Wave Functions and Uncertainty

Physicists use laser beams to create an "atom trap" in which atoms are confined within a spherical region of space with a diameter of about 1 mm. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately 1 nK, which is extremely close to absolute zero, but it would be interesting to know if this temeprature is close to any limit set by quantum physics. We can explore this issue with a 1-D model of a sodium atom in a 1-mm-long box. a) Estimate the smallest range of speeds you might find for a sodium atom in this box. b) Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in ... click for more

Wave Functions and Uncertainty

Please see attached problem. --- The Wave function of a particle is: if -1 mm mm if 0 mm mm and 0 elsewhere. a) Assuming that this function is continuous, what can you conclude about the relationship between b and c? b) Draw graphs of the wave function and the probability density over the interval -2mm <= x <= 2mm. c) What is the probability that the particle will be found to the right of the origin? ---

1-D Quantum Mechanics

A basic model of a hydrogen atom is a finite potential well with rectangular edges. A more realstic model of a hydrogen atom, although still a 1-Dimensional model, would be the electron + proton potential enrgy in one dimension: U(x) = -e^2/(4pi epsilon_0)|x|) a) Draw a graph of U(x) versus x. Center your graph at x = 0. b) Despite the divergence at x= 0, the Schrodinger Equation can be solved to find energy levels and wave functions for the electron in this potention. Draw a horizontal line across your graph of part a) about one-third the way from the bottom to the top. Label this line E2, the on this line, sketch a plausible graph of the n = 2 wave function. c) Redraw your g... click for more

1-D Quantum Mechanics

Please see attached PDF problem description. Thanks!!

Normalized wavefunction

Consider a particle of mass m moving in a tube of length L. At time t=0, its normalized wavefunction is psi(x,0)=(8/5L)^(1/2)*(1+cos(pi*x/L))*sin(pi*x/L) inside the well (0<=x<=L) and zero outside. (Hint, look for trig identities, as a superposition of eigenstates) a. If the energy of the probability is measured , what possible energies could be found, and what are the probabilities of each? b. From the answers in part (a), compute (E) ( without doing any integrals), and find the wavefunction psi(x,t) at any later time t>0. c. What is the probability that the particle is found in the left half of the tube (0<=x<= L/2) at time t?