19.2 Homework
- Show that ∞
∫
−∞ψ∗(x)xψ(x)dx=∫∞
−∞φ∗(p)(
i ̄h∂
∂p)
φ(p)dp.Remember that the wave functions go to zero at infinity.- Directly calculate the the RMS uncertainty inxfor the stateψ(x) =
(a
π)^14
e−ax^2(^2) by computing
∆x=
√
〈ψ|(x−〈x〉)^2 |ψ〉.- Calculate〈pn〉for the state in the previous problem. Use this to calculate ∆pin a similar way
to the ∆xcalculation. - Calculate the commutator [p^2 ,x^2 ].
- Consider the functions of one angleψ(θ) with−π≤θ≤πandψ(−π) =ψ(π). Show that the
angular momentum operatorL=h ̄idθd has real expectation values. - 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? You may find it useful to know
that ∫
sin(Ax) sin(Bx)dx=
sin ((A−B)x)
2(A−B)−
sin ((A+B)x)
2(A+B).
- A particle is initially in thentheigenstate of a box of length L. Suddenly the walls of the box
are completely removed. Calculate the probability to find that the particle has momentum
betweenpandp+dp. Is energy conserved? - A particle is in a box with solid walls atx=±a 2. The state att= 0 is constantψ(x,0) =
√
2
a
for−a 2 < x <0 and theψ(x,0) = 0 everywhere else. Write this state as a sum of energy
eigenstates of the particle in a box. Writeψ(x,t) in terms of the energy eigenstates. Write the
state att= 0 asφ(p). Would it be correct (and why) to useφ(p) to computeψ(x,t)?- The wave function for a particle is initiallyψ(x) =Aeikx+Be−ikx. What is the probability
fluxj(x)? - Prove that the parity operator defined byPψ(x) =ψ(−x) is a hermitian operator and find its
possible eigenvalues.