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PROBLEM 2
1. The Lagrangian L for a single particle in a potential V(2) in one...
Lagrangian and Hamiltonian mechanics The Lagrangian for a particle with velocity i = y and electric...
Lagrangian and Hamiltonian mechanics
The Lagrangian for a particle with velocity i = y and electric charge q in an electromagnetic field with Coulomb potential o(i) and vector potential A(7,1)is: L= mü? -9(0-v. A) a) Show that the canonical momentum is: p=mü +GĀ. This is an example of the generalised momentum not being the Newtonian momentum. b) Show that the Hamiltonian corresponding to the Lagrangian, H = P.y-L, is the energy of the particle in the field defined by O...
Question 1: Hamiltonians 1. Find the Hamiltonian for a single particle in cylindrical coordinates...
Question 1: Hamiltonians 1. Find the Hamiltonian for a single particle in cylindrical coordinates in an arbitrary potential V (r, θ, z) via Legendre transformation. 2. Find the Hamiltonian for a single particle in spherical coordinates in an arbitrary potential V (r, θ, φ) via Legendre transformation.
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
2. A particle is confined to the interval (-L/2. L/2) by infinite potentials for rs -L/2 and * 1/2 - Votol - ܘܐ V(x...
2. A particle is confined to the interval (-L/2. L/2) by infinite potentials for rs -L/2 and * 1/2 - Votol - ܘܐ V(x) - [+ o 0 (+ for Is-L/2 for -L/2<x<L/2 for r 1/2 ܝ 02 This is the same as the "particle-in-one-dimensional-box" model of Problem 1, except the origin of the coordinate is taken at the midpoint of the interval. With this choice of the ori gin, potential energy function V () of the particle-in-one-dimensional-box" model becomes...
Due April 19th, 2019 1. (3 pts) Consider two particles of mass mi and m2 (in one dimension) that ...
Due April 19th, 2019 1. (3 pts) Consider two particles of mass mi and m2 (in one dimension) that interact via a potential that depends only on the dstance between the particles V(l 2), so that the Hamiltonian is Acting on a two-particle wave function the translation operator would be (a) Show that the translation operator can be written as where P- p p2 is the total momentum operator of the syste (b) Show that the total momentum is conserved...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential,...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
3. Consider a particle of mass m moving in a potential given by: W (2, y,...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
particle in a cylindrically symmetric potential: do only C please 3. Particle in a cylindrically symmetrical...
particle in a cylindrically symmetric potential:
do only C
please
3. Particle in a cylindrically symmetrical potential: Let pw. be the cylindrical coordinates of a spinless particle (z = pcos y, y psiny: P 20, OS <2m). Assume that the potential energy of this particle depends only one, and not on yor: Vin-V ). Recall that & P R 1 18 dr2 + dy? - apa pap + 2 day? (a) Write, in cylindrical coordinates, the differential operator associated with...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
Lagrangian and Hamiltonian mechanics
The Lagrangian for a particle with velocity i = y and electric charge q in an electromagnetic field with Coulomb potential o(i) and vector potential A(7,1)is: L= mü? -9(0-v. A) a) Show that the canonical momentum is: p=mü +GĀ. This is an example of the generalised momentum not being the Newtonian momentum. b) Show that the Hamiltonian corresponding to the Lagrangian, H = P.y-L, is the energy of the particle in the field defined by O...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
2. A particle is confined to the interval (-L/2. L/2) by infinite potentials for rs -L/2 and * 1/2 - Votol - ܘܐ V(x) - [+ o 0 (+ for Is-L/2 for -L/2<x<L/2 for r 1/2 ܝ 02 This is the same as the "particle-in-one-dimensional-box" model of Problem 1, except the origin of the coordinate is taken at the midpoint of the interval. With this choice of the ori gin, potential energy function V () of the particle-in-one-dimensional-box" model becomes...
Due April 19th, 2019 1. (3 pts) Consider two particles of mass mi and m2 (in one dimension) that interact via a potential that depends only on the dstance between the particles V(l 2), so that the Hamiltonian is Acting on a two-particle wave function the translation operator would be (a) Show that the translation operator can be written as where P- p p2 is the total momentum operator of the syste (b) Show that the total momentum is conserved...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
particle in a cylindrically symmetric potential:
do only C
please
3. Particle in a cylindrically symmetrical potential: Let pw. be the cylindrical coordinates of a spinless particle (z = pcos y, y psiny: P 20, OS <2m). Assume that the potential energy of this particle depends only one, and not on yor: Vin-V ). Recall that & P R 1 18 dr2 + dy? - apa pap + 2 day? (a) Write, in cylindrical coordinates, the differential operator associated with...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
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