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  1. N

    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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    Foldy-Wouthusien velocity operator

    If one takes the derivative of the position operator in the Dirac Hamiltonian, the result is \dot{\vec{x}} = c \vec{\alpha}. This, however, disagrees with the classical limit in which \dot{\vec{x}}\sim \dot{\vec{p}}/m. I'm trying to show that the time derivative of the position operator...
  17. N

    Heisenberg's equation of motion

    Are you saying that the transformed operator satisfies the first equation but not the second?
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    Heisenberg's equation of motion

    The equation of motion for an observeable A is given by \dot{A} = \frac{1}{i \hbar} [A,H]. If we change representation, via some unitary transformation \widetilde{A} \mapsto U^\dag A U is the corresponding equation of motion now \dot{\widetilde{A}} = \frac{1}{i \hbar}...
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    Commutation relations in relativistic quantum theory

    Given the Hamiltonian H = \vec{\alpha} \cdot \vec{p} c + \beta mc^2, How should one interpret the commutator [\vec{x}, H] which is supposedly related to the velocity of the Dirac particle? \vec{x} is a 3-vector whereas H is a vector so how do we commute them. Is some sort of tensor product in...
  20. N

    Noncommuting operators and uncertainty relations

    Ahh, If A\psi belongs to the b-eigenspace then we can express it as A\psi = a_1\psi_1 + a_2\psi_2 A(\psi_1 + \psi_2) = a_1\psi_1 + a_2\psi_2 so A\psi_i = a_1\psi_i by linear independence.
  21. N

    Noncommuting operators and uncertainty relations

    In the degenerate case, I'm not quite understanding what A\psi being an eignestate of B has to do with being able to diagonalize A in the b-eigenspace? Could someone please help me understand this?
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    Commutator math help

    Does the relation [f(\hat{A}),\hat{B}] = df(\hat{A})/d\hat{A} follow when A commutes with [A,B]? or is this only valid when [A,B]=1?
  23. N

    Understanding Lorentz invariance

    Since no one has yet replied, let me see if I can get any further withe the first part. First of all the definition of Lorentz invariance is that the equation remains true in arbitrary intertial reference frames. The quantity A_{\mu\lambda} = \sum_{a}...
  24. N

    Potential central field

    Yes, I think there is. Note that you have two differential equations: one first order and one second order (the Lagrange equation). Hint: use the second order. But since you are interested in the shape, you need to change from time derivatives to derivatives wrt \phi. Question: what is the...
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    QFT in a nutshell: Propagators

    Hang on. It just occured to me that the implication \vec{k} \to - \vec{k} \implies d\vec{k} \to -d\vec{k} might not be valid since we're doing volume integrals here. Perhaps this saves me?
  26. N

    QFT in a nutshell: Propagators

    Hi nrqed, Thanks for your reply. The page reference is p. 23 Eq. (23). We want to show that D(x) = - i\int \frac{d^3\vec{k}}{(2\pi)^32\omega_k}[e^{-i(\omega_k t - \vec{k}\cdot\vec{x})}\theta(x^0) + e^{i(\omega_k - \vec{k}\cdot\vec{x})}\theta(-x^0)]. Although Zee states the contour...
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    QFT in a nutshell: Propagators

    Homework Statement I'm trying to show that the general form of the propagator is D(x) = - \int \frac{d^3k}{(2\pi)^32\omega_k}[e^{-i(\omega_k t - \vec{k}\cdot\vec{x})}\theta(x^0) + e^{i(\omega_k - \vec{k}\cdot\vec{x})}\theta(-x^0)] but my answers always seem to differ by a sign. Homework...
  28. N

    QFT question

    Sigh. Nevermind, there was a typo in my second integral, Eq. (15) is actually Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}\varphi(\partial^2+m^2)\varphi + J\varphi]} which can be obtained easily by integration by parts on the (\partial \varphi)^2 term: \int d^4 x\, (\partial \varphi)^2 =...
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    QFT question

    Homework Statement I'm studying from Zee's QFT in a nutshell. On page 21, I don't understand how he uses integration by parts to get from Eq (14) to Eq (15), ie from Z = \int D \varphi e^{i \int d^4 x \{ \frac{1}{2}[(\partial \varphi)^2 - m^2 \varphi^2] + J\varphi \}} to Z = \int D \varphi...
  30. N

    Zee question I.2.2

    I figured it out: For future knowledge, the trick is to use that the matrix A must be symmetric, and thus the derivative of the \frac{1}{2}\mathbf{x}^t A^{-1} \mathbf{x} with respect to x_i, say can be written \sum_n (A^{-1})_{in} x_n.
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