know that∫
sin(Ax) sin(Bx)dx=sin((2(AA−−BB))x)−sin((2(AA++BB))x)
AnswerP = |〈u( 0 L)|u(3 0 L)〉|^2〈u( 0 L)|u(3 0 L)〉 =∫L
0√
2
L
sinπx
L√
2
3 L
sinπx
3 L=
2
√
3 L
∫L
0sinπx
Lsinπx
3 Ldx=
2
√
3 L
[
sin^23 πxL(^232) Lπ
−
sin^43 πxL(^243) Lπ
]L
0=
2
√
3 L
3 L
4 π(
sin2 π
3−
1
2
sin4 π
3)
=
√
3
2 π(√
3
2
−
1
2
−
√
3
2
)
=
9
8 πP =81
64 π^2- A particle of massmis in a 1 dimensional box of length L. The particle is in the ground state.
The size of the box is suddenly (symmetrically) expanded to length 3L. Find the probability
for the particle to be in the ground state of the new potential. (Your answer may include an
integral which you need not evaluate.) Find the probability to be in thefirst excited state of
the new potential. - Two degenerate eigenfunctions of the Hamiltonian are properly normalized and have the fol-
lowing properties.
Hψ 1 = E 0 ψ 1
Hψ 2 = E 0 ψ 2
Pψ 1 = −ψ 2
Pψ 2 = −ψ 1What are the properly normalized states that are eigenfunctions of H and P? What are their
energies?- Find the first (lowest) three Energy eigenstates for a particle localized in a box such that
0 < x < a. That is, the potential is zero inside the box and infinite outside. State the
boundary conditions and show that your solutions satisfy them. Normalize the solutions to
represent one particle in the box. - A particle is in the first excited state of a box of lengthL. What is that state? Now one
wall of the box is suddenly moved outward so that the new box has lengthD. What is the
probability for the particle to be in the ground state of the new box?What is the probability
for the particle to be in the first excited state of the new box?
6.*Assume thatφ(p) =δ(p−p 0 ). What isψ(x)? What is< p^2 >? What is< x^2 >?
- For a free particle, the Hamiltonian operator H is given byH=p^2 op/ 2 m. Find the functions,
ψ(x), which are eigenfunction of both the Hamiltonian and ofp. Write the eigenfunction
that has energy eigenvalueE 0 and momentum eigenvaluep 0. Now write the corresponding
eigenfunctions in momentum space.