{\displaystyle \scriptstyle {\vec {\text{ F }}}}
is the force, F is a vector valued force function, F is a scalar valued force function, r is the position vector, ||r|| is its length, and
r
^
{\displaystyle \scriptstyle {\hat {\mathbf {r} }}}
= r/||r|| is the corresponding unit vector.
Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.
In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...
I have one problem with this question that I've been struggling with. Initially, the total energy should be given by E =m1* v0^2/2 (as U goes to zero, and m2 is at rest). However, if we write r = r1 - r2, we get E = mu*rdot^2/2 + U_eff(r), U_eff(r) also goes to 0, where mu is the reduced mass...
Homework Statement
Two particles move in a central potential. The potential has the form V(r1, r2)=-a/(/r1-r2/^1.5)...
Homework EquationsThe Attempt at a Solution
I am having trouble understanding what it means for two particles to move in a central potential. From what I understand a central...
1. Homework Statement
A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##.
How is the general form for the eigenfunctions?
What is the boundary conditions for this eigenfunctions?
Find the degeneracy level for the energy, when it is ##E<V_0##
2...
Homework Statement
A particle of mass m and spin s, it's subject at next central potential:
##
\begin{equation*}
V(\mathbf{r})=
\begin{cases}
0\text{ r<a}\\
V_0\text{ a<r<b}\\
0\text{ r>b}
\end{cases}
\end{equation*}
##
Find the constants of motion of the system and the set of...
Homework Statement
Here is a copy of the pdf problem set {https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU} the problem in question is problem number 1 which asks you to prove the orthonormality of the spherical Harmonics Y_1,1 and Y_2,1.
Homework Equations
Y_1,1 =...
Hello!
When we are dealing with a free particle in spherical coordinates,the position eigenfunction of the free particle is \psi_{klm}(r,\phi,φ)=\langle r\phiφ|klm\rangle=J_{l}(kr)Y_{lm}(\phi,φ). Here appears that the wavefunction describe a free particle of energy Ek of well-defined angular...
The problem says: Consider an ideal gas of N particles in a spherical vessel of radius R. A force acts directly over the molecules and is directly proportional to the distance to the center of the sphere ##V(r)=\alpha r##. Calculate the pressure of the gas, and the density of particles at the...
Hello,
This latest homework I have been doing has been very confusing to me and I have spent hours trying to complete it. Here is a problem that I really don't know where to start. If anyone could just point me in the right direction, or let me know if my ideas are correct or not, it would...
Homework Statement
Show that for a particle in a central potential; V=f(|r|)
H is conserved.
Homework Equations
THe hamiltonian is
H=\sum(piq'i)-L
It is conserved if dH/dt=0
Euler-Lagrange equation
d/dt(dL/dq')=dL/dq
Noether's Theorem
For a continuous transformation, T such...
It is a well known fact that in a central potential (spherically symmetric) both Lz and L^2 commutes with H and the expectaitonvalues of these are therefore constants of the motion. On the other hand the proof of this fact seems, in the most cases, to be done in the position basis where it is...
Could anyone tell me where can i find more about the fact that if your Hamiltonian is non-central then the total energy is not dependent only on principle quantum number n but also on l.
Homework Statement
A particle of mass m is under a central potential of the form U(r)=-\frac{\alpha }{r^2} where alpha is a positive constant.
At time t=0, the spherical coordinates of the particle are worth r=r_0, \theta = \pi /2 and \phi=0. The corresponding time derivatives are given by...
Homework Statement
A particle of mass m and charge e moves in a central field with potential V(r) = -\alpha r^{-3/2} and in a constant magnetic field \vec{B} = B_0 \hat{e}_z
1) Write the Lagrangian and the Hamiltonian
2) Write the first integral H_{orb} for the equation of the orbit u(\phi )...
Homework Statement
Determine the possible trajectories of a particle into the following central potential: U(r)=U_0 for r< r_0 and U(r)=0 for r>r_0.Homework Equations
Not sure. What I used: Lagrangian+Euler/Lagrange equations.The Attempt at a Solution
I used polar coordinates but I'm not sure...
Hello,
I am trying to compute the potential for a central force of the form: F(r) = f(r)r
where r=|r|
Using the conservative force information, equation1 comes for potential V(r):
equation1: V(r) = \int (-F(r))= \int (-f(r) r)
In http://en.wikipedia.org/wiki/Central_force" it is stated...
Homework Statement
Give the number of states (energy of the state smaller than E<0) \Phi(E) of a spinless particle with mass m in the central potential V(\vec{r})=-\frac{a}{\left|\vec{r}\right|}.
Homework Equations
The Attempt at a Solution
Hi,
the hamiltonian of this problem...
Hey guys,
I have a question regarding solving a radial wavefunction DE which i have written up in Mathematica and saved as a pdf http://members.iinet.net.au/~housewrk/PFpost.pdf" as I was already doing the work in MM and writing it all up again in LaTeX seemed a bit of a waste of time.
If...
Homework Statement
What minimal speed do you need to throw a stone with so that it starts orbiting the Earth? What minimal speed do you need to throw a stone with so that it flies off to infinity (forgetting about the Sun)?Homework Equations
The motion of the stone takes place in an effective...
Homework Statement
Homework Equations
(this is ~Fetter & Walecka Quantum theory of many-particle systems problem 1.2b)
Homogeneous system of spin 1/2 particles, potential V.
Expectation value of Hamiltonian in the non interacting ground state is
E^{(0)} + E^{(1)} = 2 \sum_k^{k_F} \frac{\hbar^2...
Homework Statement
Consider the following parametrization of an orbit in polar form,
\ell u = 1 + e \cos[(\phi -\phi_0)\Gamma]
where u = 1/r.
I'm trying to find the shift in the angular position of the periapsis after one complete orbit.
The Attempt at a Solution
Choose axes so that the...
When we solve the Schrodinger equation for a central potential, such as the coulomb potential for the hydrogen atom, we get eigenstates which are not, in general, spherically symmetric. So they depend on the choice of coordinates.
If we can get two distinct solutions, just by choosing...
Hi all,
I'm new to his forum. I'm having trouble with the following question on central potentials. It's an exercise from (Bransden and Joachain, Introduction to quantum mechanics).
Homework Statement
Suppose V(r) is a central potential, expand around r=0 as V(r)=r^p(b_0+b_1r+\ldots). When...
How can I compute the differential scattering cross section \sigma (\theta) d \theta for scattering in the central potential
V(r) = \frac{k}{r^2}
using classical mechanics?
In Symon's book 'Mechanics', he writes that for a body of mass m and angular momentum L in an orbit that does not intersect itself (i.e. a simple curve), the period of revolution T is related to the area A of the orbit by
T=\frac{2Am}{L}
Is this exact as he seems to be implying? It seems to me...
this is the last part of a long question.
We first have to serparate the TISE for a central potential into two parts using separation of variables (i.e into and R(r) and Y(theta, phi). Then we take the total differential equation in R(r) and substitute R(r) = f(r)/r and get:
-(hbar^2)/2m *...
Hi,
A particle of mass M is moving in an uknown central potential. Its orbit is a circle of raduis R, that passes through the origin. I need to reconstruct the potential from this information.
Thanks,
Chen
Hi,
A particle is subjected to a central potential of:
V(r) = -k\frac{e^{-\alpha r}}{r}
Where k, \alpha are known, positive constants.
If we make this problem one-dimensional, the effective potential of the particle is given by:
V_{eff}(r) = -k\frac{e^{-\alpha r}}{r} + \frac{l^2}{2 m...
How can I calculate the AM and energy of each level in a resulting multiplet of a particle of spin=1/2 with orbital AM quantum number, L=2 subject to a spin orbit potential,
V=lamda(L.S)?
i am at my wits end! :cry: :cry: :cry: