ## χ = 1/5 \begin{pmatrix} i \\ 3 \\ \end{pmatrix} ## , calculate the expectation value <Sy>(b) If you measured the observable Sy on the particle in spin state given in (a), what values might you get and what is the probability of...

Homework Statement A particle is moving in a one-dimensional harmonic oscillator, described by the Hamilton operator: H = \hbar \omega (a_+ a_- + \frac{1}{2}) at t = 0 we have \Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x)+i\psi_1(x)) Find the expectation value and variance of harmonic oscillator...

Homework Statement Suppose that { |ψ1>, |ψ2>,...,|ψn>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψj> = qj|ψj> for all j = 1...n. A particle is in the state |Φ>. Show that for this particle the expectation value of <Q> is ∑j=1nqj |<Φ|...

Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...

1. The problem statement Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by H = \frac{p^2}{2m} + \frac{m \omega ^2 x^2}{2} Knowing that the ground state of the particle at a certain instant is described by the wave...

If Eψ = Hψ, then why is expected energy ∫ψ*Hψ dx? It makes more sense if I see the ψ on the right side of H as the ψ in ∫Q(ψ*ψ) dx, where Q is some quantity we want to measure the expectation of. But if true, then since H is defined as (h2/2m) (d2/dx) + V, then what does it mean to calculate...

Homework Statement Homework Equations I know that there are two eigenstates of the operator C: |B> = (1 0) as a column vector with eigenvalue 1 |R> = (0 1) also a column vector with eigenvalue -1 The Attempt at a Solution My work is shown here: If anyone could point me in the right...

If w[n] are samples of the white gaussian noise process, I know that E[w[n1] w[n2]] = 0 for a WGN process. what would the following expression lead to: E[w[n1] w*[n2]] = ? Would it also be zero? Thanks a lot!

Imagine a particle in an equally weighted superposition of being located in three distant regions P, Q, and R, and imagine you stand in region P with a measuring device. The probability of finding the particle there is 1/3. Now imagine a large number N of particles prepared in that same state...

I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...

In 1D QM: I understand that if a given potential well, U(x), is symmetric about x = L, then the expectation value for operator [x] would be <x> = L. (I am not even entirely sure why this is, guessing that the region where x<L and x>L are equally probable) Is it possible to draw conclusion...

Homework Statement At t=0, the system is in the state . What is the expectation value of the energy at t=0? I'm not sure if this is straight forward scalar multiplication, surprised if it was, but we didn't cover this in class really, just glossed through it. If someone could walk me through...

Homework Statement An article in Business Week reports profits and losses of firms by industry. A random sample of 100 firms is selected, and for each firm in the sample, we record whether the company made money or lost money, and whether or not the firm is a service company. The data are...

Homework Statement Show that: d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket Homework Equations Liouville theorem The Attempt at a Solution <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ}) So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can...

I'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus...

Homework Statement Homework EquationsThe Attempt at a Solution I tried to solve (a), but i don't know which approach is right ((1) or (2)) and how to solve (b).[/B]

Homework Statement Homework Equations The Attempt at a Solution When I take the second formula, multiply by it's conjugate and then by x and do the integral of the first formula, I get 0, and not L/2, for <x>. Am I missing a formula ? The complex conjugate of the exponential part...

I have to calculate the Expectation Value of an Energy Eigenstate : < En > The integral is ∫ ψ* En ψ dx I have : A ) ψ = √L/2 sin nπx/L , a single standing wave of the wave function B ) ψ = BsinBcosD , the wave function of the particle C ) ψ = ΣCn ψn = C , sum of all the...

Hello, I'm a beginner at quantum mechanics. I'm working through problems of the textbook A Modern Approach to Quantum Mechanics without a professor since I am not going to college right now, so I need a brief bit of help on problem 1.10. Everything else I have gotten right so far, but I am...

Homework Statement A particle of mass m, is in an infinite square well of width L, V(x)=0 for 0<x<L, and V(x)=∞, elsewhere. At time t=0,Ψ(x,0) = C[((1+i)/2)*√(2/L)*sin(πx/L) + (1/√L)*sin(2πx/L) in, 0<x<L a) Find C b) Find Ψ(x,t) c) Find <E> as a function of t. d) Find the probability as a...

Homework Statement Show the mean position and momentum of a particle in a QHO in the state ψγ to be: <x> = sqrt(2ħ/mω) Re(γ) <p> = sqrt (2ħmω) Im(γ) Homework Equations ##\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})##The Attempt at a Solution I put ψγ into...

Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?

Homework Statement Given ##\psi = AR_{21}[BY_1^1 + BY_1^{-1} + CY_1^0]##, find ##\left<L_z\right>## and ##\left<L^2\right>##. (This is not the beginning of the homework problem, but I know my work is correct up to here. I am not looking for a solution, only an answer as to whether or not my...

Homework Statement Let ##\left|\psi\right\rangle## be a non-degenerate stationary state, i.e. an eigenstate of the Hamiltonian. Suppose the system exhibits symmetry for time inversion, but not necessarily for rotations. Show that the expectation value for the angular momentum operator is zero...

Homework Statement In a coherent state ##|\alpha\rangle##, letting ##P(n)## denote the probability of finding ##n^{\text{th}}## harmomic oscillator state. Show that $$\displaystyle{\langle\hat{n}\rangle \equiv \sum\limits_{n}n\ P(n)=|\alpha|^{2}}$$ Homework Equations The Attempt at a...

Homework Statement Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, ##A##, is given (in the same basis) by \begin{bmatrix} 0 &a \\ a & 0 \end{bmatrix} where ##a\in\mathbb{R}^+##. The initial state of the system...

It would be really appreciated if somebody could clarify something for me: I know that stationary states are states of definite energy. But are all states of definite energy also stationary state? This question occurred to me when I considered the free particle(plane wave, not a Gaussian...

Hello! Could somebody please tell me how i can compute the expectation value of the momentum in the case of a free particle(monochromatic wave)? When i take the integral, i get infinity, but i have seen somewhere that we know how much the particle's velocity is, so i thought that we can get it...

It says in Susskind's TM: ##\langle L \rangle = Tr \; \rho L = \sum_{a,a'}L_{a',a} \rho_{a,a'}## with ##a## the index of a basisvector, ##L## an observable and ##\rho## a density matrix. Is this correct? What about the trace in the third part of this equation?

Homework Statement √[/B] A particle in an infinite square well has the initial wave function: Ψ(x, 0) = A x ( a - x ) a) Normalize Ψ(x, 0) b) Compute <x>, <p>, and <H> at t = 0. (Note: you cannot get <p> by differentiating <x> because you only know <x> at one instance of time)Homework...

The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...

Homework Statement (a) Suppose we flip a fair coin until two Tails in a row come up. What is the expected number, NTT, of flips we perform? Hint: Let D be the tree diagram for this process. Explain why D = H · D + T · (H · D + T). Use the Law of Total Expectation (b) Suppose we flip a fair...

There is another topic for this but I didn't quite see it and I don't know how I've gone so far through my course not asking this simple question. So what's the difference? My thought process for hydrogen. I know it can have quantised values of energy, the energy values are the Eigen values of...

As it says; I was looking over some provided solutions to a problem set I was given and noticed that, in finding the expectation value for the momentum operator of a given wavefunction, the following (constants/irrelevant stuff taken out) happened in the integrand...

I'm having a hard time following the arguments of how the Higgs gives mass in the Standard Model. In particular, the textbook by Srednicki gives the Higgs potential as: $$V(\phi)=\frac{\lambda}{4}(\phi^\dagger \phi-\frac{1}{2}\nu^2)^2 $$ and states that because of this, $$\langle 0 | \phi(x)...

Homework Statement [/B] Particle in one dimensional box, with potential ##V(x) = 0 , 0 \leq x \leq L## and infinity outside. ##\psi (x,t) = \frac{1}{\sqrt{8}} (\sqrt{5} \psi_1 (x,t) + i \sqrt{3} \psi_3 (x,t))## Calculate the expectation value of the Hamilton operator ##\hat{H}## . Compare it...

So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...

Homework Statement The Hamiltonian of an electron in solids is given by H. We know that H is an Hermitian operator, it satisfies the following eigenvalue equation: H|Φn> = εn|Φn> Let us define the following operators in terms of H as: U = e^[(iHt)/ħ] , S = sin[(Ht)/ħ] , G = (ε -...

Homework Statement ## H ## is the Hamiltonian of an electron and is a Hermitian operator. It satisfies the following equation: ##H |\phi_n\rangle = E_n |\phi_n\rangle ## Let ## U = e^{\frac {iHt}{\hbar}} ##. Find the expectation value of U in state ##|\phi_n\rangle## Homework Equations ##...

I am following Griffiths' intro to quantum mechanics and struggling(already) on page 16. When a particle is in state ##\Psi##, $$\frac{d<x>}{dt} = \frac{i\hbar}{2m}\int_{-\infty}^{\infty} x\frac{\partial}{\partial t}\bigg (\Psi^*\frac{\partial \Psi}{\partial x}-\frac{\partial \Psi^*}{\partial...

It's my first post so big thanks in advance :) 1. Homework Statement So the question states "By interpreting <pxΨ|pxΨ> in terms of an integral over x, express <Ekin> in terms of an integral involving |∂Ψ/∂x|. Confirm explicitly that your answer cannot be negative in value." ##The 'px's should...

I am not sure why a factor of (½) appears in front of the summation over orbitals, i, j to N, of the Coulomb and exchange integrals in the HF energy expectation value.

Srednicki page 65 it says "Let us compute the vacuum expectation value of the field $$\phi(x)$$ which is given by $$\langle 0| \phi (x)|0 \rangle = \frac{\delta}{\delta J(x)} Z_{1}(J) |_{J=0}$$ This expression is then the sum of all diagrams that have a single source, with the source removed."...

Homework Statement Show that for a two spin 1/2 particle system, the expectation value is \langle S_{z1} S_{n2} \rangle = -\frac{\hbar^2}{4}\cos \theta when the system is prepared to be in the singlet state...

Homework Statement Consider the bipartite observable O_AB = (sigma_A · n) ⊗ (sigma_B · m) Where n and m are three vectors and sigma_i = (sigma_1_i, sigma_2_i, sigma_3_i) with i = [A,B] are the Pauli vectors. Compute using abstract and matrix representation the expectation value of O_AB...

Homework Statement If X1 has mean -3 and variance 2 while X2 has mean 5 and variance 4 and the two are independent find a) E(X1 - X2) b) Var(X1 - X2)The Attempt at a Solution I am not very clear on what I am supposed to be doing for this problem. I don't fully understand this expectation value...

I'm given an operator $\mathcal{L}$ is Hermitian, and asked to show $<\mathcal{L}^2>$ is $\ge 0$ I believe $<\mathcal{L}>$ is the expectation value, $=\int_{}^{}\Psi^* \mathcal{L} \Psi \,d\tau $ (Side issue: I am not sure what $d\tau $ is, perhaps a small region of space? And the interval?) I...

Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...

Hey, I'm stuck halfway through the solution it seems. I could use some tips on how to continue. 1. Homework Statement I have to determine a linear combination of the states |0\rangle, |1\rangle, of a one dimensional harmonic oscillator, so that the expectation value \langle x \rangle is a...