[stam]

Suppose that the wave function for a system can be written as

psi (x) = (1/2) phi₁(x) + (1/4) phi₂(x) + (3+ sq root(2)i)/4 phi₃(x)

and that phi₁(x),  phi₂(x),  phi₃(x) are normalized eigenfunctions of the operator E_kinetic with eigenvalues E₁, 3E₁, 7E₁ respectively


b) What are the possible values that you could obtain measuring the kinetic energy on identically prepared systems?

c) What is the probability of measuring each of these eigenvalues?

d) What is the avg value of E_kinetic that you would obtain from a large number of measurements?


so from what's given, i have this set up from the definitions of e-values and e-fxns:

E_kinetic (phi₁(x)) = E₁ phi₁(x)

E_kinetic (phi₂(x)) = 3E₁ phi₂(x)

E_kinetic (phi₃(x)) = 7E₁ phi₃(x)



but after that I really do not have any idea how to approach this. I think the lack of concrete variables is throwing me off, plus I don't have a good grasp of the concept that wave functions can be expanded as a sum of eigenfunctions. I'm confused about what the problem is even asking for, particularly (b).


for part (c)..
Do i have to consider the probabilty as integral of (psi*psi) or integral of (phi*phi)? since the eigenvalues correspond to the phi's and not wave function psi?



--> for d) <E_kinetic> = integral [psi(x)* psi (Ekinetic)] dx

Any help with getting started on this problem would be a major help. Thank you

[Enthalpy]

Hints to
b) How do you relate a measurement (here the kinetic energy) with the operator and its eigenvalues?
c) Do probabilities relate with amplitudes, or with their squares? [And could this be what makes interferences possible?]
Are eigenfunctions orthogonal to another? Could this be a useful aspect of eigenfunctions?
d) Once you know each possible result and its probability, this computation isn't specifically quantic!