Expectation values Definition and 118 Threads

Hi, Please refer to this book (in google archive), and go to section 7.7 (page 85)...

Homework Statement Show that < l,m | Lx2 - Ly2 | l,m > = 0 Homework Equations L2 = Lx2 + Ly2 + Lz2 [ Lx, Ly ] = i h Lz [ L, Lz ] = i h Lx [ Lz, Lx ] = i h Ly The Attempt at a Solution I tried substituting different commutation values in place of Lx and Ly, but I'm...

Homework Statement The probablity density function of the n-state of an electron is proportional to fn=(\frac{rz}{a_{0}})^{2n}e^ \frac{-2Zr}{\large na_{0}} show that the expectation value of the potential energy of the electron in the n-th quantum state of the hydrogen atoms is...

Homework Statement Here's a link to an image of the exam question. It appears in the exam every couple of years, and it's due in my exam this coming week. I've looked in both the textbook and the course notes, and they simply *state* the conclusion, so I don't have a way of proving it, and...

I'm a bit confused about the nature of probability conservation and expectation values. According to probability conservation, \frac{∂P(r,t)}{∂t}=0. Does that mean that expectation values e.g. <x>, <p> and <E> depend only on the position of the particle and not on time? Thanks

I know that the formula for the expectation value is: <Q(x,p)> = ∫ψ*Q(x,(h/i)d/dx)ψ dx For instance, the expectation value for momentum is. -ih∫ψ*(dψ/dx)dx But, why? How is it derived?

Is it possible to express ANY observable A(X,P) in terms of the ladder operators? I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be...

So I'm a little confused on the notation when working with wave functions constructed as a linear combination of an orthornormal basis set. Like on the form: \Phi=Ʃn cnψn If I want to find the expectation value represented by the operator O for the state described by \Phi, I would...

Homework Statement A wave function ψ is A(eix+e-ix) in the region -π<x<π and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x=0 and x=π/8, and (b) between x=0 and x=π/4. Homework Equations The Attempt at a Solution So to...

Hello everybody, I'm looking for a proof of the following equation: <x6> = <x>6+15s2<x>4 where the brackets denote an expectationvalue and s is the standard deviation. Thanks in advance!

Homework Statement Determine for the hydrogen atom states 1s and 2p the expectation value of the radius r and the associated mean square error Δr. Homework Equations Wave Functions for 1s and 2p from Demtroeder's Experimental Physics Volume 3 (it says "The normalized complete...

Homework Statement For the Harmonic Oscillator, the state |ψ> = (|0> + |1>) / √(2) Find \overline{x} = <ψ|x|ψ> \overline{p} = <ψ|p|ψ> \overline{x^2} = <ψ|x^{2}|ψ> and \overline{p^2} = <ψ|p^{2}|ψ> and <ψ| (x - \overline{x})^2 |ψ><ψ| (p - \overline{p})^2 |ψ> [b]2. Homework Equations...

Homework Statement Homework Equations I have that p(\theta)d\theta = \frac{1}{\pi}d\theta, this is definitely correct. Also y=r*sin(theta) so dy=r*cos(theta)*d(theta). Substituting d(theta) in above and simplifying, I have: p(y)dy = \frac{1}{pi} \frac{dy}{\sqrt{r^2-y^2}} The problem is...

I have a question about expectation values in quantum mechanics. When calculating <x>=\int\Psi*x\Psi does x always make this functions odd? If \Psi is odd then \Psi* I would assume is odd as well and then <x> would be odd*odd*odd, if \Psi is even then I again assume it would be even*odd*even...

I have a question about expectation values in quantum mechanics. Since calculating <x>=\int\Psi*x\Psi does x always make this functions odd? If \Psi is odd then \Psi* I would assume is odd as well and then <x> would be odd*odd*odd, if \Psi is even then I again assume it would be...

Homework Statement Calculate the expectation values of p and p2 for a particle in state n=2 in a square well potential. Homework Equations \Psi(x,y) = (2/L)*sin(n1\pix/L)*sin(n2\piy/L) p= -i\hbar\partial/\partialx The Attempt at a Solution \int\Psip\Psidxdy...

Can anyone explain to me why the only time that the expectation of L^2 operator and the expectation value of L_3^2 are equal only when there is no angular dependence? And what does this mean? Does this have something to do with being restricted to the z-axis which is what L_3 is associated...

Hi. First post. I'm trying to understand if electronic energy levels have fixed values, or merely fixed expectation values (in the latter case, orbital electrons could have any energy and it's only the average that would be fixed). Here's my argument for the latter. If it's incorrect, could...

Hi, I've found the expectation value of Sz, which is hbar/2 (|\psiup|2 - |\psidown|2) by using the formula: <Si> = <\psi|Si\psi> where i can bex, y or z and \psi is the 'spinor' vector. I tried to find Sx using the same formula, however, I could only get as far as: hbar/2 ((\psiup)*\psidown...

Homework Statement A particle of mass m that is confined to a harmonic oscillator potential V(x) = \frac{1}{2} m \omega^2 x^2 is described by a wave packet having the probability density, |\Psi (x,t) |^2 = \left(\frac{m\omega}{\pi\hbar} \right )^{1/2}\textrm{exp}\left[-\frac{mw}{\hbar}(x -...

Homework Statement A particle is represented(at t=0) by the wavefunction u(x,0) = A(a^2 - x^2) if -a<x<a = 0 otherwise Determine <x> & <p>. It is given in the book that in this case <p> \neq m*d/dt<x>. Could someone please tell me the reason...

Homework Statement Hi Say I have the following number: \left\langle {\psi _i |A|\psi _j } \right\rangle 1) First of all, am I correct when saying that \left\langle {\psi _i |A|\psi _j } \right\rangle = \left\langle {\psi _j |A^\dag |\psi _i } \right\rangle ^* where...

Homework Statement (a) Let Q be an operator which is not a function of time, abd Let H be the Hamiltonian operator. Show that: i(hbar)(\delta<q> / dt =<[Q,H]> Here <q> is the expectation value of Q for any arbirtary time-dependent wave function Psi, which is not necessarily an...

In quantum mechanics, when is this true \langle\psi|AB|\psi\rangle=\langle\psi | A|\psi\rangle\langle\psi |B|\psi\rangle ? In probability theory, when the two variables are independent, the mean value of the product is the product of the mean values. What about QM?

Show that the expectation value of Lz is -2h for the radial wavefunction Y2,-2. ? Can someone do this?

I have small question computing vacuum expectation values here http://www.cns.gatech.edu/FieldTheory/extras/SrednickiQFT03.pdf" from Mark Srednicki. My problem is with equation 210 on the pdf page 69. In the second line of 210, where does the second term come from? Z(J) and W(J) are defined...

Homework Statement I need to find <x>, <x2>, <p>, and <p2> for a particle in the first state of a harmonic oscillator. Homework Equations The harmonic oscillator in the first state is described by \psi(x)=A\alpha1/2*x*e-\alpha*x2/2. I'm using the definition <Q>=(\int\psi1*Q*\psi)dx...

I'm looking at a question... The last part is this: find the expectation values of energy at t=0 The function that describes the particle of mass m is A.SUM[(1/sqrt2)^n].\varphi_n where I've found A to be 1/sqrt2. The energy eigenstates are \varphi_n with eigenvalue E_n=(n + 1/2)hw...

I've never seen an expectation value taken and would greatly appreciate seeing a step by step of how it is done. Feel free to use any wavefunction, this is the one I've been trying to do: In the case of \Psi=c1\Psi1 + c2\Psi2 + ... + cn\Psin And the operator A(hat) => A(hat)\Psi1 =...

Homework Statement i. Confirming the wavefunction is normalised ii. Calculating the expectation values: <\hat{x}> , <\hat{x^{2}}> , <\hat{p}> , <\hat{p^{2}}> as a function of \sigma iii. Interpreting the results in regards to Heisenberg's uncertainty relation. Homework Equations...

Hello, Can someone explain to me how the expectation values are calculated in the following picture: I mean , What did they do after the brackets? What did they multiply with what? thanks

Homework Statement Using the fact that ,\left\langle \hat{L}_{x}^{2} \right\rangle = \left\langle \hat{L}_{y}^{2} \right\rangle show that \left\langle \hat{L}_{x}^{2} \right\rangle = 1/2 \hbar^{2}(l(l+1)-m^{2}. The Attempt at a Solution L^{2} \left|l,m\right\rangle = \hbar^{2}l(l+1)...

Homework Statement Express Lx in terms of the commutator of Ly and Lz and, using this result, show that <Lx>=0 for this particle. The Attempt at a Solution [Ly,Lz]=i(hbar)Lx <Lx>=< l,m l Lx l l,m> then what?

Homework Statement Evaluate the expectation value of p and p² using the momentum-space wave function Homework Equations Momentum-space wave function: \sqrt{\frac{d}{\hbar\sqrt{\pi}}}e^{\frac{-\left(p'-\hbar k\right)^2d^2}{2\hbar^2}} The Attempt at a Solution I can get \langle...

Homework Statement Consider a particle in an infinite one-dimensional box that has a length L and is centered at the origin. (Use h for Planck's constant, n, and L, as necessary.) Evaluate <x^2> for <x^2> at n=1. Homework Equations <x^2>= (2/L)[(L^3/24)-(L^3/4n^2*pi^2)cos(n*pi)] The...

I'm trying to check that the expectation value <E> is E for the wavefunction sqrt(2/L) sin(2pix / L) I know the shortcut way of doing it by saying that the hamiltonian multiplied by the function is just the eigenvalue E multiplied by the function, and since the function is normalized the...

I want to find <x> and ,<x^2>, <p>, and <p^2> of a particle in an infinite well where: V(x)=0, \frac{-a}{4}<x<\frac{3a}{4} Using the usual method, I found the wavefunction to be: \psi(x)=\sqrt{\frac{2}{a}}sin[\frac{n\pi}{a}(x+\frac{a}{4})] I also found...

Just a quick question. I finished an expectation value sum and noticed that the given solution had me stumped. Ive attached a quick picture of the simplifying process which was given as the solution. The only thing i don't understand is how to get the value iCm/(pi*hbar)^1/2. I don't know...

Homework Statement The variance of an observable Qhat in a state with wavefunction psi is, (delta Qhat)2=<(Qhat-<Qhat>)2> Show that this can be written as, (delta Qhat)2=<Qhat2>-<Qhat>2 Homework Equations As above. The Attempt at a Solution (delta...

Homework Statement calculate <x>, when \Psi(x,t)=A*exp(-(\sqrt{}Cm/2h)x^{}2 Homework Equations <x>=\int\Psi^{}*x\Psidx over all space.. \intexp(-\alphax^{}2)=\sqrt{}\pi/\alpha The Attempt at a Solution ok know how to do this but how do i do the intergral... my maths isn't so good...

Hi all. I have a question which arose from the answer of a homework problem. A particle is in the state given by \left| \psi \right\rangle = \frac{1}{{\sqrt 3 }}\left[ {\left| \psi \right\rangle _1 + \left| \psi \right\rangle _2 + \left| \psi \right\rangle _3 } \right], where {\left|...

Is the expectation value of momentum/position/energy the value that we're most likely to measure? So suppose we measure 100 particles with the same wavefunction, would we expect most of them to have momentum/position/energy that's equal to the expectation value? And I was wondering, how do we...

Homework Statement The expectation value <r> of the electron-nucleus separation distance 'r' is: <r> = ʃ r |ψ|² dV. (a) Determine <r> for the 1, 0, 0 state of hydrogen. The Attempt at a Solution Right, I've obtained the value for ψ = (1/πa³)^1/2 exp(-r/a) I then...

Homework Statement I need to find the expectation value for E but I don't know how b acts on the vacuum state. Homework Equations b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t) | \psi(t)\rangle = b^\dagger| 0\rangle The Attempt at a Solution \langle \psi(t) |...

Quantum homework - Average Expectation Values?? Hi people, I'm struggling with my quantum mechanics homework - I've included links to photographs of my attempts at solutions, but i know they are wrong because I am given what the answers are supposed to be. Can somebody help me spot where I am...

I seem to be having a rather difficult time understand all the details of the notation used in this quantum material. If I'm given the eigenvalue equations for L^{2} and L_z and L_{\stackrel{+}{-}} for the state |\ell,m>, how do I compute <L_{x}> using bra-ket formalism? I know that L_x =...

Homework Statement Hi all. The expectation value for S_x (spin in x-direction) is: \left\langle {S_x } \right\rangle = \left\langle {\phi |S_x \phi } \right\rangle = \phi ^\dag S_x \phi where \phi is the state and \phi^"sword" is the hermite conjugate. My question is: I...

I am having some great difficulty getting intuition out of the standard quantization of the Klein-Gordon Lagrangian. consider the H operator. In QM, the expectation values for H in any eigenstates |n> is just <n|H|n> now, in QFT, suppose I have a state |p> in the universe, what do I get if I...

Homework Statement Hi all. Let's say that i have a wave function \Psi (x,t) = A \cdot \exp ( - \lambda \cdot \left| x \right|) \cdot \exp ( - i\omega t) I want to find the expectation value for x. For this I use \left\langle x \right\rangle = \int_{ - \infty }^\infty x \left| \Psi...

The state \Psi = \frac{1}{\sqrt{6}}\Psi-1 + \frac{1}{\sqrt{2}}\Psi1 + \frac{1}{\sqrt{3}}\Psi2 is a linear combination of three orthonormal eigenstates of the operator Ô corresponding to eigenvalues -1, 1, and 2. What is the expectation value of Ô for this state? (A) 2/3 (B)...