Expectation Definition and 654 Threads

Consider a family of densitites $f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}}$. Let $X_{1}$ be a single observation from this family. I have shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator. Now consider $n$ observations $X_{1},..X_{n}$. I have shown that...

consider a density family $f(x,\theta)=\frac{exp(-{\sqrt{x}}/{\theta})}{2{\theta}^2}$. Let $X_{1}$ have the density above. Compute $E(X_{1}^\frac{1}{2})$. Integration by parts doesn't work since the derivative of ${\sqrt{x}}$ never vanishes, so how do I compute the expectation?

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>...

Hi, I'd like to know if the following statement is true: Let \hat{A}, \hat{B} be operators for any two observables A, B. Then \langle \hat{A} \rangle_{\psi} = \langle \hat{B} \rangle_{\psi} \forall \psi implies \hat{A} = \hat{B} . Here, \langle \hat{A} \rangle_{\psi} =...

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 \Psi (x) = C e^{i k_{0} x} e^{\frac{-x^{2}}{2 a^{2}}} Find \left\langle x \right\rangle, \left\langle x^{2} \right\rangle, \left\langle p \right\rangle, \left\langle p^{2} \right\rangle.Homework Equations Operators make a "psi-sandwich": \left\langle x \right\rangle =...

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 A coin is flipped repeatedly with probability p of landing on heads each flip. Calculate the average <n> and the variance \sigma^2 = <n^2> - <n>^2 of the attempt n at which heads appears for the first time. Homework Equations \sigma^2 = <n^2> - <n>^2 The...

Hi, i was wondering if the following is valid: E[x/y] = E[x] / E[y], given that {x,y} are non-negative and independent random variables and E[.] stands for the expectation operator. Thanks

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 The vector \psi =\psi_{n} is a normalized eigenvector for the energy level E=E_{n}=(n+\frac{1}{2})\hbar\omega of the harmonic oscillator with Hamiltonian H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}. Show that...

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...

Hello, I 'm trying to express the following in integral form: E[a/X-b/Y], where E[.] stands for the expectation operator. Let a,b be some nonnegative constants and X,Y are independent nonnegative Gamma distributed random variables. Any help would be useful. Thanks in advance

Assume we have a number ##S_0##. For ##i=1..n## define$$S_i=\begin{cases}(1+b)S_{i-1}\text{ with probability }p\\(1+a)S_{i-1}\text{ with probability }1-p\end{cases}$$. What is the expected value of ##S_n##?

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 Hey guys, so here's the question: The energy eigenstates of the hydrogen atom \psi_{n,l,m} are orthonormal and labeled by three quantum numbers: the principle quantum number n and the orbital angular momentum eigenvalues l and m. Consider the state of a hydrogen atom at t=0...

I'm having trouble working out a few details from my probability book. It says if P(An) goes to zero, then the integral of X over An goes to zero as well. My book says its because of the monotone convergence theorem, but this confuses me because I thought that has to do with Xn converging to X...

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 -...

Homework Statement What is the expectation value of \hat{S}_{x} with respect to the state \chi = \begin{pmatrix} 1\\ 0 \end{pmatrix}? \hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}Homework Equations <\hat{S}_{x}> = ∫^{\infty}_{-\infty}(\chi^{T})^{*}\hat{S}_{x}\chi...

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...

What is the expectation, E(log(x-a)), when x is log normally distributed? Also x-a>0. I am looking for analytical solution or good numerical approximation. Thanks

Homework Statement sup guys! I think I've solved this set of problems, but I was just wondering if I've done it right - I have no way to tell. I'll put all the questions and answers here - plus the stuff I used. So could you please tell me if there's any mistakes? Here it is - using Word...

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 |...

How do I read, interpret the following definitions for the expectation of a random variable X? Assume the integral is over the entire relevant space for X. (1) E(X) = ∫ x dP (2) E(X) = ∫ x dF(X) If I asked you to calculate (1) or (2) for an arbitrary X, how does it look? My only other...

Homework Statement Calculate <i(\hat{a} - \hat{a^{t}})> Homework Equations |\psi > = e^{-\alpha ^{2}/2} \sum \frac{(\alpha e^{i\phi })^n}{\sqrt{n!}} |n> \hat{a}|n> = \sqrt{n}|n-1> I derived: \hat{a}|\psi> = (\alpha e^{i\phi})^{-1}|\psi> The Attempt at a Solution...

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

Homework Statement A one-dimension system is in a state described by the normalisable wave function Ψ(x,t) i.e. Ψ → 0 for x → ±∞. (a) Show that the expectation value of the position ⟨x⟩ is a real quantity. [1] (b) Show that the expectation value of the momentum in the x-direction ⟨p⟩...

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...

(please refer to attached image) The question appears to be simple enough, but i have two queries A) does E[X1 X2] mean the same as E[X1 | X2] B) If not/so, how exactly do I go about computing this. I've seen a few formulas in my lectures notes for computing conditional expectations for...

Homework Statement f(x,y)=6a^{-5}xy^{2} 0≤x≤a and 0≤y≤a, 0 elsewhere Show that \overline{xy}=\overline{x}.\overline{y} Homework Equations \overline{x}=\int^{∞}_{-∞}{x.f(x)dx} The Attempt at a Solution \overline{x}=\int^{∞}_{-∞}{x.f(x)dx} =\int^{a}_{0}{x.6a^{-5}xy^{2}dx}...

Homework Statement The expectation value of the time derivative of an arbitrary quantum operator \hat{O} is given by the expression: d\langle\hat{O}\rangle/dt\equiv\langled\hat{O}/dt\rangle=\langle∂\hat{O}/∂t\rangle+i/hbar\langle[\hat{H},\hat{O}]\rangle Obtain an expression for...

We have for two random variables X and Y (one sum runs over j and one over k): E(X+Y) = ƩƩ(xj+yk)P(X=xk,Y=yk) = ƩƩxjP(X=xk,Y=yk) + ƩƩykP(X=xk,Y=yk) Now this can be simplified to obtain E(X+Y)=E(X)+E(Y) if we use that: P(X=xk,Y=yk) = P(X=xk)P(Y=yk), because then (and same goes the other...

Here is a proof question: For two random variables X and Y, we can define E(X|Y) to be the function of Y that satisfies E(Xg(X)) = E(E(X|Y)g(Y)) for any function g. Using this definition show that E(X1 + X2|Y) = E(X1|Y) + E(X2|Y) So what I did was I plugged into X = X1 + X2 E(E(X1 +...

The below gives all the information I was given. I'm pretty sure my answer is right, but a part of me isn't, and that's why I'm asking here. Homework Statement Let (X,Y) have the joint pdf: f_{XY}(x,y)=e^{-y}, 0 < x < y < \infty Find E(XY). Homework Equations...

If X is a continuous random variable and E(X) exists, does the limit as x→∞ of x[1 - F(x)] = 0? I encountered this, but so far I have neither been able to prove this, nor find a counterexample. I have tried the mathematical definition of the limit, l'Hopital's rule, integration by parts, a...

why we find expectation value? why do we take expectation value of sime entity? and how come we know that our expected value is correct

Hello, first post here. I am preparing for my Introductory Quantum Mechanics course, and in the exam questions, we are asked to use Ehrenfest's theorem to show that \frac{d}{dt}\langle \vec{r}\cdot \vec{p} \rangle = \langle 2T-\vec{r}\cdot \nabla V \rangle Now, from other results...

Homework Statement In a QM problem I must calculate the expectation value of the kinetic energy, namely ##\langle \Psi _c ,\frac{ \hat { \vec p } ^2}{2m} \Psi _c \rangle##. Where ##\Psi _c=ce^{-\alpha x^2}##. Homework Equations ##\int _{-\infty}^\infty \exp \{ -a x^2 +bx \} dx =...

I have found what I think is the correct answer I just want to check an assumption. The magnetic field points in the +ve z-direction. We are given the initial state vector \left| A \right\rangle_{initial}=\frac{1}{5}\left[ \begin{array}{c}3\\4\end{array} \right] Am I right in thinking that...