Expectation value Definition and 337 Threads

I just have a simple question to get me started. If I am given an initial value wavefunction ψ(x,0) and I am asked to find <P> at t = 0 can I use this: <P> = -ih∫ψ*(x,0)\frac{∂}{∂x}ψ(x,0)dx or do I need to find ψ(x,t) before I find <P>?

Homework Statement Given an observable quantity A, when will it happen that the same value for A will be measured every time? What is the relationship between the operator \hat{A} and \Psi for this case? and What is the relationship between \widehat{A} and \widehat{H}, the...

Homework Statement I am given ψ(x), want to calculate <x^{2}>. Homework Equations \psi(x) = a\exp(ibx-(c/2)(x-d)^2) <x^2> = \int\limits_{-∞}^∞ \psi^*x^2\psi \mathrm{d}x The Attempt at a Solution Well, I normalized the wave function and found a = (\frac{c}{\pi})^{1/4}. So, the...

In a linear superposition, what is the relationship between expectation value of, say, energy and the amplitude coefficients of the eigenfunctions?

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 I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...

I've managed to get myself confused on a seemingly simple point of mathematics. When I calculate the expectation value of momentum in quantum mechanics <p>=\int{\psi* \frac{\hbar}{i} \frac{d}{dx} \psi dx} To what should I be applying the derivative? \psi?

Homework Statement given a certain state |ψ> that is an eigenstate of L^2 and Lz Calculate <Lx^2> and <Lx> Homework Equations L^2|ψ> = l(l+1)h^2 Lz|ψ> = mh|ψ> Lx = YPz - ZPy The Attempt at a Solution <Lx^2> = (1/2)(h^2)[l(l+1)-(m1)^2] for Lx i got <Lx> = ∫ψ(YPz-ZPy)ψ dx

I understand that the the Higgs field has a vacuum expectation value of 246 GeV. I think that means that the expectation of the Hamiltonian energy operator applied to the vacuum state is 246 GeV. What does this imply for the energy density of the Higgs field in the vacuum (i.e. in Joules /...

Hi all! Homework Statement If we consider the hydrogen atom as a spinless particle. Let this system in the state \Psi ( \vec{r} )= \frac{1}{6} [4 \Psi_{100} ( \vec{r} )+ 3 \Psi_{211}- \Psi_{210} ( \vec{r} ) + \sqrt{10}\Psi_{21-1} ( \vec{r} )] Calculate: 1) Expectation value of...

Homework Statement Show that the expectation value of angular momentum <Lx> is zero Homework Equations L±|l,m⟩ = SQRT(l(l+1)−m(m±1)h|l,m±1⟩ L± = Lx ± iLy The Attempt at a Solution I'm supposed to use ladder operators here to show <Lx> is zero. I start with...

Just to check something: If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ? Or can you not split up the operators like <a|A (B|a>) ? Thanks.

Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...

Homework Statement The ground state wave function for a particle of mass m moving with energy E in a one-dimensional harmonic oscillator potential with classical frequency omega is: u(subscript 0) (x)= N(subscript 0) exp((-alpha^2)(x^2)/2) and alpha=sqrt (m *omega/h-bar) where...

Ψ(x)=(2/a)^(1/2) [csin(nπx/a)] The Expectation value of momentum <P>=∫Ψ＊(x)[-ih d/dx ] Ψ(x) dx = 0 the average momentum is zero.It means the particle is moving equally in the +x and -x. And if Ψ(x)=(2/a)^(1/2) {csin(πx/a)+dsin(2πx/a)} I calculate the average momentum is also...

Homework Statement What is the expectation valueof the Sχ for a system in the time-dependent state |Ψ> = 2e-2iωt |z+> -ieiωt |z-> Homework Equations maybe the state must be normalised first i.e 1/√5 times the initial ψ The Attempt at a Solution And then say<ψ|Sχ|ψ> where ψ...

Find the expectation value of (px)2, keeping in mind that ψ0(x) = A0e−ax2 where A0 = (2mω0/h)^1/4, and <x2> = ∫x2|ψ|2dx = h_bar / 2mω0 <ψ(x)|px2|ψ(x)> = ∫ψ(x)(pop2)ψ(x) dx pop = [hbar / i] (\delta/\deltax) I'm not going to attempt to type out me solving the integral because it...

"Expectation value", a few questions I've read that in quantum mechanics we use the term "expectation value" for example for the energy of a system. Despite its name, the expectation value of the energy of for instance the quantum harmonic oscillator is not the most probable measured energy of...

Homework Statement I'm re-hashing a problem from my notes; given the wave function \psi(x)=Ne^{-(x-x_0)/2k^2} Find the expectation value <x>. Homework Equations The normalization constant N for this is in my notes as N^2=1/\sqrt{2 \pi k^2} N=1/(2\pi k^2)^{(1/4)} It should be...

Homework Statement 1. What is <x^{2}>, in terms of position and expectation values. 2. How can I use the correspondence principal to explain the quantum vs classical results (below). My textbook (Serway, Modern Physics) uses <x> as the expectation value, meaning the average position of a...

Homework Statement Show that, if [H,A] = 0 and dA/dt = 0, then <&Delta;A> is constant in time. Homework Equations d<A>/dt = <i/ℏ[H,A] + dA/dt> The Attempt at a Solution I am trying to use the above equation to show that d<&Delta;A>/dt is 0, and I can get to d&Delta;A/dt = 0, but I...

Homework Statement Calculate <x> for the Gaussian wave packet \psi(x)=Ne-(x-x0)/2k2 Homework Equations \left\langle x \right\rangle = \int dx x|\psi(x)|2 The Attempt at a Solution So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm...

The wavefunction of hydrogen is given by \psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta, \phi) If I am only given the radial part, and asked to find the expectation value of the radial part I integrate the square of the wavefunction multiplied by r cubed allowing r to range from 0 to...

Homework Statement The expectation value of the position observable x is <x> = ∫ψ*xψdx. The expectation value of the expectation value, <<x>>=<x>, is still the expectation value...why? Homework Equations The Attempt at a Solution All I can think of is that the expectation value...

1. What is the expectation value, <x>, for the given distribution over the interval from – to + infinity of the function: f(x)=e^(-.5(x-mu)^2(sigma^-2)) 2. This is a statistics problem i think. I just need to know how this type of problem is worked out because it is relevant to my...

Homework Statement If x is the position of a particle then the expectation value of x, <x> is : (I got lazy and just copied an image from Wiki, just pretend <x> is on the lhs of the eqn) When Griffith derives an expression for d<x> / dt, he uses the fact that dx/dt is zero, since "the wave...

Homework Statement Find the expectation value of position as a function of time. Homework Equations This is in the latter half of a multi-part question, previously we were given that: Eqn 1: Ψ(x, t) = A(ψ1(x)e−iE1t/h¯ + iψ2(x)e−iE2t/h¯) and in an even earlier part: Eqn 2: ψn(x) =...

Given some state \left|\psi\right\rangle, and two operators \hat{A} and \hat{B}, how do you prove that if \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}| \psi\rangle then \hat{A} = \hat{B} ?

Given a stationary state H \psi = E \psi \Rightarrow \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right)\psi = E\psi Firstly is it true that \left<p\right> = \frac{\hbar}{i}\int\psi^* \frac{\partial \psi}{\partial x} dx= 0 ?? If it is, how do we prove it?

Homework Statement Kindly take a look at the attachment for the problem statement. Homework Equations Schrodinger Equation: H psi = E psi The Attempt at a Solution For Part A: H psi = E psi S= 1/2 So energy of ground state is -1/2.K.H.hcross Could you tell me if I am on the...

Homework Statement how do you delete a post?

Hello, I want to find <xftf|x(t)|xiti> in harmonic oscillator. I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff. Thank you

Homework Statement given a wavefuntion \Psi = (1/sqrt50) (3\mu1 + 5\mu2 - 4\mu3) what is the expectation value of the total energy? My thoughts were to calculate <\Psi|\hat{}H|\Psi> but the previous part to the question asks for the probability of each outcome(which I know how to...

Homework Statement I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0) \left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0 and \left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0...

Homework Statement Hi Say I have an operator O, and I find its expectation value <O>. Now, if I wish to find the expectation value of O† († denoting Hermitian conjugate), then will this just equal the complex conjugate of <O>? Niles.

Homework Statement Compute the complex conjugate of <p> using eq 1.35 (<p>=∫ψ*(h/i)∂/∂x ψ dx) and prove that <p> is real (<p>=<p>*) Homework Equations equation 1.35 is given above The Attempt at a Solution to take the c.c. don't i just add a minus to the i and switch the stars like...

Homework Statement Hey forum, I copied the problem from a pdf file and uploaded the image: http://img232.imageshack.us/img232/6345/problem4.png What is the probability that the measurement of L^{2} will yield 2\hbar^{2} Homework Equations \left\langle L^{2} \right\rangle = \left\langle \Psi...

Homework Statement Prove that for a particle in a potential V(r) the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque: d/dt <L> = <N> Where N = r x(-del V) N, r, and L are vectors. Homework Equations...

Homework Statement Show: <Jx>=<Jy>=0 Homework Equations Jx=1/2(J++J-) Jy=1/2(J+-J-) The Attempt at a Solution <jm l Jx l jm> = < jm l 1/2 J+ l jm> + < jm l j- l jm > = < jm l h/2 sqrt [(j-m)(j+m+1)] + h/2sqrt[(j+m)(j+m+1) l jm > i am not sure how to apply the next step

Homework Statement The expectation value of the sum of two random variables is given as: \langle x + y \rangle = \langle x \rangle + \langel y \rangle My textbook provides the following derivation of this relationship. Suppose that we have two random variables, x and y. Let p_{ij}...

My understanding was that the expectation value of an observable H for a state |a> is just <a|H|a>. But in a homework problem, my prof. used <H> = <a|H|a>/<a|a>. I'm a little confused by the discrepancy, why the discrepancy?

Homework Statement Evaluate <x^2> for the wave function \psi(x)=\int_{-\infty}^{\infty}dk exp(-|k|/k_0)exp(ikx) My calculation yields a negative answer and I can't find my error Homework Equations...

Homework Statement |O> = k |R1> + 1/9 |R2> a) Find k if |O> has already been normalized, and b) then the expectation value. The Attempt at a Solution a) To Normalise: |(|O>)|2 = (1/9 |R2> + k |R1>).(1/9 |R2> - k|R1>) = 1/81|R2>2 - k2|R1>2 = 1 I just assumed that |k| = (1-(1/81))0.5, but...

Homework Statement Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix). Homework Equations I think the only one relevant is the expectation value: <a> = integral[psi*(a)psi] / integral[psi*psi] where psi* is...

Hi, I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable...

Homework Statement Psi(x) = Ax -a<x<a I am trying to find the probability that my measured momentum is between h/a and 2h/a Homework Equations I have normalized A= sqrt(3/(2a^3)) I know that if I was finding the expected momentum I would use \int\Psi * p \Psi dx The...

Homework Statement Show that the expectation value of the Coulomb potential v(\vec{r_1},\vec{r_2})=\frac{e^2}{|\vec{r_1}-\vec{r_2}|}, between two electrons depends on the relative orientation of spin of the two electrons. Assume each electron is in the product state form...

Is there any way of proving <p> = 0 for a discrete (bound) state given it's wave function? I've seen proofs using the hermitian properties of p but I'm interested in proving that the integral of Psi*(x) Psi'(x) is identically zero regardless of Psi(x) as long as it's a solution of Schroedinger's...

hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...

If I have H=p^2/2m+V(x), |a'> are energy eigenkets with eigenvalue E_{a'}, isn't the expectation value of [H,x] wrt |a'> not always 0? Don't I have that <a'|[H,x]|a'> = <a'|(Hx-xH)|a'> = <a'|Hx|a'> - <a'|xH|a'> = 0 ? But if I calculate the commutator, I get: <a'|[H,x]|a'> = <a'|-i p \hbar /...