1.) (30 points) Throughout this problem, consider the following spherically symmetric three dimensional potential: V(r) =...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V(r)2r2. The particle is in the I-0 state. Noting that all eigenfunetions must be finite everywhere, find the ground-state radial wave-function R() and the ground-state energy. You do not have to nor oscillator is g (x) = C x exp(-8x2), where C and B are constants) harmonic malize the solution....
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Consider an electron moving in a spherically symmetric potential V = kr, where k>0. (a) Use the uncertainty principle to estimate the ground state energy. (b) Use the Bohr-Sommerfeld quantization rule to calculate the ground state energy. (c) Do the same using the variational principle and a trial wave function of your own choice. (d) Solve for the energy eigenvalue and eigenfunction exactly for the ground state. (Hint: Use Fourier transforms.) (e) Write down the effective potential for nonzero angular...
Consider a particle of mass in a 10 finite potential well of height V. the domain – a < x < a. a) Show that solutions for – a < x < a take the form on (x) = A cos(knx) for odd n, and on (x) = A sin(knx) for even n. . Show a) Match the boundary conditions at x = a to prove that cos(ka) = Bk where k is the wave vector for -a < x...
Find parts a and b.
Consider the three-dimensional cubic well V = {(0 if
0<x<a, 0<y<a, 0<z<a), (infinity otherwise).
The stationary states are psi^(0) (x, Y, z) =
(2/a)^(3/2)sin(npix/a)sin(npiy/a) sin(npiz/a), where nx, ny , and
nz are integers.
The corresponding allowed energies are E^0 =
(((pi^2)(hbar^2))/2m(a^2))(nx^2+ny^2+nz^2).
Now let us introduce perturbation V={(V0 if
0<x<(a/2), 0<y<(a/2)), (0 otherwise)
a) Find the first-order correction to the ground state
energy.
b) Find the first-order correction to the first
excited state.
1. Consider the...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
cosmology
(1) 1. (10 points] In a simple model for a spherically symmetric dark matter halo, the dark matter density is given by M PDM = 4ar, where M, is the enclosed mass at the virial radius r. Let v(r) be the velocity of a dark matter particle that is executing a circular orbit in the halo with radius r centered at r=0. Show that v(r) is independent of radius and obtain its value in terms of M, and ty....
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,...
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...