課程名稱︰物理化學二
課程性質︰必修
課程教師︰鄭原忠
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2010/3/27
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.For a particle with mass m in a one-dimensional box with lenth a, the
ground-state wavefunction is φ=(2/a)^1/2sin(pix/a). Consider the particle
in the ground state.
(a)(5%)Show that φ is normalized. Hint:sin^2(x/2)={1-cos(x)}/2
(b)(5%)What is the probability that the particle is the middle third of
the box?
(c)(5%)What is the kinetic enerrgy of the particle?
2.(10%)A hermitian operator A satisfies ∫φ*Aψτ = ∫ψ(A φ )* dτ for
arbitrary wavefunctions ψ and φ. Show that the momentum operator
Px=-ihd/2pidx is hermitian.
3.For a particle in a two-dimensional square box, the total energy
eigenfunctions are ψ(x,y)=Nsin(Nxpix/a)sin(Nypiy/a)
(a)(5%)What is the pamiltonian of this system?
Define additional parameters and also identify the boundary conditions
for the eigenstates.
(b)(5%)Obtain an expression for eigen-energies E(nxny) in terms of nx, ny,
and a.
(c)(5%)Contour plots of four eigenfunctions are shown on the right. The x
and y directions of the box lie along the horizontal and vertical
directions, respectively. Identify the quantum numbers Nx amd Ny for
states a-d.
(d)(5%)Give the degeneracies of the states a-d. Sort the four states in the
order of increasing energy.
(a)l-------- (b)l+++--- (c)l++--++ (d)l++--
l---+---- l---+++ l++--++ l++--
l--+++--- l+++--- l++--++ l++--
l---+---- l---+++ l++--++ l--++
l-------- l+++--- l++--++ l--++
l l l l--++
--------- ------- -------- -------
4. A quantum particle with mass m in a harmonic potential is described by the
hamiltonian H=p^2/2m+1mw^2x^2/2. Define the non-hermitian ladder operators:
a=(mw2pi/2h)^1/2(x+ip/mw)
a*=(mw2pi/2h)^1/2(x-ip/mw)
We have showed that H can be re-written in terms of a and a* as H=
hw(a*a+1/2)/2pi. On addition, a ψn(x)=n^1/2ψn-1(x) and a*ψn(x)=
(n+1)^1/2ψn+1(x), whereψn(x) denotes the eigenfunctions of H with
vibrational quantum number n=0,1,2.........Answer the following questions
using the porperties of ladder operators:
(a)(5%)Evaluate Hψn(x) using ladder operators to find the energy levels
En.
(b)(5%)The ground state state wavefunction is ψo(x)=(alfa/pi)^1/4*
exp(-alfax^2/2) with alfa=mw2pi/h. What is the wavefunction
of the first excited state ψ1(x)? Hint:use Px=-ihd/2pidx to
evaluate a*ψo(x).
(c)(10%)Consider the state ψ that is the equal superposition of ψo and
ψ1:ψ=c{ψo(x)+ψ1(x)}.What is the value of the normalization
constant c (assuming a real number)? What is the average ebergy
and the standard deviation in energy?
(d)(10%)Calculate the expectation values <x> ad<x^2> for the state ψ.
(e)(5%)Sketch rough graphs of ψo(x), ψ1(x) in the harmonic potential.
Label the energy levels. Use the inerference of ways to explain
what you found in (d).
5.Answer true or false for the following statements(3 points each):
(a)The zero point energy is lower for a He atom in a box than fot an
electron.
(b)Molecules with a longer pi-conjugated system tend to absorb photons with
higher energyies.
(c)If g(x) is an eigenfunction of the linear operator A, then cg(x) is also
an eigenfunction of A, where c is an arbitrary constant.
(d)According to the superposition principle, if g1(x) and g2(x) are both
eigenfunctions of the linear operator A< then their linear combinations
are also eigenfunctions of A.
(e)The wavefunction of a system must satisfy the time-independent Schrodinge
equation.
(f)If we measure the observable A when the system's wavefunction is not an
eigenfunction of A, then we can get an outcome that is not an eigenvalue
of A.
(g)For the n=25 harmonic oscillatoreigenfunction, the sign of ψ in the
right=hand classical forbidden region is opposite the sign in the left-
hand classical forbidden region.
6. Bonus questions(5 points each):
(a)Explain why a harmonic osccillatorwhose energy expectation value equals
to zero must violate the Heisenberg's uncertainty principle. Hint: you
can take it for granted that if A is a hermitian operator, then the
expectation value of A^2 for any wavefunctions must be greater or equal
to zero. i.e.(A^2) >= 0
(b)Electron tunneling occurs in the scanning tunneling mucroscope, which
makes possible atomic resolution of surfaces. Explain why? Hint:use the
distance dependence of electronic tunneling probabilities.
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