In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.
Hi,
In Birrel and Davies ch4 they write:
\langle \psi|:T_{ab}:|\psi \rangle =\langle \psi|T_{ab}|\psi \rangle -\langle 0|T_{ab}|0 \rangle
this is for the usual Mink field modes and vac state. Why does normal ordering reduce to this expression, could anybody point me the way to...
Homework Statement
In my homework assignment I have a wavefunction defined as \Psi(x)=N\exp(-|x|/a) and I am asked to find the expectation value of momentum squared in configuration space.
Homework Equations
\int\Psi*(x)\hat{p^2}\Psi(x)dx
The Attempt at a Solution
N is 1/\sqrt{a}...
Homework Statement
Consider an observable A associated to an operator A with eigenvalues an.
Using the formula <A> = ∫ψ*Aψ compute the expectation value of A for the following wave function:
\Psi=\frac{1}{\sqrt{3}}\phi_{1}+\frac{1}{\sqrt{6}}\phi_{2}+\frac{1}{\sqrt{2}}\phi_{3}
where...
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
An electron starts in a spin state |\psi(t=0)\rangle = |z \uparrow \rangle and evolves in a magnetic field B_0(\hat{x} + \hat{z}). The Hamiltonian of the system is \hat{H} = \alpha \vec{B}\cdot\vec{S}. Evaluate \langle \psi (t_{1/2}) | S_x | \psi(t_{1/2}) \rangle...
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...
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
1. For the ground-state of the 2D rigid rotor what is the expectation value of the angular momentum? And what is the corresponding uncertainty in this value?
2. Describe in words what the uncertainty in position is.
3. Explain why the rigid rotor can have vanishing...
Homework Statement
Show that for the expectation values the following relations hold: d \langle x \rangle /dt =\langle p \rangle /m and d \langle p \rangle /dt = - \langle d V/dx \rangle.
Homework Equations
\langle x \rangle = \int _{- \infty}^{\infty} \Psi ^* x \Psi dx.
The Attempt...
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 <ΔA> 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<ΔA>/dt is 0, and I can get to dΔA/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...
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
the origial question given is: Show that the difference in energies between 2s and 2p radial wavefunctions is equal to the energy of the angular part of the 2p wavefunction, and thus that they have the same overall energy.
hints given:a)use virial theorem to determine...
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...