What is Expectation values: Definition and 122 Discussions
In probability theory, the expected value of a random variable
X
{\displaystyle X}
, denoted
E
(
X
)
{\displaystyle \operatorname {E} (X)}
or
E
[
X
]
{\displaystyle \operatorname {E} [X]}
, is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of
X
{\displaystyle X}
. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.
By definition, the expected value of a constant random variable
X
=
c
{\displaystyle X=c}
is
c
{\displaystyle c}
. The expected value of a random variable
X
{\displaystyle X}
with equiprobable outcomes
{
c
1
,
…
,
c
n
}
{\displaystyle \{c_{1},\ldots ,c_{n}\}}
is defined as the arithmetic mean of the terms
c
i
.
{\displaystyle c_{i}.}
If some of the probabilities
Pr
(
X
=
c
i
)
{\displaystyle \Pr \,(X=c_{i})}
of an individual outcome
c
i
{\displaystyle c_{i}}
are unequal, then the expected value is defined to be the probability-weighted average of the
c
i
{\displaystyle c_{i}}
s, that is, the sum of the
n
{\displaystyle n}
products
c
i
⋅
Pr
(
X
=
c
i
)
{\displaystyle c_{i}\cdot \Pr \,(X=c_{i})}
. The expected value of a general random variable involves integration in the sense of Lebesgue.
I am not able to use Latex for some reason. It is very glitchy and if I do one backspace then it fills my whole screen with multiple copies of the same equation. Thus I am pasting a screenshot of handwritten equations instead. Apologies for any inconvenience.
In Introduction to Quantum...
I've make progress in obtaining the values for the mutual information using the following:
$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$
I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since...
Apart from the usual integral method, are there any other ways to find expectation value of momentum? I know one way is by using ehrenfest theorem, relating it time derivative of expectation value of position operator.
Even using the uncertainty principle, we might get it if we know the...
I first normalized the given wavefunction and found the value of n that satisfies the normalization condition. I then used E = <E> = pi^2* h_bar^2* n^2/(2*m) to get the expectation value of energy. Assuming that this was the right process, I'm now trying to find <E^2> using the same equation...
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...
I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?
I've been studying quantum mechanics, and working problems to get a feel for expectation values and what causes them to be real.
I was working the problem of finite 1D wells, when I came across a situation I did not understand.
A stationary state solution is made up of a forward and reverse...
The expectation value of any function ##f(x)## is given by <f(x)>= \int_{-\infty}^{\infty}f(x)\psi^2(x) dx
But what is ##f(x)## actually? In a physical sense.
For example if ##f(x)=x## or ##f(x)=x^2##, what do these functions represent on a physical level?
Homework Statement
Can someone explain to me what particles (fermions, scalar/vector bosons, gravitons, ...) can have their vacuum expectation values and why? Which components of these fields can have VEV-s?
The Attempt at a Solution
I am assuming only scalar boson fields have it (like Higgs...
Hi!
I want to know under what conditions the operator expectation values of a product of operators can be expressed as a product of their individual expectation values. Specifically, under what conditions does the following relation hold for quantum operators (For my specific purpose, these are...
Working on a homework at the moment involving spinors. The algebra isn't hard at all, I just want to make sure my understanding is right and I'm not doing this incorrectly.
1. Homework Statement
An electron in a one-dimensional infinite well in the region 0≤x≤a is described by the spinor ψ(x)...
Homework Statement
A particle moving in a periodic potential has one-dimensional dynamics according to a Hamiltonian ## \hat H = \hat p_x^2/2m+V_0(1-cos(\hat x))##
a) Express ## \frac{d <\hat x>}{dt}## in terms of ##<\hat p_x>##.
b) Express ## \frac{d <\hat p_x>}{dt}## in terms of ##<sin(\hat...
Homework Statement
A few questions:
Q1) How does 1.29 flow to 1.30 and 1.31? How was the integral-by-parts done?
Q2) The author states that <v> = d<x>/dt represents the expectation value of velocity. What does this actually mean? I tried to rationalise that d<x>/dt represented the velocity...
Hi.
I came across the following in the solution to a question I was looking , regarding expectation values of momentum in 3-D
< p12p22p32 > = < p12 > < p22 > <p32 >
ie. the expectation value has been factorised. I can't figure out why this is true and also why it doesn't apply to the following...
Hi. I'm trying to prove that
[\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p)
where
\rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar})
is the Wigner function, being \rho a density matrix. On the other hand...
Homework Statement
I am trying to calculate the expectation value of ##\hat{P}^3## for the harmonic oscillator in energy eigenstate ##|n\rangle##
Homework EquationsThe Attempt at a Solution
[/B]
##\hat{P}^3 = (i \sqrt{\frac{\hbar \omega m}{2}} (\hat{a}^\dagger - \hat{a}))^3 = -i(\frac{\hbar...
Homework Statement
Write down a spinor that represents the spin state of the particle at any time t > 0. Use the expression to find the expectation values of ##S_x## and ##S_y##
Homework Equations
The particle is a spin-##\frac 1 2## particle, the gyromagnetic ratio is ##\gamma_s \lt 0##, and...
Homework Statement
See Image, Sorry Its easier for me to attach images than writing all equation on the forum's keyboard! I only need to check if I'm working it out correctly up to the position expectation value because I don't want to dive in the rest on wrong basis !
Homework Equations...
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 into...
hello :-)
here is my problem...:
1. Homework Statement
For a linear harmonic oscillator, \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2x^2
a) show that the expectation values for position, \bar{x}, and momentum \bar{p} oscillate around zero with angular frequency \omega. Hint...
I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...
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...
Homework Statement
Given the following wave function valid over -a \le x \le a and which is 0 elsewhere,
\psi(x) = 1/\sqrt{2a}
Find the uncertainty in \left<\left(\Delta p\right)^2\right> momentum, and the uncertainty product \left<\left(\Delta x\right)^2\right>\left<\left(\Delta...
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 -...
An animation of the CHSH experiment to generate correlated photons is at: http://www.animatedphysics.com/games/photon_longdistance_chsh.htm
@georgir has a program to show the calculations using the formula for photon detection
return Math.random() < (Math.cos(r(p-a)*2)+1)/2;
yields the...
Homework Statement
Homework Equations
$$ \psi_{100} = \frac {1}{\sqrt{\pi a^{3}}} e^{-r/a} $$
The Attempt at a Solution
a)
$$\langle r \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{3} e^{-2r/a} dr$$
This comes out to be ##\frac {3}{2}a##...
Why does the expectation values of some operators, such as 'number' operator ##a^\dagger a## and atomic population operator ##\sigma^\dagger\sigma##, are always nonnegative? Can we prove this from a mathematical point? For example, are these operators positive semidefinite?
If I have the following expectation value for a general operator A < psi | cA | psi > where c is a complex constant and I want to take c outside the bracket does it go as c or its complex conjugate ?
If you have some wave function of some particle, say...
|¥>
And you calculate the expectation value of momentum, say...
<¥|p|¥>
What ensures that that spatial integral is real valued?
Separately, all the components of the integral are complex valued
Suppose there's an object within a sphere of radius 5-metres from a given point P=(x_0,y_0,z_0). The probabilities of the object being within 0-1, 1-2, 2-3, 3-4 and 4-5 metres of P are given to be respectively p_1,p_2,p_3,p_4 and p_5. With this information, is it possible to find the expected...
Homework Statement
The position-space representation of the radial component of the momentum operator is given by
## p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right ) ##
Show that for its expectation value to be real:## \left \langle \psi|p_r|\psi \right...
Homework Statement
The question is as stated:
"The ##H_2## molecule has oscillatory excitations. In classical physics the energy can be approximated to \begin{equation} E = \frac{p^2}{2m} + \frac{m \omega^2 x^2}{2} \end{equation}where m is the reduced mass. Quantum mechanics can be applied to...
Homework Statement
Find the expectation values of x and p for the state
\vert \alpha \rangle = e^{-\frac{1}{2}\vert\alpha\vert^2}exp(\alpha a^{\dagger})\vert 0 \rangle, where ##a## is the destruction operator.
Homework Equations
Destruction and creation operators
##a=Ax+Bp##...
I am reading an intriguing article on rigged Hilbert space
http://arxiv.org/abs/quant-ph/0502053
On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator A, corresponding to some observable in space of square integrable functions...
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} =...
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...
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...
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...
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⟩...
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}...
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...
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...
I am slightly confused on how do we calculate vacuum expectation values of product of creation and annihilation operators for bosons, e.g. ##\langle 0| a_{k_1} a^\dagger_{k_2} a_{k_3} a^\dagger_{k_4} |0 \rangle##
If i commute ##k_3## and ##k_4##:
$$\langle 0| a_{k_1} a^\dagger_{k_2}...
I'm stuck on a question in atkins molecular quantum mechanics 4e (self test 1.9).
If (Af)* = -Af, show that <A> = 0 for any real function f.
I think you are expected to use the completeness relation sum,s { |s><s| = 1.
I'm sure the answer is simple but I'm stumped.
Ok, so I'm a little confused about why <p> = 0 for Hydrogen in the ground state. If someone explain the reasoning behind this, I'd greatly appreciate it.
Also, and more importantly, does that mean that <p> = 0 for Hydrogen in other states as well? If not, how would you go about finding <p>...
Consider a quantum system with angular momentum 1, in a state represented by the vector
\Psi=\frac{1}{\sqrt{26}}[1, 4, -3]
Find the expectation values <L_{z}> and <L_{x}>
I'm reviewing my quantum mechanics; I had a pretty horrible course on it during undergrad. I feel like this should be...
Homework Statement
Homework Equations
The Attempt at a Solution
I have totally no idea how to solve this question. But I find it somehow similar to the Larmor precession problem. Therefore I try to solve my problem by referring to that.
Are there any mistakes if I do it like...