Consider a particle that is free (U=0) to move in a two-dimensional space. Using polar coordinates...

Consider a particle that is free (U=0) to move in a two-dimensional space. Using polar coordinates as generalized coordinates, solve the differential equation for rho and demonstrate that the trajectory is a straight line.

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genius_generous answered 2 years ago

Solve the two-dimensional wave equation in polar coordinates and discuss the eigen values of the problem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)=...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)= 1/2 mω^2 (x^2+y^2 ) a. Use separation of variables in Cartesian coordinates to solve the Schroedinger equation for this particle. b. Write down the normalized wavefunction and energy for the ground state of this particle. c. What is the energy and degeneracy of each of the lowest 5 energy levels of this particle? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational...

Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational acceleration ? exists. Take the vertical axis ?. Using Heisenberg's equation of motion, find the position-dependent operators ? ? and momentum arithmetic operator ?? in Heisenberg display that depend on time. In addition, calculate ??, ?0, ??, ?0.

Consider a particle of mass m that can move in a one-dimensional box of size L...

Consider a particle of mass m that can move in a one-dimensional box of size L with the edges of the box at x=0 and x = L. The potential is zero inside the box and infinite outside. You may need the following integrals: ∫ 0 1 d y sin ⁡ ( n π y ) 2 = 1 / 2 , for all integer n ∫ 0 1 d y sin ⁡ ( n π y ) 2 y = 1...

3. Geodesics in R2 : Consider 2D flat space in polar coordinates r and θ. Find...

3. Geodesics in R2 : Consider 2D flat space in polar coordinates r and θ. Find the curves parametrized by r(s) and θ(s) that satisfy the geodesic equation, and show that they correspond to straight lines in R2

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line element...

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line element is defined by: ds2 = -xdv2 + 2dvdx i) Assuming that the nature and properties of the spacetime line element still hold, show that a particle in the negative x-axis can never venture into the positive x-axis. That is, show that a particle becomes trapped if it travels into the negative x-axis. ii) Draw a representative light cone that illustrates how a particle is...

5. Consider a particle in a two-dimensional, rigid, square box with side a. (a) Find the...

5. Consider a particle in a two-dimensional, rigid, square box with side a. (a) Find the time independent wave function φ(x,y)describing an arbitrary energy eigenstate. (b)What are the energy eigenvalues and the quantum numbers for the three lowest eigenstates? Draw the energy level diagram

how to find out analytical solution of 2-dimensional wave equation in polar coordinates? Also interpret the...

how to find out analytical solution of 2-dimensional wave equation in polar coordinates? Also interpret the eigenvalues of the solution?

Consider a system of ? point particles obeying quantum mechanics constrained to move in a two-dimensional...

Consider a system of ? point particles obeying quantum mechanics constrained to move in a two-dimensional plane (1 2-D ideal quantum gas) in a microcanonical ensemble. Find a formula for the entropy ? via Ω, the number of microstates with energy ≤ ? and show that it is an extensive property of the system.

1. Consider a free, unbound particle with V (x) = 0 and the initial state wave...

1. Consider a free, unbound particle with V (x) = 0 and the initial state wave function: Ψ(x, 0) = Ae−a|x| a>0 (a) (3 pts) Normalize the initial state wave function. (b) (5 pts) Construct Ψ(x, t). (c) (5 pts) Discuss the limiting cases, i.e. what happens to position and momentum when a is very small and when a is very large?

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