The solution to the TDSE for a certain particle of mass m is:
*(x,t) = A x exp(-yx + i (hy**/4***m)t) for x>0
=0 for x <****0.
where y is real and postive, h is Planck's constant, and A is a normalization constant.
KEY (sorry I didn't have a greek keyboard):
**this is squared.
***this is Pi.
****this is less than or equal to.
a). Is wave function odd, even, or neitiher odd nor even (in x)?
b). Show that * is an eigenfunction of energy, and find its energy eigenvalue.
c). Find the expectation value for the particle's position.
d). If Px is the usual 1D momentum operator, find the function that results when Px operates on *(x,t)
e). Find the potential V(x) in which the particle moves.
f). Does the given wave function describe a particle that is bound or unbound?
g). If * describes a bound particle: which energy state is it in (ground, first, excited, etc.)? If * describes an unbound particle, what is the expectation value of the momentum?
h). If * describes a bound particle: consider a beam of light in which each photon has energy equal the binding energy of the particle. Find a formula for the wavelength of the light.
If * describes an unbound particle: Consider a beam of light in which each photon has momentum equal to the particle's expectation value for momentum. Find a formula for the wavelength of the light.
That's quite the problem. What have you done on it? We're not here to do that huge amount of work for students, but we are glad to help if you show what you've done and where you are stuck.