PLEASE, ANSWER ALL 5 QUESTIONS AND ALL PARTS POSSIBLE. EXPRESS ALL ANSWERS IN TH

Question

PLEASE, ANSWER ALL 5 QUESTIONS AND ALL PARTS POSSIBLE.

EXPRESS ALL ANSWERS IN THEIR SIMPLEST FORM 1.(6pt)Using the definitions for x, y, and z to the right to transform the Cartesiarn components of angular momentum below into spherical polar components angle and polar angle are allowed to vary: a. Transform: Lx = | y c. What do l and Ly become for motion exclusive to the xy-plane, where not only is r sinOsin ar b.Transform: Ly = ( -x- r constant, but = 90° = constant as well. + acting on the eigenfunction 3.(5pt)Find the normalization constant N for the following wavefunctions a. = Nx2e_3x" over the range:-oo xs+00 N(sin )-icons ))over the range: Ostst over the range: 0 x 12 4.(5pt) Find the overlap of the following wavefunctions (Find a numeric value of the overlap) 16 xe-r and Y, = xover the range: 0sxs +o 4 b.) Y,-76 +1 sin( 4x) and Y,--t1 4.(5pt)Evaluate the following expectation values for the (normalized) wavefunctions: a. Find r> for the wavefunction: --sin 3 2 (x) over the range: O s b. Find using the kinetic energy operator: O for the wavefunction --xer over the range: - oo S r3 too sin(2x) over the range:--x h2 d2 2m dx2 4

Explanation / Answer

1. for cartesean coordinates, (x,y,z)

angular momentum = m * (r x v)

now, v = dx i /dt + dy j / dt + dz k /dt [ where i , j and k are unit vectors in x,y and z directions respectively]

and r is position vector r = xi + yj + zk

so L = m*[(xi + yj + zk) x (dx i /dt + dy j / dt + dz k /dt )]

L = m*[i(ydz/dt - zdy/dt) + j(zdx/dt -xdz/dt) + k(xdy/dt - ydx/dt)]

so Lx = m[ydz/dt - zdy/dt]

now in polar coordinates

x = rsin(theta)cos(phi)

y = rsin(theta)sin(phi)

z = rcos(theta)

so, Lx = m[r*sin(theta)sin(phi)*r(-sin(theta)*theta') - r*cos(theta)*r*[sin(theta)cos(phi)*phi' - cos(theta)sin(phi)theta']]

so, Lx = mr^2[-sin^2(theta)sin(phi)(theta') - cos(theta)sin(theta)cos(phi)*phi' + cos^2(theta)sin(phi)theta']

2. Ly = m(zdx/dt - xdz/dt)

substituting

Ly = m[rcos(theta)*r[sin(theta)(-sin(phi))(phi') + cos(theta)cos(phi)(theta')] - rsin(theta)cos(phi)(-rsin(theta)*(theta'))]

Ly = mr^2[-sin(theta)cos(theta)(sin(phi))(phi') + cos^2(theta)cos(phi)(theta') + sin^2(theta)cos(phi)(theta'))]

3. given r and theta = 90 deg, so theta' = 0

Lx = mr^2[-sin^2(theta)sin(phi)(theta') - cos(theta)sin(theta)cos(phi)*phi' + cos^2(theta)sin(phi)theta']

Lx = mr^2[ - cos(90)sin(90)cos(phi)*phi' ] = 0

Ly = mr^2[-sin(theta)cos(theta)(sin(phi))(phi') + cos^2(theta)cos(phi)(theta') + sin^2(theta)cos(phi)(theta'))]

Ly = 0

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