課程名稱︰物理化學二
課程性質︰化學系必修
課程教師︰金必耀
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2013/4/16
考試時限（分鐘）：160分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(20%) Consider the famous two-slit experiment for the classical particles,
waves and quantum particle. In each of the following problems, you perfoem
the experiment with only one slit open first, and the repeat the experiment
with both slits open. Sketch and comment if necessary one the patterns you
would see for each of these experiments.
(a) Classical particles such as bullets.
(b) Classical wave such as water wave.
(c) High intensity electron beams with many electrons fired at the same time
.You should discuss the importance of the size of two-slit device
relative to the de Broglie's wavelength of electrons on the resulting
pattern you could observe.
(d) Very low intensity electron beams with only one electron in the device
at any time. You should comment one the short-time and long-time
behavior.
(e) Very low intensity electron beam with observation. You should comment
on the situation when one can really distinguish the slit the slit
electron in the device the slit electron pass through (the resolution is
high) and the situation when one cannot tell the slits electron pass
through (the resolution is low).
2.(10%) Use the raising and lowering operator formalism to calculate the
uncertainty relation,ΔχΔΡ, for all states of the harmonic oscillator.
Recall that
(Δχ)^2 = ＜χ^2＞-＜χ＞^2
(ΔΡ)^2 = ＜Ρ^2＞-＜Ρ＞^2
What happens to the ΔχΔΡ products as the quantum number increases?
3.(15%) The Hamiltonian of an electron bound to a harmonic well in the
presence of an external electric field can be written as
Λ
Λ Ρ^2 1 Λ Λ
Η= ── + — mω^2 χ^2 + eεχ
2m 2
where ε is the strength of external electric field. Solve the Schrodinger
equation for this system. You don't need to write down the position
representation of wavefunctions. You can simply express your results in
terms of the eigenvectors of undisplaced harmonic oscillator using Dirac's
bracket notation.
︿
Define the polarizability of a system as α=μ/ε, where μ=＜0︱eχ｜0＞
is the dipole moment of the system in the ground state. Calculate the
polarizability of an electron bound to a harmonic well.
4.(10%) Consider two distinguishable particles (masses: m1 = m2 = m) confined
in a one-dimensional box. The potential has the form
╭
V(χ1,χ2) =│0 0＜χ1,χ2＜Ｌ
│∞ outside the box.
╰
We also assume that these two particles can not cross each other, i.e.
V(χ1,χ2) = 0 if χ1＜χ2, and V goes to infinity if χ1＞χ2. Write down
the Schrodinger equation for this two-particle system in the position
representation. Derive the eigenvalues and eigenfunctions for this system.
Sketch the contour plot for the lowest three eigenfunctions in the χ1-χ2
coordinate system.
5.(10%) Using the uncertainty relation, ΔχΔΡ≧h-bar/2, to estimate the
ground state energy of the following systems:
(a) A particle in a box of length L.
(b) A harmonic oscillator of classical frequency ω.
6.(15%)
The electronic states of metal surfaces have attractted a lot of attention
recently. In this problem, you are asked to find out the lowest energy state
of an electron interacting with a metal surface as shown in the figure.
Metal │ e^2
／／／／│ F(x)= -───
／／／／│ (2x)^2
／ x／／│ x
────┼────
／／／／│ ／¯¯
／／／／│ ／
／／／／│ ∕
／／／／│∕
／／／／│∣
(a) Show that the interation force between the electron and the metal
surface can be approximated as
e^2
F(x)= -───
(2x)^2
by treating the effect of metal as a mirror charge located at -x.
(b) Write down the Schrodinger equation for this problem. You should notice
that this equation corresponds exactly to that of a one-dimensional
hydrogen atom.
(c) Find out the energy in atomic unit for the lowest eigenstate.※ (Hint:
Since we are interested only in the lowest eigenstate, we can try a
simple nodeless exponential (e^-αχ) with α to be determined from
previous equation. Can you find a solution with this kind of functional
form ? If not, the next good guess for the trial wavefunction is of the
form (a+bχ)e^-αχ)
(d) Guve a guess for the general solution for the energy of this system.
7.(10%) What is the magnitude of the angular momentum for the electrons in
3s,3p and 3d orbitals ? How many radial and angular nodes are there for each
of these orbitals.
8.(15%)
(a) State the Pauli exclusion principle interms of the interchange of
identical particles.
(b) Construct the approximate total wavefunction that is consistent with
Pauli exclusion principle for the ground state of helium using orbital
approximation.
(c) Find the wavefunctions and the corresponding term symbols for the excited
states of helium atom that can be constructed by the product of 1s and 2s
hydrogenlike orbitals. Which are the lowest excited states ?
───────────────
※Watch out this is probably the hardest problem of this test. It may take
some time work out the solution. Or, if you prefer, you can just give a
guess.
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06/25 23:41, , 1F
06/25 23:41, 1F