Sorry, I think my reasoning was erroneous (it still might be)
I think it is better to say that the initial state of the system can be described by the singlet state (because they point in opposite directions at t = 0, and the wave function needs to be antisymmetric):
\Psi(0) = 1/ \sqrt{2} (...
Thank you very much for your reply. I am getting some problems with the algebra:
As the particles are fixed, the hamiltonian is going to be equal to the potential energy:
\hat{H} = \hat{V}
Generally speaking,
\hat{\sigma_1} = \sigma_{1x} \hat{i} + \sigma_{1y} \hat{j} + \sigma_{1z} \hat{k}...
Homework Statement
Consider two spin 1/2 particles interacting through a dipole-dipole potential
\hat{V} = A\frac{(\hat{\sigma_1} \cdot \hat{\sigma_2})r^2 - (\sigma_1 \cdot \vec{r})(\sigma_2 \cdot \vec{r})}{r^5}
If both spins are fixed at a distance d between each other, and at t = 0 one of...
Homework Statement
A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by \vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}, where it has been emphasized that the basis ε is not necessarily orthonormal...
Homework Statement
Considering the Hamiltonian for a harmonic oscillator:
H=\frac{p^2}{2m}+\frac{mw^2}{2}q^2
We have seen that the equations of motion are significantly simplified using the canonical transformation defined by F_1(q,Q)=\frac{m}{2}wq^2cot(Q)
Show explicitly that between both...
Such a beautiful and elegant explanation! This was indeed a very interesting problem. It had me thinking for a long time, dealing with logarithmic solutions and trying to understand what they really meant... By the due date of the homework (yesterday), my professor commented on how this...
Hmm, I found a similar example of electromagnetic interaction in order to find the Lorentz Force. Here \dot{p}\neq mv, and the force F actually corresponded to the derivative of the linear momentum, which differed form \dot{p}.
At this point I'm kind of lost, I was thinking about a...
Hello, thanks for your replies.
Yes this is exactly what I did. You are right, I commited a manipulation failure, but as far as I can tell it doesn't affect the equation that much. This is the procedure I followed:
From eq. (1), we have
p= m\dot{q} e^{q/a} .
Deriving this equation...
Thanks for your reply.
The Hamilton equations of motion read:
\dot{q}=\frac{∂H}{∂p}
\dot{p}=-\frac{∂H}{∂q}
From which I obtained the following equations:
\dot{q}=\frac{p}{m}e^{-\frac{q}{a}} (1)
\dot{p}=\frac{p^2}{2am}e^{-\frac{q}{a}} (2)
From (1), I obtained an expression for p...
Homework Statement
Consider the following Hamiltonian
H=\frac{p^2}{2m}e^{\frac{-q}{a}}
a: constant
m: mass of the particle
q corresponds to the coordinate, and p its momentum.
note: q' stands for the derivative of q.
a) Prove that for p(t) > 0 this system seems to describe a particle...
Homework Statement
Considering the following integral,
I = \int^\infty_{-\infty} \frac{x^2}{1+x^4}
I can rewrite it as a complex contour integral as:
\oint^{}_{C} \frac{z^2}{1+z^4}
where the contour C is a semicircle on the half-upper plane with a radius which extends to infinity. I can...
Hmm, I think I solved the problem.
We can take the Lagrange equations in the form that follows from the D'Alembert principle, and supose that the generalized force is derivable not from the potential V but from a more general function U, which is usually called generalized potential. Then...
Homework Statement
Consider a particle of mass m and electric charge e subject to a uniform electromagnetic field (E(x,t),B(x,t)). We must remember that the force they exert is given by:
F = cE(x,t) + ex' \times B(x,t)
A principle of action that represents such particle subject to the...