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1) Consider P3, the vector space of all polynomials (with real coefficients) of degree
at most three (i.e. up to and including cubics). We can turn P3 into a (real) Hilbert
space by giving it an inner product. Choose the inner product to be
(flg) III S+1 1 dxfg
Where f and g are polynomials in X. With respect to this inner product, find an
orthonormal basis for which the first basis vector is a constant, the second basis vector
is a linear polynomial, the third basis vector is a quadratic polynomial, and the fourth
basis vector is a cubic polynomial. What are these polynomials called?
2(a) Consider a 3-dimensional quantum-mechanical Hilbert space and the operator O
represented in some basis by the matrix:
O = 0 1 1 0 1
1
Could O correspond to a physical observable? Why or why not? Find the eigenvalues
and eigenvectors of O. What is the probability for measuring each of these eigenvalues
if the state-vector for the system is 16) = 1/3 (|1)
+
|2)]
?
b) Two observables A1 and A2, which do not involve time explicitly, are known not to
commute: [A1,A2] 0. Suppose we also know that they both commute with the
Hamiltonian, H: [Ai,H = 0; i = 1,2. Prove that the energy eigenstates are in general
degenerate. Are there any exceptions?
3a) Consider particle with Gaussian wavefunction 4(x)
a
where pois its momentum and d is a parameter with dimensions of length. What are the
expectation values of X and p for the particle? Calculate xAp. With respect to the
uncertainty principle, what is special about a Gaussian?
b) You run a STOP sign and get pulled over by a police officer. Suppose the law
requires that your vehicle stop within one meter of the STOP sign. What is the
maximum speed you can go (in km per hour) for you to still be able to argue that, by the
uncertainty principle, you were actually at rest at the STOP sign? Take the mass of the
vehicle to be (i) 1000 kg, (ii) the possible mass of a neutrino, 1 eV/c²
4) To have a probabilistic interpretation of the wave function (x; t) it is
normalized: / dxy (x, t) (4)x,t) = 1. The RHS is a constant (=1) and
does not change with time. However, it is not obvious that the LHS
is
constant in time because the wave function depends on time.
a) Prove (using the Schrodinger equation) that the normalization of the wave
function is indeed independent of time.
b) We have established that a wave function, once normalized, will remain
normalized. However, conservation of total probability could
in principle be achieved by having probability jump between very
distant points. This is unphysical: we would like probability to
be conserved locally. That is, we want the probability to increase
at say point X only if the probability simultaneously decreases at
some adjacent point X + €. A local conservation law is expressed in
physics as a continuity equation. Show that in quantum mechanics,
the probability density, p(x,t) = 4*(x,t) 4 (x,t) also satisfies a continuity equation:
ap
=
of
Find J(x,t).
at
5) Consider a particle of mass m confined to a three-dimensional cube of side a.
Suppose that there is secretly an extra (fourth) spatial dimension. The extra dimension
is curled-up into a circle of circumference 2mR i.e. functions on this space are periodic
with period 2mR.
a) Solve the stationary state Schrodinger equation for the wave function,
where us is the coordinate over the
extra dimension. Besides the usual spectrum of a particle in a
box, there are now additional energy eigenvalues corresponding to
excitations in the hidden extra dimension.
b) Determine the lowest new energy level that indicates the presence
of the extra dimension. Compare this energy level with the energy
levels of a particle in a box of size a without the extra dimension.
If R<<a, argue="" that="" one="" can="" observe="" the="" effects="" of="" this="" extra="" curled-up="" dimension="" if="" the<br="">probing energy is sufficiently high.
6) Consider a harmonic oscillator in two dimensions with
= 2m 2
Px
mw
â =
+
i
=
mw
2h
a) Evaluate all six commutators between â, at, b and b+ Express A in terms of
â, at, b and bt
b) Label the states by eigenvalues of at â and 6+b i.e. = m|m,n and
6+b m, n) = n/m,n) What is the energy of the state m,n)? What is the
degeneracy?
c) Degenerate energies indicate the presence of a symmetry. That is, there exists
an operator that commutes with H but not with at â and 6tb. Consider the
operator ih(6ta-at6). - Show that this operator commutes with the Hamiltonian.
How does it transform the state mn) ? Express the operator in terms
of
</a,>X, Px.D.Py and thereby explain what operator it is and the physical reason for why
it commutes with the Hamiltonian.
7) Suppose a particle can be in only two possible positions, given by the position
eigenstates |1) and [2). Suppose the Hamiltonian is A = where E is a
real constant with dimensions of energy.
a) Express the operator H as a matrix in the {(1), (2)} basis.
b) Find the energy eigenvalues and the corresponding normalized energy
eigenstates.
c) Suppose that, at time t = 0, the particle is known to be in position 2. What is the
probability of finding the particle in position 1 as a function of time?
d) For a general state 14), how would you define the position-space wave function
for this discrete system?

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