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# Mathematical Physics

Mathematical physics is the study of mathematical methods for application of problems in physics. There are five distinct branches of mathematical physics: geometrically advanced formulation of classical mechanics, partial differential equations, quantum theory, Relativity and Quantum Relativistic Theories and statistical mechanics. Each branch has its own section of physics problems in which it corresponds to.

Mathematical physics is denoted by the nature of research. The research is aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Therefore, mathematical physics can cover a broad range of topics from pure mathematics to theoretical physics. Mathematical physicist does not conduct experiments like “regular” physicists do. The advent of the computer has created a new application of mathematical physics however that allows highly complex simulations being conducted. These simulations are run off of complex mathematical models to model certain phenomena.

## BrainMass Categories within Mathematical Physics

### Noise reduction using interferometry

Consider a Mach-Zehnder interferometer where in one arm we place a filter with a transmission coefficient f << 1. The system is set up such that at one detector site we have constructive interference and at the other side we have destructive interference. Demonstrate that the expected number of detected photons N needed before a

### Einstein Summation and Levi-Civita Tensor

Please help answer this question. It requires you to give proof that two definitions of angular momentum are equal. The question is in the attachment. It is taken from this site. http://farside.ph.utexas.edu/teaching/qmech/Quantum/node80.html

### Lagrangian interaction and Feyman propagator

1. Consider a real scalar field phi with interaction Largrangian L_int = (p/(3!))phi^3. What is the mass dimension of u? Evaluate the leading u-dependent contributions to the following equation: (see attached file)

### Summation of Divergent Series

In physics one often needs to sum divergent series. Here we give a simple example of how one can estimate the limit of x to infinity of a function f(x) given by its Taylor expansion around x = 0 when its radius of convergence is only 1.

### Problems with Variable Names

Provided is a list of variable names. Determine what is wrong (if anything) with each variable name and how to fix it. - two_wheels - BTUs-2-watts - loop - Another*variable - stDDev - mean.

### Residues Theorem and Integration of the function 1/(1+x^2)

Using the Residues Theorem, calculate the following: Integral limits between negative infinity and positive infinity [dx/(1+x^2)]. Please the file attached for the formula in its adequate notation.

### Proving Factorial Equations Involving Double Factorials

In many problems in mathematical physics, particularly in connection with Legendre polynomials (Chapter 12), we encounter products of the odd positive integers and products of the even positive integers. For convenience, these are given special labels as double factorials: 1*3*5***(2n + 1) = (2n + 1)!! 2*4*6***(2n) = (2n)!

### Time Required.

How much time is required before a 10 mCi sample of 99mTc (T1/2=6.0 hours) and a 25 mCi sample of 113m In (T1/2=1.7 hours) possess equal activities? Please see the attach file.

### Interference of 2 Waves.

Two loudspeakers are placed side by side a distance d=4.00m apart. A listener observes maximum constructive interference while standing in front of the loudspeakers, equidistant from both of them. The distance from the listener to the point halfway between the speakers is l=5.00m. One of the loudspeakers is then moved dir

### Determining the Moment of Inertia of a Pendulum

The pendulum shown in the figure (see attachment) consists of a thin disk and two slender rods. The disk has a mass of 2 kg, the longer rod AB has a mass of 6.5 kg and the shorter rod CD has a mass of 2.5 kg. Determine the moment of inertia of the pendulum about an axis perpendicular to the page passing through (a) point O, and

### Center of mass of meter sticks

Three uniform meter sticks, each of mass m, are placed on the floor as follows: stick 1 lies along the y axis from y = 0.350 m to y = 1.35 m, stick 2 lies along the x axis from x = 0.130 m to x = 1.13 m, stick 3 lies along the x axis from x = 1.41 m to x = 2.41 m. Calculate the location of the center of mass of the meter sticks.

### As a result of a chemical spill, benzene is evaporating at the rate of 1 gram per minute into a room that is 6 m * 6 m * 3 m in size and has a ventilation of 10 m3/min.

As a result of a chemical spill, benzene is evaporating at the rate of 1 gram per minute into a room that is 6 m * 6 m * 3 m in size and has a ventilation of 10 m3/min. a- Compute the steady state concentration of benzene in the room. b- Assuming the air in the room is initially free of benzene, compute the time necessary fo

### Dynamics: Springs and Oscillations

The 0.5 lb weight is suspended from a rigid frame as shown in the attached diagram. Pin A at the end of the rotating arm OA engages a slot in the frame, causing the frame to oscillate in the vertical direction. The arm is accelerated uniformly from rest at t = 0 and θ = 0 at the ω.(dot) = 100 rad/s^2. (2) Use numerical int

### Tuning Frequency Changes

In each problem one of its numerical parameters depends on 3504529. Divide 3504529 by 8 and depending on a value of a remainder, R, choose a corresponding value of this parameter. Use C = 4 5 6 7 8 9 10 11 For R = 0 1 2 3 4 5 6 7 The tuning frequency f of an

### The Period of Pvhysical Pendulum

A large bell is hung from a wooden beam so it can swing back and forth with negligible friction. The center of mass of the bell is 0.50m below the pivot, the bell has mass 38.0kg , and the moment of inertia of the bell about an axis at the pivot is 20.0kg*m^2. The clapper is a small, 1.8kg mass attached to one end of a slender

### Showing that a Matrix is Unitary

1) Show that the matrix is unitary. (Please see attached document for matrix.) 2) Find the eigenvalues of this matrix (they are not real numbers) and the eigenvectors. Please see the attached file for full question.

### Legendre Polynomials and Equations

Show that the Legendre polynomials satisfy the equation. Please see the attached file. Please show each step in the solution.

### Find the Fourier Series Expansion

Find the Fourier series expansion of f(theta) = theta^2 on the domain - pi < theta < pi, and use it to show that the following: see attached file for equations.

### Four Masses

Please see problem attached. The four masses shown in Figure Ex13.17 are connected by massless, rigid rods, with m = 184 g. (a) Find the coordinates of the center of mass. (b) Find the moment of inertia about a diagonal axis that passes through masses B and D.

### Pendulum Motion: Lagrangian Generalized Coordinates

Consider the pendulum of figure 7.4, suspended inside a railroad car, but suppose that the can is oscillating back and forth, so that the point of suspension has position x=Acos(wt), y=0. Use the angle Φ as the generalized coordinate and write down the equations that give the Cartesian coordinates of the bob in terms of Φ and

### Gaussian Distribution and Fourier Transforms

1. The Fourier Transform of the probability density, P(x) is + T(k) =  (e^(ikx)}*P(x) dx - and is called the characteristic function of the random variable x. Let F(k) = log (T(k)) and show that a) F(0) = 0 b) F'(0) = i<x> c) F'' (0) = i<(x)^2> 2. Take P(x) to be the G

### Recoil Force of a Gun

A soldier on a firing range fires an 8-shot burst from an assault weapon at a full automatic rate of 1000 rounds per minute. Each bullet has a mass of 7.42 g and a speed of 295 m/s relative to the ground as it leaves the barrel of the weapon. a). Calculate the average recoil force (in N) exerted on the weapon during that burs

### Two Dimensional Surfaces Described by a Metric

Please show me how to integrate g(x) into the metric and how to solve. See attached file for full problem description.

### Derive the Equation of Motion

Derive the Equation of Motion. See attached file for full problem description.

### Simultaneous Equations for a System

X^2 + y^2 = 1 y=|x| - a find all "a" which prevent the system to have no solutions

### Simple Inequality Calculation

Show that X^4 - x^3 + 1 > 0

### Dirac delta function, divergence, and curl

See attached files

### Planetary motion: h = |dr/dt|^2/2 - μ/r is a constant.

Starting from the formula d^r/dt^2 = -μ (r/r^3) show by differentiation that the quantity h = |dr/dt|^2/2 - μ/r is a constant. It is useful to recall that |dr/dt|^2 = dr/dt . dr/dt. You will need to use your calculus skills to find an expression for the derivative of 1/r with respect to time t, after which the fact that rdr/dt

### Critical Point on Van der Waals Isotherms

The critical point is the unique point on the original van der Waals isotherms (before the Maxwell construction) where both the first and second derivatives of P with respect to V (at fixed T) are zero. Use this fact to show that: V_c = 3Nb, P_c = (1/27) (a/b2), kT_c = (8/27) (a/b).