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    Hamiltonian for a 1D-spin chain

    Homework Statement [/B] A 1D spin chain corresponds to the following figure: Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin...
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    How do I calculate the potential created by a dipole

    Homework Statement I'm given that there is a positive charge of 1 nC at x=0.25 m and a negative charge of -1 nC at x=-0.25 m. I've calculated the potential created at different points along the x-axis by the positive charge and the negative charge using the formula, $$V=\frac{kq}{|r|},$$ where...
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    Distance the hoop travels up the incline

    Homework Statement A ring (hollow cylinder) of mass 2.61kg, inner radius 6.35cm, and outer radius 7.35cm rolls (without slipping) up an inclined plane that makes an angle of θ=36.0°, as shown in the figure below. At the moment the ring is at position x = 2.19m up the plane, its speed is...
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    Find the apparent weight

    Homework Statement A car traveling on a straight road at 9.15m/s goes over a hump in the road. The hump may be regarded as an arc of a circle of radius 10.4m. What is the apparent weight of a 665N woman in the car as she rides over the hump? Homework Equations ##F=ma##; ##a=v^2/r## The...
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    Find monic generators of the ideals

    Homework Statement Let ##T## be the linear operator on ##F^4## represented in the standard basis by $$\begin{bmatrix}c & 0 & 0 & 0 \\ 1 & c & 0 & 0 \\ 0 & 1 & c &0 \\ 0 & 0 & 1 & c \end{bmatrix}.$$ Let ##W## be the null space of ##T-cI##. a) Prove that ##W## is the subspace spanned by...
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    Find the velocity when the ball rolls without slipping

    Homework Statement A thin spherical shell is sliding with velocity ##v_0## on a table initial until friction eventually causes it to roll without slipping. Find its translational velocity when the it rolls without slipping as a fraction of ##v_0##. Homework Equations $$I=\frac{2}{3}MR^2$$...
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    Acceleration of a uniform solid sphere rolling down incline

    Homework Statement Find the acceleration of a uniform solid sphere (of mass ##m## and radius ##R##) rolling without slipping down an incline at angle ##\alpha## using the Lagrangian method. Homework Equations Euler-Lagrange equation which says, $$\frac{\partial\mathcal{L}}{\partial...
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    Moment of inertia of a half disk about an axis

    Homework Statement Consider a half disk (of uniform density) with the flat end lying on the x-axis, symmetric about the y-axis (i.e. being cut into two quarters by the y-axis). Calculate the moments of inertia about each of the axes. Homework Equations $$I_{rr}=\sum_{i}m_ir_i^2$$ The Attempt...
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    Show that a series is divergent

    Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...
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    Lagrangian for a bead on a wire

    Homework Statement A bead of mass ##m## slides (without friction) on a wire in the shape, ##y=b\cosh{\frac{x}{b}}.## Write the Lagrangian for the bead. Use the Lagrangian method to generate an equation of motion. For small oscillations, approximate the differential equation neglecting terms...
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    Lagrangian for a particle in a bowl with parabolic curvature

    Homework Statement A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system. Homework Equations...
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    Exponentially driven harmonic oscillator

    Homework Statement An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion. Homework Equations ##F=m\ddot{x}## The Attempt at a Solution We have...
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    Use angular momentum to find the velocity (comet orbit)

    Homework Statement A comet orbits the sun. It's position in polar coordinates is given by, $$r(\phi)=\frac{1.8r_0}{1+0.8\cos{\phi}},$$ where ##r_0## is the position at closest approach. Its velocity at this point is given by ##v_0##. Use the concept of angular momentum to find the following...
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    Energy eigenvalues of spin Hamiltonian

    Homework Statement The Hamiltonian of the positronium atom in the ##1S## state in a magnetic field ##B## along the ##z##-axis is to good approximation, $$H=AS_1\cdot S_2+\frac{eB}{mc}(S_{1z}-S_{2z}).$$ Using the coupled representation in which ##S^2=(S_1+S_2)^2##, and ##S_z=S_{1z}+S_{2z}## are...
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    Find the point when velocity is momentarily zero

    Homework Statement [/B] A block travels in the positive x-direction with some velocity ##v_0##. It is subject to a drag force $$F(v)=-F_0e^{Kv},$$ where ##F_0,\,K## are positive constants. Find the point in time when the velocity of the block momentarily goes to zero. Find how far the block has...
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    Applying a time-dependent force to a proton

    Homework Statement A proton is initially located at the origin of some coordinate system (at rest), when a time-dependent force, $$F(t)=F_0\sin{(\omega t)},$$ is applied to it, where ##F_0## and ##\omega## are constants. a) Find the velocity and displacement of the proton as functions of...
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    Is first Brillouin zone the same as Wigner-Seitz cell?

    Homework Statement Not a homework question, but I am attempting to understand what exactly the first Brillouin zone is. Homework Equations The Attempt at a Solution From my textbook, what I'm gathering is that one constructs the first Brillouin zone by constructing a "Wigner-Seitz" type cell...
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    Fourier transform of periodic potential in crystal lattice

    Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
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    Kinematics car chase question

    Homework Statement We have that a police car is travelling at ##26.4## m/s, and that at a time, ##t=0##, a car speeds by at ##37.5## m/s. After ##1## s, the police car accelerates constantly at ##2## m/s2. We are asked to find the time, ##t##, where the police car catches the speeder. Homework...
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    Integrated track length in electromagnetic shower

    Homework Statement Show that integrated track length in EM-shower is proportional to ##E_0##. Homework Equations ##E(t)=\frac{E_0}{2^t}##, with radiation length, ##x_0##. Knowledge that shower terminates at ##E_c##. The Attempt at a Solution The total track length is naturally the total...
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    Electromagnetic shower energy spectrum

    Homework Statement The energy of a daughter particle in electromagnetic shower is approximated by, ##E(t)=\frac{E_0}{2^t}##. Show that the energy falls off like approximately ##E^{-2}##, for small ##E##. Homework Equations Nothing really. Just a matter of knowing how to differentiate. The...
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    Linear perturbation to harmonic oscillator

    Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...
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    Anharmonic oscillator first-order correction to energy

    Homework Statement I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions. Homework Equations First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$ The...
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    Find electric potential of field inside and outside nucleus

    Homework Statement Derive following expression for the electrostatic potential energy of an electron in the field of a finite nucleus of charge, ##+Ze##, and radius, ##R=r_0A^{1/3}##, where ##r_0## is a constant. (Charge density is constant.) The potential we are asked to derive is: $$ V(r) =...
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    Center of mass energy problem

    Homework Statement While not explicitly a homework question, I am having some trouble with deriving expressions for the center of mass energy in a fixed target experiment versus a collider experiment. The question is effectively, "Derive an expression for the center of mass energy in a fixed...
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    Drawing the quark flow diagram for proton-pion interaction

    Homework Statement I am given the following interaction, $$\pi^-+p\rightarrow \pi^++\pi^-+n,$$ and asked to draw the Feynman (quark flow diagram). Homework Equations None; just baryon number conservation, quark flavor conservation, etc. The Attempt at a Solution First, as baryon number and...
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    How to solve for D and sketch the wavefunctions

    Homework Statement See attached image. The potential in question is, ##-V_0## for ##0<r<a,## and ##0## for ##r\geq a.## Homework Equations $$\sinh(x)=\frac{e^x+e^{-x}}{2}$$ $$\cosh(x)=\frac{e^x-e^{-x}}{2}$$ The Attempt at a Solution I know that the wavefunction for ##r<a## is given by...
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    Eigenvectors of Ly and associated energies

    Homework Statement Consider a particle with angular momentum l=1. Write down the matrix representation for the operators L_x,\,L_y,\,L_z,for this particle. Let the Hamiltonian of this particle be H = aL\cdot L-gL_z,\,g>0.Find its energy values and eigenstates. At time t=0,we make a measurement...
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