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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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    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'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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    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

    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

    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...
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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...
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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?
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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...
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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...
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    Zee question I.2.2

    I've decided to slowly work my way through Zee's quantum field theory in a nutshell over the vacation. I'm confused by the second question of the book which deals with matrix differentiation. Homework Statement Derive the equation \langle x_i x_j \cdots x_k x_l \rangle = \sum_{\textrm{wick}}...
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    Mapping torus is an (m+1)-manifold

    Homework Statement Let X be an m-manifold. Let M(f) be the space obtained from X\times [0,1] by gluing the ends together using (x,0)\sim (f(x),1). Show that if M is an m-manifold then M(f) is an (m+1)-manifold. The Attempt at a Solution Since X has an atlas \{ (U_\alpha,\varphi_\alpha) \}...
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    Determine the energy levels, their degeneracy and wave functions of a particle

    Homework Statement Determine the energy levels, their degeneracy and wave functions (in ket notation) of a particle with spin quantum number s =1 if the Hamiltonian is AS_x^2 + AS_y^2 + B S_z^2 where A and B are constants. The Attempt at a Solution' I've spent ages thinking about this...
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    2 quantum questions

    Homework Statement 1. Consider a beam of z-oriented electrons, 80 % up, 20 % down which is passed through an x-oriented Stern-Gerlach machine. What percentage of electrons are measured in the +/- x-directions? 2. Consider deuterium. Nuclear spin = 1 with 1 electron orbiting in the n =1 state...
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    Spin-orbit coupling perturbation

    Homework Statement An electron in a hydrogen atom is in the n = 2, l = 1 state. It experiences a spin-orbit interaction H' = \alpha \mathbf{L} \cdot \mathbf{S}. Calculate the energy level shifts due to the spin-orbit interaction. Homework Equations Degenerate perturbation theory. The...
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    Hermitian operators in spherical coordinates

    Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is Hermitian? e.g. suppose i have the operator i \partial /\partial \phi. If the operator was a function of x I know exactly what to do, just check \int_\mathbb{R} \psi_l^\ast \hat{A} \psi_m dx =...
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    Expectation of an Hermitian operator is real.

    Homework Statement WTS \langle \hat{A} \rangle = \langle \hat{A} \rangle^\ast The Attempt at a Solution \langle \hat{A} \rangle^\ast = \left(\int \phi_l^\ast \hat{A} \phi_m dx\right)^\ast=\left(\int (\hat{A}\phi_l)^\ast \phi_m dx\right)^\ast= \int \phi_m^\ast \hat{A}\phi_l dx. So...
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    Canonical perturbation theory

    Homework Statement An electron is inside a magnetic field oriented in the z-direction. No measurement of the electron has been made. A magnetic field in the x-direction is now switched on. Calculate the first-order change in the energy levels as a result of this perturbation. The Attempt...
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    Please help with t-substitution

    Homework Statement \int_{0}^{2\pi} \frac{\cos^2\alpha}{A + \sin^2\alpha}d\alpha The Attempt at a Solution I believe when the trig functions all appear squared one may use the substitution t =\tan \alpha. Then t^2 + 1 = \sec^2\alpha \implies \cos^2\alpha = \frac{1}{t^2 +1} \sin^2\alpha...
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    Christoffel symbols from definition or Lagrangian

    I asked this question in the tensor analysis formum but did we did not reach a satisfactory conclusion. Here is the problem: Let \mathbf{x} : U \subset\mathbb{R}^2 \to S be a local parametrization of a regular surface S. Then the coefficients of the second derivatives of x in the basis of...
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    Hadron spin

    Homework Statement Find the the total spin of an n-quark hadron Homework Equations The Attempt at a Solution I'm totally stumped on this. Any hints would be greatly appreciated.
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    Earth's magnetic field

    Hi all, Does anyone know where I could find values for the Earth's magnetic field at major cities? Thanks
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    All eigenvalues zero => zero map

    I want to prove that if all the eigenvalues of a linear transformation T : V --> V are zero, then T = 0. I think this is obvious but I'm having difficulty putting it into words. If all the eigenvalues of T are zero, then there exists a basis B for V in which [T]_B is the zero matrix. Thus...
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    Christoffel symbols from definition or Lagrangian

    Hi, Let \mathbf{x}(u,v) be a local parametrization of a regular surface. Then the coefficients of \mathbf{x}_{uu},\mathbf{x}_{uv} etc. in the basis of the tangent space are defined as the Christoffel symbols. On the other hand, if we write the first fundamental form \langle,\rangle in...
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    Square lattice electron gas

    Hi all, I'm struggling to understand the relationship between electrical conductivity and Bragg reflection in a 2D square lattice free electron gas with lattice spacing a. Is it the case that Bragg reflection in the electron gas results in electrical resistivity? My understanding of...
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    Noncommuting operators and uncertainty relations

    Hello all, I've been thinking about the connection between commutativity of operators and uncertainty. I've convinced myself that to have simultaneous eigenstates is a necessary and sufficient condition for two observeables to be measured simultaneously and accurately. It's also clear...
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