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...
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...
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...
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},$$
where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle##
Homework Equations :[/B] are all given...
Homework Statement
How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom.
Homework Equations
The Attempt at a Solution
I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I...
Homework Statement
I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##.
Homework Equations
$$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...
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...
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 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!
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...
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...
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 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...
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...
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
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...
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?
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...
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...
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 -...