Finding the ground-state radial wave function R(r) and the ground-state energy.

A Particle of mass m is in a three-dimensional spherically-symmetric harmonic oscillator potential given by V(r)=(1/2)Kr^2 The particle is in the l=0 state. Find the ground-state radial wave function R(r) and the ground-state energy.

Determine whether or not the particle is in an eigenstate of Lz.

A particle is confined in a cubic bow with edge of length a, with V=0 inside the box. The particle is in its ground state, determine whether or not the particle is in an eigenstate of Lz. I do not know how to do this, eigenstate? Detailed solution needed, please.

Semi-infinite square well.

Consider the semi-infinite square well given by V(x)=-Vo<0 for 0<=x<=a and V(x)=0 for x>a. There is an infinite barrier at x=0. A particle with mass m is in a bound state in this potential energy E<=0. a) solve the Schroedinger Eq to derive phi(x) for x=>0. Use the appropriate boundary conditions and normalized the wave function so that the final answer does not contain any arbitrary constants. b) Show that the allowed energy levels E must satisfy the equation (attached) c)The equation in part (b) can not be solved analytically to give the allowed energy levels but simple solutions exist in certain special cases. Determine the conditions on Vo and a so that a bound state exists with... click for more

One-Dimensional Time-Independent Schrodinger

The one dimensional parity operator (pie) is defined by attached. in other wards (phi) changes x into -x everywhere in the function (a) is it a Hermitian operator? (b) For what potentials V(x), is it possible to find a set of wave functions which are high eign- functions of the parity operator in solution of the one-dimensional time independent Schrodinger Eq? I believe a) is yes, but am not sure 100%. I however do not now had to do part b). Detail solution please.

A particle in an infinite tube.

a) A particle with mass m and energy E is inside a square with tube infinite potential barriers at x=0, x=a, y=0 and y=a. The tube is infinitely long in the z direction. Inside the tube v=0. The particle is moving in the +z direction solve the Schrodinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in the +z direction. b)Energy should not be quantized in this case because the particle is not in a bound state. Use the answer from part a to show that this is indeed the case.