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doorknob
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Homework Statement
Consider a particle-in-a-box problem, involving a
particle of mass m subject to a potential:
V(x) = +∞ for X≤0
V(x) = 0 for 0<X<L
V(x) = +∞ for X≥L
|ϕn> are the eigenstates of the Hamiltonian H with the corresponding eigenvalues:
En = (n^2)*(hbar^2)*(Pi^2) / 2mL^2
The state of the particle at the instant t = 0 is:
|Ψ(0)> = c1 |ϕ1> + c2 |ϕ2> + c3 |ϕ3> + c4 |ϕ4>
where ci are complex coefficients that satisfy Σ|ci|^2 = 1 (i.e., the state is normalized).
a) When the energy of the particle in the state |Ψ(0)> is measured what is the probability of
finding a value smaller than
3*(hbar^2)*(Pi^2) /mL^2?
b) What is the mean energy <H> and the uncertainty ΔE = [<H^2> − <H>^2] in the energy of the particle in the state |Ψ(0)> ?
c) Give the expression for the state vector |Ψ(t)> at time t. Do the results found in parts (a) and
(b) for t = 0 remain valid at an arbitrary time t? Make sure to show how you arrived at your conclusion.
d) The energy was measured at time t and the result
8*(hbar^2)*(Pi^2) /mL^2?
was found. After this measurement,
what is the state of the system? What is the result if the energy is measured again?
The Attempt at a Solution
a) the probability of obtaining an eigenvalue corresponding to the eigenstate |Ψ(t)> is:
|<ϕi|Ψ(t)>|^2
However, how do I compare the value to 3*(hbar^2)*(Pi^2) /mL^2? I do not know what the complex coefficients (c1,c2..c4) are.
b) Expectation energy <H> is defined as:
<H> = <Ψ|H|Ψ>
=> <H> = c1*E1<ϕ1|ϕ1> + c2*E2<ϕ2|ϕ2> + c3*E3<ϕ3|ϕ3> + c4*E4<ϕ4|ϕ4>
since normalized, => <ϕi|ϕi> = 1
So <H> = c1*E1 + c2*E2 + c3*E3 +c4*E4
But I do not know what the complex coefficients (c1,c2..c4) are.
c) don't know since have not completed part (a) or (b)
d) I would guess 3nd excited state (I'm assuming n=1 is the ground state)? Since if n=4, then 4^2 = 16.
16*(hbar^2)*(Pi^2) /2mL^2
reduces to 8*(hbar^2)*(Pi^2) /mL^2
After a measurement has been made, the system |Ψ> collapses to one of its eigenstates where it remains forever. If the energy were measured again, it should remain the same value as when you first made the measurement (since the system collapsed).
Any comments, feedback, criticism, would be helpful. I am more interested in the concept and understanding the material. Thank you in advance!