1. We saw in class that a 3-dimensional ideal gas obeys pV = 3/2E, where p is the pressure, V is the volume, and E is the internal energy (average kinetic energy). Derive the corresponding formula for a two-dimensional ideal gas, i.e. a collection of noninteracting particles that moves on a plane. (Hint: suppose a two-dimensional ideal gas of N particles, with internal energy E, is confined by one-dimensional "walls" to an area A; the pressure is defined as the force per unit length).
2. Each r seconds a particle, which was initially x = 0, jumps either left or right (each with probability 1/2) a distance a. At time tn= nr, the particle is at location xk = ka with probability P(n, k). Calculate P(n, k), <k> and <(delta(k))^2>.

Solution Summary

This solution provides calculations for various questions involving particles and dimensions.

Disspative Kepler's Potential in One Dimension. Question 1: (i) Write down the equation of motion for the point particle of mass m moving in the Kepler ...

... to the equation y = at - bt^2. The dimensions of the ... is 3.4+5.4t>3.4 All four particles are all ... then the acceleration is positive, but the particle is slowing ...

... in a nucleus can be represented (in one dimension) by a ... The solution determines the energy of particles in the ... well the probability to find the particle is zero ...

... electrons in conjugated pi systems as particles in one ... to approach the problem: First, determine the dimensions of the ... m is the mass of the particle (here, an ...

... each of mass m, moving in one dimension and bound ... as well as the coordinate of each particle relative to ... makes sense since the coupling of the particles is all ...

Problems Involving the Schrodinger Equation. (a) The time-independent Schrodinger equation for a free particle moving in one dimension (x) is written as (d^2 Psi ...

... is a significant source of sand particles that is ... Human activity adds yet another dimension to coastal ... is dissipated because the large particle size allows ...

... As the angle is a dimension less quantity angular velocity is also having the dimensions [M0L0T-1 ... hence a particle at a distance of 30 in from the axis ...

... the accelerated frame of reference of the particle. ... Therefore these particles move towards the closed end ... quantities and their formulae, units and dimensions. ...

... Since it only has macroscopic dimensions along two directions ... to calculate the energy of a particle in such ... we take away one more macroscopic dimension and form ...