PHYS 451/510/710 Quantum Mechanics (Spring 2024)
Homework #2, due Sunday January 28 at 23:59
Schrödinger equation, wave function, probability density, expectation values, particle in the box,
Heisenberg uncertainty principle
1. Consider a particle
that is constrained to move in the region 0 < x < ∞. Its wave function
√
is Ψ(x, t) = A x exp[−βx + iγt], where A, β, and γ are some real constants.
(a) What is the most probable position of the particle?
(b) Find ∆x, the standard deviation of x.
(c) What is the probability that the particle is found within ±∆x interval from its average
position?
2. The wave function of a particle of mass m is given by
Ψ(x, t) = Ce−amx −ibt /h̄ ,
4
4
where a and b are some real numbers, C is a normalization constant, x is in meters and t is
in seconds.
(a) Is this is a stationary state?
(b) What are the units of a, b, and C?
(c) Find the potential in which the particle moves.
3. Consider a particle of mass m in the infinite square well 0 < x < a. At the moment of time
t = 0 its wave function is given by
{
B, a/4 < x < 3a/4
Ψ(x, t = 0) =
0, otherwise
where B is a real constant.
(a) Find the wave function at t > 0.
(b) What is the probability that the particle is found in the ground and first excited state
of the infinite square well?
4. Verify whether the Heisenberg uncertainty principle holds for the n = 1 and n = 2 eigenstates
of the infinite square well Hamiltonian.
5. Problem 1.7 in Griffiths.
Additional problems for PHYS 510/710 students only
6. A particle of mass m moves in an infinite square well (0 < x < a). The initial wave function
is given by
3πx
πx
sin
.
Ψ(x, t = 0) = A cos
a
a
(a) Determine the normalization constant A.
(b) Expand the wavefunction at the initial time in terms of the eigenfunctions ϕn (x) of
the infinite square well, i.e. determine the coefficients cn that define Ψ(x, 0) as a
superposition of eigenstates of the infinite square well.
(c) Determine the time-dependent wave function, Ψ(x, t).
(d) Is the motion periodic? If so, what is the period?
(e) If a measurement of the energy is made, what will be the outcome(s) and, with what
probability will those value(s) be measured?
(f) What is the average energy of the particle? How does it change with time (assuming
that no measurement is performed)?
(g) Will the particle’s energy change if an energy measurement is made?
2
0
