Expectation value Definition and 337 Threads

Homework Statement [/B] Sakurai problem 1.20: find the linear combination of spin-up and spin-down S_z eigenkets that maximizes the uncertainty product \langle(\Delta S_x)^2\rangle\langle(\Delta S_y)^2\rangle. Homework Equations [/B] In general, we can write a normalized spin-space ket as...

I have an inner product ## \langle \alpha|f| \beta \rangle## where ##f## is an operator that is a function of position ##x## operator (1D). According to the book I read (and I'm sure in any other book as well), that inner product can be written in position representation as ## \int...

Homework Statement I will denote operators by capital letters. The question is calculate <p | XXPP | x> / <p | x > Homework Equations X |x> = x |x> P |p> = p |p> P |x> = -i(hbar)d/dx X |p> = i(hbar)d/dp The Attempt at a Solution If I start on the RHS and take PP out I get...

Homework Statement The expectation value of <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx For the Guassian wave-packet ψ(x)=(1/(π^1/4)(√d))e^-((x^2)/(2d^2)) Limits on all integrals are ∞ to -∞. Homework Equations <P^2>= -ħ∫ψ* ∂^2ψ/∂x^2 dx ψ(x)=(1/(π^1/4)(√d))e^(iKx)-((x^2)/(2d^2)) The Attempt at a Solution Ok...

Can I ask a basic question. This was a question in a test, I could not solve this. When is it true that the result of a single measurement for a dynamical variable is equal to the expectation value of the operator corresponding to that dynamical variable? Thank you for your help. Sincerely...

Where one can find a proof of the expectation value of operator expression. <A> = < Ψ | A | Ψ > or <A> = integral( Ψ* A Ψ dx ) Thanks.

Homework Statement The problem asks me to find the expectation value of W. Homework Equations The given ψ[x,t] is Asin(πx/a) e^((-i Eot)/ħ). By QM postulate 2 the QM operator of W is: iħ δ/δt or equivalently -ħ/i δ/δt. The Attempt at a Solution <w>=∫ψ*iħδ/δtψ= iħδ/δt 1/(2e^(-iEot/)ħ)...

Homework Statement I'm curious in proving that expectation value of momentum for any bound state is zero. So the problem is how to prove this.Homework Equations $$ \langle \mathbf{p_n} \rangle \propto \int \psi^*(\mathbf{r_1}, \dots ,\mathbf{r_N}) \nabla_n \psi(\mathbf{r_1}, \dots...

I have come across a bit of conflict in wording of some physics and chemistry textbooks about the probability of finding particles in certain places. To be more specific, I have come across 3 different statements: 1. $$\int_a^b {| \psi(x) |^2 dx}$$ The above integral is said to give the...

Homework Statement It's an old assignment for exam, but the solution manual gives little help: Describing a particle of mass m moving in one dimension (x) the wave function at time t=0 is: ## \Psi(x,t=0) = A \frac{1}{\sqrt{(x-x_0)^4 + l^4}} ## ##x_0## and ##l## are positive constants...

Homework Statement Consider a one-dimensional particle subject to the Hamiltonian H with wavefunction \Psi(r,t) =\sum_{n=1}^{2} a_{n}\Psi _{n}(x)e^{\frac{-iE_{n}t}{\hbar}} where H\Psi _{n}(x)=E_{n}\Psi _{n}(x) and where a_{1} = a_{2} = \frac{1}{\sqrt{2}}. Calculate the expectation value of the...

Homework Statement Hello, I'm a bit confused about the calculation of the expectation values. Normally, when I have a wave function of sort and I want to calculate the expectation value of some operator, I just insert it into the braket <ψ|A|ψ>, where ψ for example is a wave function composed...

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

When calculating the expectation value of momentum of a real wavefunction is it always zero ? The momentum operator introduces an i into the integral and with real wavefunctions there is no other i to cancel and all Hermitian operators have real expectation values.

Homework Statement A particle at time zero has a wave function Psi(x,t=0) = A*[phi_1(x)-i*sin(x)], where phi_1 and phi_2 are orthonormal stationary states for a Schrodinger equation with some potential V(x) and energy eigenvalues E1, E2, respectively. a) Compute the normalization constant A. b)...

we have a wavefunction \psi (x) the question asks for \psi (p) and says to use this to calculate the expectation value of momentum. The problem is the expectation value of momentum is integrated over dx so after transforming how do you get the integral to be over dp? thanks for any help with...

Homework Statement System of 2 particles with spin 1/2. Let \vert + \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \\ \vert - \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} singlet state \vert \Phi \rangle = \frac{1}{\sqrt{2}} \Big( \vert + \rangle \otimes \vert - \rangle - \vert - \rangle...

CAn someone help me calculate expectation value of energy of electron in conduction band for semi conductor

I know the difference between the expectation value and probability density, but how do you calculate the probability density of an observable other than position? For position, the probability of the particle being in a particular spot is given by |\Psi|^2, which is the probability density, and...

Homework Statement The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction ##ψ=e^(-αr) ## show that Homework Equations ##<ψ|H|ψ>/<ψ|ψ> = (\hbarα)^2/2m + 3mω^2/2α^2## The following...

Homework Statement Given the following normalised time-independent wave function the question asks for the expectation value of the energy of the particle. The well has V(x)=0 for 0<x<a Homework Equations ψ( x ) = √(1/a) ( 1+2cos(∏x/a) )sin(∏x/a) The Attempt at a Solution I...

Homework Statement I've been reading Leonard Susskind's Theoretical Minimum volume on QM, and enjoying it quite a bit - the book doesn't include exercise solutions at the end though, and if they exist online for this volume I haven't been able to find them. (Perhaps if such solutions...

Homework Statement Hello, I need to calculate the expectation value for position and momentum for a wavefunction that fulfills the following relation: ψ0(-x)=ψ0(x)=ψ*0(x) The wave function is normalised. Homework Equations There is also a second wave equation that is orthogonal...

Homework Statement A particle is under a central potential. Initially its wave function is an eigenfunction ##\psi## such that ##\hat {\vec L ^2} \psi = 2 \hbar ^2## , ##\hat L_3 \psi =0##. Calculate the expectation value of ##\hat {\vec L}## for all times. Homework Equations...

Homework Statement Why can't you do integration-by-parts directly on the middle expression in equation 1.29--pull out the time derivative over onto x, note that \displaystyle \frac{\partial x}{\partial t} = 0, and conclude that \displaystyle \frac{d \langle x \rangle }{dt} = 0Homework Equations...

Homework Statement Find ##\langle L_z \rangle##. What is ##\langle L_Z \rangle## for one atom only? Homework Equations The Attempt at a Solution Using ##L_z = -i\hbar \frac{\partial }{\partial \phi}##, I get: \langle L_z\rangle = \frac{32}{3} \pi k^2 \hbar a_0^3 Not...

Why, in a Gaussian wave function the position and momentum expectation value coincide to be zero? Does it have any physical interpretation? I had an idea that expectation value is the average value over time on that state. But, for Gaussian it tells that it vanishes. Can you please explain.?

When we say expectation value of an operator like the pauli Z=[1 0; 0 -1], like when <Z> = 0.6 what does it mean? What is difference between calculating expectation value of Z and its POVM elements{E1,E2}? thanks

Homework Statement I really do not understand why the expectation value of an observable such as position is <x> = \int\Psi*(x)\Psi Homework Equations If Q is an operator then <Q> = = \int\Psi*(Q)\Psi cn = <f,\Psi> The Attempt at a Solution What I understand this is saying is...

Hi, suppose that the operators $$\hat{A}$$ and $$\hat{B}$$ are Hermitean operators which do not commute corresponding to observables. Suppose further that $$\left|A\right>$$ is an Eigenstate of $$A$$ with eigenvalue a. Therefore, isn't the expectation value of the commutator in the eigenstate...

Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...

Question and symbols: Consider a state|ε> that is in a quantum superposition of two particle-in-a-box energy eigenstates corresponding to n=2,3, i.e.: |ε> = ,[1/(2^.5)][|2> + |3>], or equivalently: ε(x) = [1/(2^.5)][ψ2(x) + ψ3. Compute the expectation value of momentum: <p> = <ε|\widehat{}p|ε>...

Hello, I've been trying to define <p2> in terms of <x2>, much the same way that you can write <p> = m d<x>/dt, because it would be easier in my calculations. Is this possible, or am I on a fools errand? Edit: For Gaussian distributions.

Homework Statement A particle is in a 1D harmonic oscillator potential. Under what conditions will the expectation value of an operator Q (no explicit time dependence) depend on time if (i) the particle is initially in a momentum eigenstate? (ii) the particle is initially in an energy...

This is not a homework problem. Just a curiosity. But my statistics is way rusty. Suppose a binomial probability distribution with probability p for a success. What is the expected number of trials one would have to make to get your first success? In practice, this means if we took a large...

Homework Statement What is the average momentum for a packet corresponding to this normalizable wavefunction? \Psi(x) = C \phi(x) exp(ikx) C is a normalization constant and \phi(x) is a real function. Homework Equations \hat{p}\rightarrow -i\hbar\frac{d}{dx}The Attempt at a Solution...

Homework Statement This is a question I had in my Quantum Mechanics class but my problem is with the calculus which is why i am posting it here. The question is to find the expectation value of x given the wave function equals Ax^3 where 0 ≤ x ≤ a, 0 otherwise. The solution given in class is...

Homework Statement ψ = x3 when 0≤ x ≤a and 0 otherwise find <x> Homework Equations ∫ψ*ψdx=1The Attempt at a Solution So first I multiplied x3 times A, to get Ax3, then plugging that into the equation, I get ∫A2x6dx=1 Then I solve that for A, getting A = \sqrt{\frac{7}{a^{7}}} So I plug that...

when calculating the momentum expectation value the term i(h-bar)d/dx goes inbetween the complex PSI and the 'normal' PSI, so do you differentiate the normal PSI and then multiply by the complex PSI? or do you differentiate the product of the two PSI's i.e. the modulus of PSI? thanks for any...

Homework Statement Given the wave function psi(x,0) = 3/5 sqrt(2/L) sin(xpi/L) + 4/5 sqrt(2/L) sin(5xpi/L) in an infinite potential well from 0 to L, what is the expectation value <x> and rms spread delta E = sqrt(<E^2>-<E>^2) Homework Equations <x> = integral from 0 to L of psi*xpsi dx...

Homework Statement Given the following hypothetic wave function for a particle confined in a region -4≤X≤6: ψ(x)= A(4+x) for -4≤x≤1 A(6-x) for 1≤x≤6 0 otherwise Using the normalized wave function, calculate the...

Is there anyone out there that knows how to define the p operator for a 2-d box. Please can you give a full answer, and not only a hint. I think that no one on this planet knows what it is. I have looked all over the internet. If there is no answer. Why don't people just say it? I think nobody...

Homework Statement Let ##\Psi(x,0)## be the wavefunction at t=0 described by ##\Psi(x,0) = \frac{1}{\sqrt{2}}\left(u_1(x) + u_2(x)\right)##, where the ##u_i## is the ##ith## eigenstate of the Hamiltonian for the 1-D infinite potential well. The energy of the system is measured at some t -...

Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...

if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?

Homework Statement Hey everyone. Imma type this up in Word as usual: http://imageshack.com/a/img577/3654/q9ey.jpg Homework Equations http://imageshack.com/a/img22/3185/pfre.jpg The Attempt at a Solution http://imageshack.com/a/img703/8571/xogb.jpg

Homework Statement A hydrogen like ion (with one electron and a nucleus of charge Ze) is in the state ψ = ψ_{2,0,0} - ψ_{2,1,0} What's the expectation value of \hat{r} (position operator) as a function of Z? Assuming origin at nucleus. Homework Equations for Z=1 < ψ |...

Homework Statement Hey guys! So this is a bit of a long question, I've done most of it but I need a few tips to finish the last part, and I'm not sure if I've done the first one correctly. I'll be typing it up in Word cos Latex is long! http://imageshack.com/a/img5/8335/n7iw.jpg...

<x>= ∫ complex ψ x ψ dx How do we get this formula? And why must the complex ψ must be placed in front? Please guide or any link to help,not really understand this makes me difficult to start in quantum mechanics. Your help is really appreciated. Pls

Hi everyone, I was just working on some problems regarding the mathematical formalism of QM, and while trying to finish a proof, I realized that I am not sure if the following fact is always true: Suppose that we have two linear operators A and B acting over some vector space. Consider a...