Recent content by noospace

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    2nd quantization

    Have I done this much right? I think I might have made a mistake because I was expecting the contribution to the expectation coming from the kinetic energy to vanish.
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    2nd quantization

    Suppose I have a system of N identical bosons interacting via pairwise potential V(\vec{x} - \vec{x}'). I want to show that the expectation of the Hamiltonian in the non-interacting ground state is \frac{N(N-1)}{2\mathcal{V}}\widetilde{V}(0) where \widetilde{V}(q) = \int d^3 \vec{x}...
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    Easier for self-study: Analysis or Algebra?

    Do analysis. I have the same interests as you and I would say that in retrospect taking algebra over analysis was a bad decision. Yes, lie groups and lie algebras play an important role in advanced theoretical physics like particle theory, but you won't be covering that; just finite group...
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    Rabi oscillations and spin 1/2 systems.

    Any two-level system can be written in the form e^{-i\phi/2}\cos(\theta/2) | 0 \rangle + \sin\theta(\theta/2) e^{i\phi/2}|1\rangle justifying the Bloch sphere interpretation. The density operator of the two-level system can be expanded in the basis of Pauli matrices...
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    Rabi oscillations and spin 1/2 systems.

    Hi all, Can anybody please explain to me the connection between Rabi oscillations and spin-1/2 systems? I believe the connection lies in the bloch sphere and the ability to represent the spin-1/2 system by a superposition of Pauli matrices but I'm just not getting it. Thanks
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    Jeans' theorem

    I suppose what I want to show is that the term \sum_{\vec{k},\vec{k}',\alpha,\alpha'}\int d^3 \vec{x} c^\ast_{\vec{k}'\alpha'}c_{\vec{k}\alpha} (\vec{k}\cdot \vec{u}^\ast_{\vec{k}'\alpha'})(\vec{k'}\cdot \vec{u}_{\vec{k}\alpha}) vanihses. For then, \frac{1}{2}\int d^3...
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    Jeans' theorem

    I'm trying to get from the magnetic vector potential \vec{A}(\vec{x},t) = \frac{1}{\sqrt{\mathcal{V}}}\sum_{\vec{k},\alpha=1,2}(c_{\vec{k}\alpha}(t) \vec{u}_{\vec{k}\alpha}(\vec{x}) + c.c.) where c_{\vec{k}\alpha}(t) = c_{\vec{k}\alpha}(0) e^{-i\omega_{\vec{k}\alpha}t}...
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    Easier for self-study: Analysis or Algebra?

    What material do they cover and what are your interests. Personally, I majored in physics/mathematics and took algebra over analysis. The main use of analysis in physics is probably residue calculus which I taught myself when I needed it.
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    Conceptual question about wavefunctions/momentum

    Hi all, If I have the wave function of a system, then the expectation of position is easily visualized as the centroid of the distribution. Does anyone know how to visualize the expectation of velocity given just the postion-space wavefunction (real and imaginary parts)
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    Time evolution of spin state

    Homework Statement An +x-polarized electron beam is subjected to magnetic field in the y-direction. What is the probablity of measuring spin +x after a period of time t. Homework Equations Time evolution operator U = e^{-i/\hbar \hat{H} t} The Attempt at a Solution Since the...
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    Number theory problem

    Hi all, Consider the the number of distinct permutations of a collection of N objects having multiplicities n_1,\ldots,n_k. Call this F. Now arrange the same collection of objects into k bins, sorted by type. Consider the set of permutations such that the contents of any one bin after...
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    Squeezed gaussian expectation

    I'm trying to evaluate the expectation of position and momentum of \exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle} where \hat{a},\hat{a}^\dag are the anihilation/creation operators respectively. Recall \hat{x}...
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    Question about frequency versus wavelength

    Conservation of energy eh? I like that explanation. What assurance do we have that the photon does not exchange energy with its surroundings in passing from one medium to another? Is it it possible to `bump up' the energy of a photon that is part of a self-propagating electromagnetic...
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    Question about frequency versus wavelength

    This is something I really should know but found I was unable to explain it to myself. When a ray of light passes from one medium to another its frequency remains invariant, but it slows down, forcing the wavelength to decrease according to c = \nu\lambda. The frequency of the wave will...
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    Foldy-Wouthusien velocity operator

    Thanks for replying. How do you define x',p' etc.? You also say that i[H,X] is simply related to i[H',x] which is indeed easy to compute. In fact i[H',x] = \frac{c^2\vec{p}}{E} I'm afraid I don't say what the simply relationship is? Could you please expand upon that?
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